mat1202 set theory :week2 · thanatyod jampawai, ph.d. mat1202 set theory :week2 4 / 1..... the...
TRANSCRIPT
![Page 1: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/1.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
MAT1202 SET THEORY :Week2
Thanatyod Jampawai, Ph.D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 1 / 1
![Page 2: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/2.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory
Review Week 1
The Existential Axiom There is a set at least one.
The Axiom of ExtensionalityTwo sets are equal (are the same set) if they have the same elements.
∀x ∀y [∀z (z ∈ x↔ z ∈ y)→ (x = y)]
The Axiom of SpecificationTo every set A and to every condition p(x) there corresponds a set B whose elements areexactly those elements x of A for which p(x) holds.
x ∈ B ↔ (x ∈ A ∧ p(x) holds )
Subset
A ⊂ B ↔ ∀x(x ∈ A→ x ∈ B)
A = B ↔ ∀x(x ∈ A↔ x ∈ B)
A = B ↔ (A ⊆ B) ∧ (B ⊆ A)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 2 / 1
![Page 3: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/3.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory
Outline for Week 2
Chapter 2 The Axiomatic Set Theory2.2 The Axiom of Operations2.3 The Power set Axiom
ConclusionAssignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 3 / 1
![Page 4: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/4.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2.2 The Axiom of Operations
Theorem (2.2.1)
Let A and B be sets. Then there is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x ∈ B. By the axiom of specification, there is a set Csuch that
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x ∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1
![Page 5: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/5.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2.2 The Axiom of Operations
Theorem (2.2.1)
Let A and B be sets. Then there is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Proof .
Let A and B be sets.
Let p(x) be statement x ∈ B. By the axiom of specification, there is a set Csuch that
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x ∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1
![Page 6: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/6.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2.2 The Axiom of Operations
Theorem (2.2.1)
Let A and B be sets. Then there is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x ∈ B.
By the axiom of specification, there is a set Csuch that
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x ∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1
![Page 7: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/7.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2.2 The Axiom of Operations
Theorem (2.2.1)
Let A and B be sets. Then there is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x ∈ B. By the axiom of specification, there is a set Csuch that
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x ∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1
![Page 8: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/8.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2.2 The Axiom of Operations
Theorem (2.2.1)
Let A and B be sets. Then there is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x ∈ B. By the axiom of specification, there is a set Csuch that
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x ∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1
![Page 9: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/9.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2.2 The Axiom of Operations
Theorem (2.2.1)
Let A and B be sets. Then there is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x ∈ B. By the axiom of specification, there is a set Csuch that
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x ∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1
![Page 10: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/10.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2.2 The Axiom of Operations
Theorem (2.2.1)
Let A and B be sets. Then there is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x ∈ B. By the axiom of specification, there is a set Csuch that
x ∈ C ↔ (x ∈ A ∧ x ∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x ∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1
![Page 11: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/11.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Definition (2.2.1)
The set C in theorem 2.1.1 is intersection of A and B, denoted A ∩B ,i.e.,
A ∩B = {x : x ∈ A ∧ x ∈ B}
orx ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 5 / 1
![Page 12: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/12.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.2)
Let A,B and C be sets. Then
1. A ∩∅ = ∅
2. A ∩A = A (idempotent law)
3. A ∩B = B ∩A (commutative law)
4. A ∩ (B ∩ C) = (A ∩B) ∩ C (associative law)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 6 / 1
![Page 13: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/13.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∩ ∅ = ∅For any x
x ∈ ∅ ↔ x ∈ ∅↔ x ∈ A ∧ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∩ ∅
Thus, A ∩ ∅ = ∅.2. Idempotent law A ∩A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A ∩A
Thus, A ∩A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 7 / 1
![Page 14: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/14.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∩ ∅ = ∅
For any x
x ∈ ∅ ↔ x ∈ ∅↔ x ∈ A ∧ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∩ ∅
Thus, A ∩ ∅ = ∅.2. Idempotent law A ∩A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A ∩A
Thus, A ∩A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 7 / 1
![Page 15: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/15.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∩ ∅ = ∅For any x
x ∈ ∅ ↔ x ∈ ∅↔ x ∈ A ∧ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∩ ∅
Thus, A ∩ ∅ = ∅.2. Idempotent law A ∩A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A ∩A
Thus, A ∩A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 7 / 1
![Page 16: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/16.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∩ ∅ = ∅For any x
x ∈ ∅ ↔ x ∈ ∅↔ x ∈ A ∧ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∩ ∅
Thus, A ∩ ∅ = ∅.
2. Idempotent law A ∩A = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A ∩A
Thus, A ∩A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 7 / 1
![Page 17: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/17.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∩ ∅ = ∅For any x
x ∈ ∅ ↔ x ∈ ∅↔ x ∈ A ∧ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∩ ∅
Thus, A ∩ ∅ = ∅.2. Idempotent law A ∩A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A ∩A
Thus, A ∩A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 7 / 1
![Page 18: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/18.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∩ ∅ = ∅For any x
x ∈ ∅ ↔ x ∈ ∅↔ x ∈ A ∧ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∩ ∅
Thus, A ∩ ∅ = ∅.2. Idempotent law A ∩A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A ∩A
Thus, A ∩A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 7 / 1
![Page 19: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/19.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∩ ∅ = ∅For any x
x ∈ ∅ ↔ x ∈ ∅↔ x ∈ A ∧ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∩ ∅
Thus, A ∩ ∅ = ∅.2. Idempotent law A ∩A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A ∩A
Thus, A ∩A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 7 / 1
![Page 20: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/20.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∩ ∅ = ∅For any x
x ∈ ∅ ↔ x ∈ ∅↔ x ∈ A ∧ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∩ ∅
Thus, A ∩ ∅ = ∅.2. Idempotent law A ∩A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∧ x ∈ A
↔ x ∈ A ∩A
Thus, A ∩A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 7 / 1
![Page 21: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/21.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∩B = B ∩A
For any x
x ∈ A ∩B ↔ x ∈ A ∧ x ∈ B
↔ x ∈ B ∧ x ∈ A
↔ x ∈ B ∩A
Hence, A ∩B = B ∩A.
4. Associative law A ∩ (B ∩ C) = (A ∩B) ∩ C
For any x
x ∈ A ∩ (B ∩ C) ↔ x ∈ A ∧ x ∈ B ∩ C
↔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
↔ x ∈ (A ∩B) ∩ C
So, A ∩ (B ∩ C) = (A ∩B) ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 8 / 1
![Page 22: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/22.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∩B = B ∩AFor any x
x ∈ A ∩B ↔ x ∈ A ∧ x ∈ B
↔ x ∈ B ∧ x ∈ A
↔ x ∈ B ∩A
Hence, A ∩B = B ∩A.
4. Associative law A ∩ (B ∩ C) = (A ∩B) ∩ C
For any x
x ∈ A ∩ (B ∩ C) ↔ x ∈ A ∧ x ∈ B ∩ C
↔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
↔ x ∈ (A ∩B) ∩ C
So, A ∩ (B ∩ C) = (A ∩B) ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 8 / 1
![Page 23: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/23.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∩B = B ∩AFor any x
x ∈ A ∩B ↔ x ∈ A ∧ x ∈ B
↔ x ∈ B ∧ x ∈ A
↔ x ∈ B ∩A
Hence, A ∩B = B ∩A.
4. Associative law A ∩ (B ∩ C) = (A ∩B) ∩ C
For any x
x ∈ A ∩ (B ∩ C) ↔ x ∈ A ∧ x ∈ B ∩ C
↔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
↔ x ∈ (A ∩B) ∩ C
So, A ∩ (B ∩ C) = (A ∩B) ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 8 / 1
![Page 24: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/24.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∩B = B ∩AFor any x
x ∈ A ∩B ↔ x ∈ A ∧ x ∈ B
↔ x ∈ B ∧ x ∈ A
↔ x ∈ B ∩A
Hence, A ∩B = B ∩A.
4. Associative law A ∩ (B ∩ C) = (A ∩B) ∩ C
For any x
x ∈ A ∩ (B ∩ C) ↔ x ∈ A ∧ x ∈ B ∩ C
↔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
↔ x ∈ (A ∩B) ∩ C
So, A ∩ (B ∩ C) = (A ∩B) ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 8 / 1
![Page 25: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/25.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∩B = B ∩AFor any x
x ∈ A ∩B ↔ x ∈ A ∧ x ∈ B
↔ x ∈ B ∧ x ∈ A
↔ x ∈ B ∩A
Hence, A ∩B = B ∩A.
4. Associative law A ∩ (B ∩ C) = (A ∩B) ∩ C
For any x
x ∈ A ∩ (B ∩ C) ↔ x ∈ A ∧ x ∈ B ∩ C
↔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
↔ x ∈ (A ∩B) ∩ C
So, A ∩ (B ∩ C) = (A ∩B) ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 8 / 1
![Page 26: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/26.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∩B = B ∩AFor any x
x ∈ A ∩B ↔ x ∈ A ∧ x ∈ B
↔ x ∈ B ∧ x ∈ A
↔ x ∈ B ∩A
Hence, A ∩B = B ∩A.
4. Associative law A ∩ (B ∩ C) = (A ∩B) ∩ C
For any x
x ∈ A ∩ (B ∩ C) ↔ x ∈ A ∧ x ∈ B ∩ C
↔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
↔ x ∈ (A ∩B) ∩ C
So, A ∩ (B ∩ C) = (A ∩B) ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 8 / 1
![Page 27: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/27.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∩B = B ∩AFor any x
x ∈ A ∩B ↔ x ∈ A ∧ x ∈ B
↔ x ∈ B ∧ x ∈ A
↔ x ∈ B ∩A
Hence, A ∩B = B ∩A.
4. Associative law A ∩ (B ∩ C) = (A ∩B) ∩ C
For any x
x ∈ A ∩ (B ∩ C) ↔ x ∈ A ∧ x ∈ B ∩ C
↔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
↔ x ∈ (A ∩B) ∩ C
So, A ∩ (B ∩ C) = (A ∩B) ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 8 / 1
![Page 28: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/28.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∩B = B ∩AFor any x
x ∈ A ∩B ↔ x ∈ A ∧ x ∈ B
↔ x ∈ B ∧ x ∈ A
↔ x ∈ B ∩A
Hence, A ∩B = B ∩A.
4. Associative law A ∩ (B ∩ C) = (A ∩B) ∩ C
For any x
x ∈ A ∩ (B ∩ C) ↔ x ∈ A ∧ x ∈ B ∩ C
↔ x ∈ A ∧ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∧ x ∈ C
↔ x ∈ (A ∩B) ∩ C
So, A ∩ (B ∩ C) = (A ∩B) ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 8 / 1
![Page 29: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/29.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 30: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/30.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 31: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/31.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A
x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 32: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/32.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 33: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/33.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B
x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 34: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/34.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 35: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/35.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 36: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/36.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→)
Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 37: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/37.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A.
By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 38: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/38.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B.
So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 39: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/39.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.
(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 40: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/40.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.
(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 41: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/41.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A)
By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 42: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/42.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 43: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/43.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B)
Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 44: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/44.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A.
Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 45: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/45.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B.
So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 46: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/46.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.3)
Let A and B be sets. Then
1. A ∩B ⊆ A and A ∩B ⊆ B
2. A ∩B = A if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ∩B ⊆ A x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ A
A ∩B ⊆ B x ∈ A ∩B ↔ (x ∈ A ∧ x ∈ B) → x ∈ B
2. A ∩B = A ↔ A ⊆ B
(→) Assume that A ∩B = A. By 1, A = A ∩B ⊆ B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∩B ⊆ A) By 1, A ∩B ⊆ A.
(A ⊆ A ∩B) Let x ∈ A. Then x ∈ B. So x ∈ A ∩B or A ⊆ A ∩B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 9 / 1
![Page 47: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/47.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.4)
Let A,B and C be sets. Then
1. A ⊆ B ∩ C if and only if A ⊆ B and A ⊆ C
2. if A ⊆ B, then (A ∩ C) ⊆ (B ∩ C)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 10 / 1
![Page 48: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/48.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ⊆ B ∩ C ↔ (A ⊆ B) ∧ (A ⊆ C)
A ⊆ B ∩ C ↔ ∀x [x ∈ A→ x ∈ B ∩ C]
↔ ∀x [x ∈ A→ (x ∈ B ∧ x ∈ C)]
↔ ∀x [(x ∈ A→ x ∈ B) ∧ (x ∈ A→ x ∈ C)]
↔ A ⊆ B ∧A ⊆ C
2. A ⊆ B → (A ∩ C) ⊆ (B ∩ C) Assume that A ⊆ B. For any x
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ C (∵ A ⊆ B)
→ x ∈ B ∩ C
Therefore, A ∩ C ⊆ B ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 11 / 1
![Page 49: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/49.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ⊆ B ∩ C ↔ (A ⊆ B) ∧ (A ⊆ C)
A ⊆ B ∩ C ↔ ∀x [x ∈ A→ x ∈ B ∩ C]
↔ ∀x [x ∈ A→ (x ∈ B ∧ x ∈ C)]
↔ ∀x [(x ∈ A→ x ∈ B) ∧ (x ∈ A→ x ∈ C)]
↔ A ⊆ B ∧A ⊆ C
2. A ⊆ B → (A ∩ C) ⊆ (B ∩ C) Assume that A ⊆ B. For any x
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ C (∵ A ⊆ B)
→ x ∈ B ∩ C
Therefore, A ∩ C ⊆ B ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 11 / 1
![Page 50: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/50.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ⊆ B ∩ C ↔ (A ⊆ B) ∧ (A ⊆ C)
A ⊆ B ∩ C ↔ ∀x [x ∈ A→ x ∈ B ∩ C]
↔ ∀x [x ∈ A→ (x ∈ B ∧ x ∈ C)]
↔ ∀x [(x ∈ A→ x ∈ B) ∧ (x ∈ A→ x ∈ C)]
↔ A ⊆ B ∧A ⊆ C
2. A ⊆ B → (A ∩ C) ⊆ (B ∩ C)
Assume that A ⊆ B. For any x
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ C (∵ A ⊆ B)
→ x ∈ B ∩ C
Therefore, A ∩ C ⊆ B ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 11 / 1
![Page 51: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/51.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ⊆ B ∩ C ↔ (A ⊆ B) ∧ (A ⊆ C)
A ⊆ B ∩ C ↔ ∀x [x ∈ A→ x ∈ B ∩ C]
↔ ∀x [x ∈ A→ (x ∈ B ∧ x ∈ C)]
↔ ∀x [(x ∈ A→ x ∈ B) ∧ (x ∈ A→ x ∈ C)]
↔ A ⊆ B ∧A ⊆ C
2. A ⊆ B → (A ∩ C) ⊆ (B ∩ C) Assume that A ⊆ B.
For any x
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ C (∵ A ⊆ B)
→ x ∈ B ∩ C
Therefore, A ∩ C ⊆ B ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 11 / 1
![Page 52: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/52.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ⊆ B ∩ C ↔ (A ⊆ B) ∧ (A ⊆ C)
A ⊆ B ∩ C ↔ ∀x [x ∈ A→ x ∈ B ∩ C]
↔ ∀x [x ∈ A→ (x ∈ B ∧ x ∈ C)]
↔ ∀x [(x ∈ A→ x ∈ B) ∧ (x ∈ A→ x ∈ C)]
↔ A ⊆ B ∧A ⊆ C
2. A ⊆ B → (A ∩ C) ⊆ (B ∩ C) Assume that A ⊆ B. For any x
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ C (∵ A ⊆ B)
→ x ∈ B ∩ C
Therefore, A ∩ C ⊆ B ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 11 / 1
![Page 53: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/53.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ⊆ B ∩ C ↔ (A ⊆ B) ∧ (A ⊆ C)
A ⊆ B ∩ C ↔ ∀x [x ∈ A→ x ∈ B ∩ C]
↔ ∀x [x ∈ A→ (x ∈ B ∧ x ∈ C)]
↔ ∀x [(x ∈ A→ x ∈ B) ∧ (x ∈ A→ x ∈ C)]
↔ A ⊆ B ∧A ⊆ C
2. A ⊆ B → (A ∩ C) ⊆ (B ∩ C) Assume that A ⊆ B. For any x
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ C (∵ A ⊆ B)
→ x ∈ B ∩ C
Therefore, A ∩ C ⊆ B ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 11 / 1
![Page 54: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/54.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ⊆ B ∩ C ↔ (A ⊆ B) ∧ (A ⊆ C)
A ⊆ B ∩ C ↔ ∀x [x ∈ A→ x ∈ B ∩ C]
↔ ∀x [x ∈ A→ (x ∈ B ∧ x ∈ C)]
↔ ∀x [(x ∈ A→ x ∈ B) ∧ (x ∈ A→ x ∈ C)]
↔ A ⊆ B ∧A ⊆ C
2. A ⊆ B → (A ∩ C) ⊆ (B ∩ C) Assume that A ⊆ B. For any x
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ C (∵ A ⊆ B)
→ x ∈ B ∩ C
Therefore, A ∩ C ⊆ B ∩ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 11 / 1
![Page 55: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/55.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.5)
Let A,B,C and D be sets. Then
1. if A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩D
2. if A = B and C = D, then A ∩ C = B ∩D
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 12 / 1
![Page 56: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/56.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∩ C ⊆ B ∩D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∩D
Hence, A ∩ C ⊆ B ∩D.
2. A = B ∧ C = D → A ∩ C = B ∩D
A = B ∧ C = D → A ⊆ B ∧ C ⊆ D
→ A ∩ C ⊆ B ∩D ( By 1 )
A = B ∧ C = D → B ⊆ A ∧D ⊆ C
→ B ∩D ⊆ A ∩ C ( By 1 )
Therefore, A ∩ C = B ∩D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 13 / 1
![Page 57: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/57.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∩ C ⊆ B ∩D
Assume that A ⊆ B and C ⊆ D.
Then
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∩D
Hence, A ∩ C ⊆ B ∩D.
2. A = B ∧ C = D → A ∩ C = B ∩D
A = B ∧ C = D → A ⊆ B ∧ C ⊆ D
→ A ∩ C ⊆ B ∩D ( By 1 )
A = B ∧ C = D → B ⊆ A ∧D ⊆ C
→ B ∩D ⊆ A ∩ C ( By 1 )
Therefore, A ∩ C = B ∩D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 13 / 1
![Page 58: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/58.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∩ C ⊆ B ∩D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∩D
Hence, A ∩ C ⊆ B ∩D.
2. A = B ∧ C = D → A ∩ C = B ∩D
A = B ∧ C = D → A ⊆ B ∧ C ⊆ D
→ A ∩ C ⊆ B ∩D ( By 1 )
A = B ∧ C = D → B ⊆ A ∧D ⊆ C
→ B ∩D ⊆ A ∩ C ( By 1 )
Therefore, A ∩ C = B ∩D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 13 / 1
![Page 59: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/59.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∩ C ⊆ B ∩D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∩D
Hence, A ∩ C ⊆ B ∩D.
2. A = B ∧ C = D → A ∩ C = B ∩D
A = B ∧ C = D → A ⊆ B ∧ C ⊆ D
→ A ∩ C ⊆ B ∩D ( By 1 )
A = B ∧ C = D → B ⊆ A ∧D ⊆ C
→ B ∩D ⊆ A ∩ C ( By 1 )
Therefore, A ∩ C = B ∩D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 13 / 1
![Page 60: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/60.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∩ C ⊆ B ∩D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∩D
Hence, A ∩ C ⊆ B ∩D.
2. A = B ∧ C = D → A ∩ C = B ∩D
A = B ∧ C = D → A ⊆ B ∧ C ⊆ D
→ A ∩ C ⊆ B ∩D ( By 1 )
A = B ∧ C = D → B ⊆ A ∧D ⊆ C
→ B ∩D ⊆ A ∩ C ( By 1 )
Therefore, A ∩ C = B ∩D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 13 / 1
![Page 61: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/61.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∩ C ⊆ B ∩D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∩ C → x ∈ A ∧ x ∈ C
→ x ∈ B ∧ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∩D
Hence, A ∩ C ⊆ B ∩D.
2. A = B ∧ C = D → A ∩ C = B ∩D
A = B ∧ C = D → A ⊆ B ∧ C ⊆ D
→ A ∩ C ⊆ B ∩D ( By 1 )
A = B ∧ C = D → B ⊆ A ∧D ⊆ C
→ B ∩D ⊆ A ∩ C ( By 1 )
Therefore, A ∩ C = B ∩D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 13 / 1
![Page 62: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/62.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Example (2.2.1)
Give counter examples for convering of the statements in theorem 2.2.5
Solution
1. if A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩D
Converse: A ∩ C ⊆ B ∩D → A ⊆ B ∧ C ⊆ D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
2. if A = B and C = D, then A ∩ C = B ∩D
Converse: A ∩ C = B ∩D → A = B ∧ C = D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 14 / 1
![Page 63: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/63.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Example (2.2.1)
Give counter examples for convering of the statements in theorem 2.2.5
Solution
1. if A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩D
Converse: A ∩ C ⊆ B ∩D → A ⊆ B ∧ C ⊆ D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
2. if A = B and C = D, then A ∩ C = B ∩D
Converse: A ∩ C = B ∩D → A = B ∧ C = D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 14 / 1
![Page 64: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/64.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Example (2.2.1)
Give counter examples for convering of the statements in theorem 2.2.5
Solution
1. if A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩D
Converse: A ∩ C ⊆ B ∩D → A ⊆ B ∧ C ⊆ D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
2. if A = B and C = D, then A ∩ C = B ∩D
Converse: A ∩ C = B ∩D → A = B ∧ C = D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 14 / 1
![Page 65: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/65.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Example (2.2.1)
Give counter examples for convering of the statements in theorem 2.2.5
Solution
1. if A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩D
Converse: A ∩ C ⊆ B ∩D → A ⊆ B ∧ C ⊆ D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
2. if A = B and C = D, then A ∩ C = B ∩D
Converse: A ∩ C = B ∩D → A = B ∧ C = D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 14 / 1
![Page 66: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/66.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Example (2.2.1)
Give counter examples for convering of the statements in theorem 2.2.5
Solution
1. if A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩D
Converse: A ∩ C ⊆ B ∩D → A ⊆ B ∧ C ⊆ D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
2. if A = B and C = D, then A ∩ C = B ∩D
Converse: A ∩ C = B ∩D → A = B ∧ C = D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 14 / 1
![Page 67: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/67.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Example (2.2.1)
Give counter examples for convering of the statements in theorem 2.2.5
Solution
1. if A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩D
Converse: A ∩ C ⊆ B ∩D → A ⊆ B ∧ C ⊆ D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
2. if A = B and C = D, then A ∩ C = B ∩D
Converse: A ∩ C = B ∩D → A = B ∧ C = D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 14 / 1
![Page 68: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/68.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Example (2.2.1)
Give counter examples for convering of the statements in theorem 2.2.5
Solution
1. if A ⊆ B and C ⊆ D, then A ∩ C ⊆ B ∩D
Converse: A ∩ C ⊆ B ∩D → A ⊆ B ∧ C ⊆ D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
2. if A = B and C = D, then A ∩ C = B ∩D
Converse: A ∩ C = B ∩D → A = B ∧ C = D
A = {1}, C = {2}, B = {3}, D = {4} → A ∩ C = ∅ and B ∩D = ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 14 / 1
![Page 69: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/69.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.6)
Let A1, A2, ..., An, B1, B2, ..., Bn be sets. Then, for any n ∈ N,
1. if A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn, then
(A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn),
2. if A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn, then
(A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 15 / 1
![Page 70: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/70.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 71: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/71.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 72: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/72.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 73: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/73.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.
Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 74: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/74.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N.
Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 75: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/75.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 76: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/76.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1.
By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 77: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/77.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1.
By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 78: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/78.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 79: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/79.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 80: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/80.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let n ∈ N and let A1, A2, ..., An, B1, B2, ..., Bn be sets.
1. A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧An ⊆ Bn → (A1 ∩A2 ∩ ... ∩An) ⊆ (B1 ∩B2 ∩ ... ∩Bn)
Let P (n) be this statement where n ∈ N.
Basic step : Since A1 ⊆ B1 → A1 ⊆ B1 is true, P (1) holds.Inductive step : Assume that P (k) holds where k ∈ N. Then
A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ...∧Ak ⊆ Bk → (A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk)
Suppose that A1 ⊆ B1 ∧A2 ⊆ B2 ∧ ... ∧Ak ⊆ Bk∧Ak+1 ⊆ Bk+1. By inductionhypothesis,(A1 ∩A2 ∩ ...∩Ak) ⊆ (B1 ∩B2 ∩ ...∩Bk) and ∧Ak+1 ⊆ Bk+1. By theorem 2.2.5,
(A1 ∩A2 ∩ ... ∩Ak)∧Ak+1 ⊆ (B1 ∩B2 ∩ ... ∩Bk)∧Bk+1
Hence, P (k + 1) holds. �
2. A1 = B1 ∧A2 = B2 ∧ ... ∧An = Bn → (A1 ∩A2 ∩ ... ∩An) = (B1 ∩B2 ∩ ... ∩Bn)
Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 16 / 1
![Page 81: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/81.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Axiom 2.2.1 (Axiom of Union)
Let A and B be sets. There is a set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Theorem (2.2.7)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Proof .
Let C1 and C2 be sets satisfying condition of theorem 2.2.7. Then
x ∈ C1 ↔ (x ∈ A ∨ x ∈ B)
x ∈ C2 ↔ (x ∈ A ∨ x ∈ B)
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 17 / 1
![Page 82: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/82.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Axiom 2.2.1 (Axiom of Union)
Let A and B be sets. There is a set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Theorem (2.2.7)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Proof .
Let C1 and C2 be sets satisfying condition of theorem 2.2.7. Then
x ∈ C1 ↔ (x ∈ A ∨ x ∈ B)
x ∈ C2 ↔ (x ∈ A ∨ x ∈ B)
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 17 / 1
![Page 83: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/83.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Axiom 2.2.1 (Axiom of Union)
Let A and B be sets. There is a set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Theorem (2.2.7)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Proof .
Let C1 and C2 be sets satisfying condition of theorem 2.2.7.
Then
x ∈ C1 ↔ (x ∈ A ∨ x ∈ B)
x ∈ C2 ↔ (x ∈ A ∨ x ∈ B)
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 17 / 1
![Page 84: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/84.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Axiom 2.2.1 (Axiom of Union)
Let A and B be sets. There is a set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Theorem (2.2.7)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Proof .
Let C1 and C2 be sets satisfying condition of theorem 2.2.7. Then
x ∈ C1 ↔ (x ∈ A ∨ x ∈ B)
x ∈ C2 ↔ (x ∈ A ∨ x ∈ B)
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 17 / 1
![Page 85: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/85.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Axiom 2.2.1 (Axiom of Union)
Let A and B be sets. There is a set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Theorem (2.2.7)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Proof .
Let C1 and C2 be sets satisfying condition of theorem 2.2.7. Then
x ∈ C1 ↔ (x ∈ A ∨ x ∈ B)
x ∈ C2 ↔ (x ∈ A ∨ x ∈ B)
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 17 / 1
![Page 86: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/86.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Definition (2.2.2)
The set C in theorem 2.2.7 is union of A and B, denoted A ∪B , i.e.,
A ∪B = {x : x ∈ A ∨ x ∈ B} or x ∈ A ∪B ↔ (x ∈ A ∨ x ∈ B).
Theorem (2.2.8)
Let A,B and C be sets. Then
1. A ∪∅ = A
2. A ∪A = A (idempotent law)
3. A ∪B = B ∪A (commutative law)
4. A ∪ (B ∪ C) = (A ∪B) ∪ C (associative law)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 18 / 1
![Page 87: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/87.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Definition (2.2.2)
The set C in theorem 2.2.7 is union of A and B, denoted A ∪B , i.e.,
A ∪B = {x : x ∈ A ∨ x ∈ B} or x ∈ A ∪B ↔ (x ∈ A ∨ x ∈ B).
Theorem (2.2.8)
Let A,B and C be sets. Then
1. A ∪∅ = A
2. A ∪A = A (idempotent law)
3. A ∪B = B ∪A (commutative law)
4. A ∪ (B ∪ C) = (A ∪B) ∪ C (associative law)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 18 / 1
![Page 88: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/88.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∪ ∅ = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∪ ∅
So, A ∪ ∅ = A.2. Idempotent law A ∪A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ A
↔ x ∈ A ∪A
Thus, A ∪A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 19 / 1
![Page 89: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/89.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∪ ∅ = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∪ ∅
So, A ∪ ∅ = A.2. Idempotent law A ∪A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ A
↔ x ∈ A ∪A
Thus, A ∪A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 19 / 1
![Page 90: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/90.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∪ ∅ = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∪ ∅
So, A ∪ ∅ = A.2. Idempotent law A ∪A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ A
↔ x ∈ A ∪A
Thus, A ∪A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 19 / 1
![Page 91: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/91.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∪ ∅ = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∪ ∅
So, A ∪ ∅ = A.
2. Idempotent law A ∪A = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ A
↔ x ∈ A ∪A
Thus, A ∪A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 19 / 1
![Page 92: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/92.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∪ ∅ = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∪ ∅
So, A ∪ ∅ = A.2. Idempotent law A ∪A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ A
↔ x ∈ A ∪A
Thus, A ∪A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 19 / 1
![Page 93: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/93.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∪ ∅ = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∪ ∅
So, A ∪ ∅ = A.2. Idempotent law A ∪A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ A
↔ x ∈ A ∪A
Thus, A ∪A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 19 / 1
![Page 94: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/94.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∪ ∅ = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∪ ∅
So, A ∪ ∅ = A.2. Idempotent law A ∪A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ A
↔ x ∈ A ∪A
Thus, A ∪A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 19 / 1
![Page 95: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/95.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A,B and C be sets.
1. A ∪ ∅ = AFor any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ ∅ (∵ x ∈ ∅ is false)↔ x ∈ A ∪ ∅
So, A ∪ ∅ = A.2. Idempotent law A ∪A = A
For any x
x ∈ A ↔ x ∈ A
↔ x ∈ A ∨ x ∈ A
↔ x ∈ A ∪A
Thus, A ∪A = A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 19 / 1
![Page 96: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/96.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∪B = B ∪A
For any x
x ∈ A ∪B ↔ x ∈ A ∨ x ∈ B
↔ x ∈ B ∨ x ∈ A
↔ x ∈ B ∪A
Hence, A ∪B = B ∪A.
4. Associative law A ∪ (B ∪ C) = (A ∪B) ∪ C
For any x
x ∈ A ∪ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∨ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∨ x ∈ C
↔ x ∈ (A ∪B) ∪ C
Therefore, A ∪ (B ∪ C) = (A ∪B) ∪ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 20 / 1
![Page 97: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/97.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∪B = B ∪AFor any x
x ∈ A ∪B ↔ x ∈ A ∨ x ∈ B
↔ x ∈ B ∨ x ∈ A
↔ x ∈ B ∪A
Hence, A ∪B = B ∪A.
4. Associative law A ∪ (B ∪ C) = (A ∪B) ∪ C
For any x
x ∈ A ∪ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∨ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∨ x ∈ C
↔ x ∈ (A ∪B) ∪ C
Therefore, A ∪ (B ∪ C) = (A ∪B) ∪ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 20 / 1
![Page 98: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/98.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∪B = B ∪AFor any x
x ∈ A ∪B ↔ x ∈ A ∨ x ∈ B
↔ x ∈ B ∨ x ∈ A
↔ x ∈ B ∪A
Hence, A ∪B = B ∪A.
4. Associative law A ∪ (B ∪ C) = (A ∪B) ∪ C
For any x
x ∈ A ∪ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∨ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∨ x ∈ C
↔ x ∈ (A ∪B) ∪ C
Therefore, A ∪ (B ∪ C) = (A ∪B) ∪ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 20 / 1
![Page 99: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/99.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∪B = B ∪AFor any x
x ∈ A ∪B ↔ x ∈ A ∨ x ∈ B
↔ x ∈ B ∨ x ∈ A
↔ x ∈ B ∪A
Hence, A ∪B = B ∪A.
4. Associative law A ∪ (B ∪ C) = (A ∪B) ∪ C
For any x
x ∈ A ∪ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∨ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∨ x ∈ C
↔ x ∈ (A ∪B) ∪ C
Therefore, A ∪ (B ∪ C) = (A ∪B) ∪ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 20 / 1
![Page 100: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/100.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∪B = B ∪AFor any x
x ∈ A ∪B ↔ x ∈ A ∨ x ∈ B
↔ x ∈ B ∨ x ∈ A
↔ x ∈ B ∪A
Hence, A ∪B = B ∪A.
4. Associative law A ∪ (B ∪ C) = (A ∪B) ∪ C
For any x
x ∈ A ∪ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∨ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∨ x ∈ C
↔ x ∈ (A ∪B) ∪ C
Therefore, A ∪ (B ∪ C) = (A ∪B) ∪ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 20 / 1
![Page 101: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/101.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∪B = B ∪AFor any x
x ∈ A ∪B ↔ x ∈ A ∨ x ∈ B
↔ x ∈ B ∨ x ∈ A
↔ x ∈ B ∪A
Hence, A ∪B = B ∪A.
4. Associative law A ∪ (B ∪ C) = (A ∪B) ∪ C
For any x
x ∈ A ∪ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∨ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∨ x ∈ C
↔ x ∈ (A ∪B) ∪ C
Therefore, A ∪ (B ∪ C) = (A ∪B) ∪ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 20 / 1
![Page 102: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/102.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∪B = B ∪AFor any x
x ∈ A ∪B ↔ x ∈ A ∨ x ∈ B
↔ x ∈ B ∨ x ∈ A
↔ x ∈ B ∪A
Hence, A ∪B = B ∪A.
4. Associative law A ∪ (B ∪ C) = (A ∪B) ∪ C
For any x
x ∈ A ∪ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∨ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∨ x ∈ C
↔ x ∈ (A ∪B) ∪ C
Therefore, A ∪ (B ∪ C) = (A ∪B) ∪ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 20 / 1
![Page 103: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/103.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Commutative law A ∪B = B ∪AFor any x
x ∈ A ∪B ↔ x ∈ A ∨ x ∈ B
↔ x ∈ B ∨ x ∈ A
↔ x ∈ B ∪A
Hence, A ∪B = B ∪A.
4. Associative law A ∪ (B ∪ C) = (A ∪B) ∪ C
For any x
x ∈ A ∪ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∨ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∨ x ∈ C
↔ x ∈ (A ∪B) ∪ C
Therefore, A ∪ (B ∪ C) = (A ∪B) ∪ C.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 20 / 1
![Page 104: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/104.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 105: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/105.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 106: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/106.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B
x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 107: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/107.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 108: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/108.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B
x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 109: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/109.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 110: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/110.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 111: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/111.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B.
By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 112: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/112.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B.
So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 113: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/113.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.
(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 114: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/114.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.
(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 115: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/115.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B)
Let x ∈ A ∪B. Then x ∈ A or x ∈ B.If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 116: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/116.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B.
Then x ∈ A or x ∈ B.If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 117: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/117.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 118: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/118.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A,
then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 119: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/119.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B.
Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 120: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/120.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.
If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 121: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/121.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done.
Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 122: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/122.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.
(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 123: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/123.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B)
By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 124: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/124.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.9)
Let A and B be sets. Then
1. A ⊆ A ∪B and B ⊆ A ∪B
2. A ∪B = B if and only if A ⊆ B
Proof .
Let A and B be sets.
1. A ⊆ A ∪B x ∈ A → (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
B ⊆ A ∪B x ∈ B ↔ (x ∈ A ∨ x ∈ B) → x ∈ A ∪B
2. A ∪B = B ↔ A ⊆ B
(→) Assume that A ∪B = B. By 1, A ⊆ A ∪B = B. So A ⊆ B.(←) Assume that A ⊆ B.(A ∪B ⊆ B) Let x ∈ A ∪B. Then x ∈ A or x ∈ B.
If x ∈ A, then x ∈ B because A ⊆ B. Thus, A ∪B ⊆ B.If x ∈ B, it is done. Thus, A ∪B ⊆ B.(B ⊆ A ∪B) By 1, B ⊆ A ∪B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 21 / 1
![Page 125: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/125.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.10)
Let A,B,C and D be sets. Then
1. if A ⊆ B and C ⊆ D, then A ∪ C ⊆ B ∪D
2. if A = B and C = D, then A ∪ C = B ∪D
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 22 / 1
![Page 126: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/126.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∪ C ⊆ B ∪D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∪ C → x ∈ A ∨ x ∈ C
→ x ∈ B ∨ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∪D
Thus, A ∪ C ⊆ B ∪D.
2. A = B ∧ C = D → A ∪ C = B ∪D
A = B ∧ C = D → A ⊆ B ∨ C ⊆ D
→ A ∪ C ⊆ B ∪D ( By 1 )
A = B ∧ C = D → B ⊆ A ∨D ⊆ C
→ B ∪D ⊆ A ∪ C ( By 1 )
Hence, A ∪ C = B ∪D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 23 / 1
![Page 127: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/127.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∪ C ⊆ B ∪D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∪ C → x ∈ A ∨ x ∈ C
→ x ∈ B ∨ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∪D
Thus, A ∪ C ⊆ B ∪D.
2. A = B ∧ C = D → A ∪ C = B ∪D
A = B ∧ C = D → A ⊆ B ∨ C ⊆ D
→ A ∪ C ⊆ B ∪D ( By 1 )
A = B ∧ C = D → B ⊆ A ∨D ⊆ C
→ B ∪D ⊆ A ∪ C ( By 1 )
Hence, A ∪ C = B ∪D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 23 / 1
![Page 128: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/128.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∪ C ⊆ B ∪D
Assume that A ⊆ B and C ⊆ D.
Then
x ∈ A ∪ C → x ∈ A ∨ x ∈ C
→ x ∈ B ∨ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∪D
Thus, A ∪ C ⊆ B ∪D.
2. A = B ∧ C = D → A ∪ C = B ∪D
A = B ∧ C = D → A ⊆ B ∨ C ⊆ D
→ A ∪ C ⊆ B ∪D ( By 1 )
A = B ∧ C = D → B ⊆ A ∨D ⊆ C
→ B ∪D ⊆ A ∪ C ( By 1 )
Hence, A ∪ C = B ∪D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 23 / 1
![Page 129: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/129.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∪ C ⊆ B ∪D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∪ C → x ∈ A ∨ x ∈ C
→ x ∈ B ∨ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∪D
Thus, A ∪ C ⊆ B ∪D.
2. A = B ∧ C = D → A ∪ C = B ∪D
A = B ∧ C = D → A ⊆ B ∨ C ⊆ D
→ A ∪ C ⊆ B ∪D ( By 1 )
A = B ∧ C = D → B ⊆ A ∨D ⊆ C
→ B ∪D ⊆ A ∪ C ( By 1 )
Hence, A ∪ C = B ∪D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 23 / 1
![Page 130: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/130.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∪ C ⊆ B ∪D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∪ C → x ∈ A ∨ x ∈ C
→ x ∈ B ∨ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∪D
Thus, A ∪ C ⊆ B ∪D.
2. A = B ∧ C = D → A ∪ C = B ∪D
A = B ∧ C = D → A ⊆ B ∨ C ⊆ D
→ A ∪ C ⊆ B ∪D ( By 1 )
A = B ∧ C = D → B ⊆ A ∨D ⊆ C
→ B ∪D ⊆ A ∪ C ( By 1 )
Hence, A ∪ C = B ∪D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 23 / 1
![Page 131: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/131.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∪ C ⊆ B ∪D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∪ C → x ∈ A ∨ x ∈ C
→ x ∈ B ∨ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∪D
Thus, A ∪ C ⊆ B ∪D.
2. A = B ∧ C = D → A ∪ C = B ∪D
A = B ∧ C = D → A ⊆ B ∨ C ⊆ D
→ A ∪ C ⊆ B ∪D ( By 1 )
A = B ∧ C = D → B ⊆ A ∨D ⊆ C
→ B ∪D ⊆ A ∪ C ( By 1 )
Hence, A ∪ C = B ∪D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 23 / 1
![Page 132: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/132.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∪ C ⊆ B ∪D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∪ C → x ∈ A ∨ x ∈ C
→ x ∈ B ∨ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∪D
Thus, A ∪ C ⊆ B ∪D.
2. A = B ∧ C = D → A ∪ C = B ∪D
A = B ∧ C = D → A ⊆ B ∨ C ⊆ D
→ A ∪ C ⊆ B ∪D ( By 1 )
A = B ∧ C = D → B ⊆ A ∨D ⊆ C
→ B ∪D ⊆ A ∪ C ( By 1 )
Hence, A ∪ C = B ∪D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 23 / 1
![Page 133: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/133.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B,C and D be sets.
1. A ⊆ B ∧ C ⊆ D → A ∪ C ⊆ B ∪D
Assume that A ⊆ B and C ⊆ D. Then
x ∈ A ∪ C → x ∈ A ∨ x ∈ C
→ x ∈ B ∨ x ∈ D (∵ A ⊆ B and C ⊆ D)
→ x ∈ B ∪D
Thus, A ∪ C ⊆ B ∪D.
2. A = B ∧ C = D → A ∪ C = B ∪D
A = B ∧ C = D → A ⊆ B ∨ C ⊆ D
→ A ∪ C ⊆ B ∪D ( By 1 )
A = B ∧ C = D → B ⊆ A ∨D ⊆ C
→ B ∪D ⊆ A ∪ C ( By 1 )
Hence, A ∪ C = B ∪D.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 23 / 1
![Page 134: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/134.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.11)
Let A,B and C be sets. Then
1. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C) (distributive law for intersection)
2. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C) (distributive law for union)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 24 / 1
![Page 135: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/135.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
x ∈ A ∩ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∧ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C)
↔ x ∈ A ∩B ∨ x ∈ A ∩ C
↔ x ∈ (A ∩B) ∪ (A ∩ C)
2. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)
x ∈ A ∪ (B ∩ C) ↔ x ∈ A ∨ x ∈ B ∩ C
↔ x ∈ A ∨ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ x ∈ C)
↔ x ∈ A ∪B ∧ x ∈ A ∪ C
↔ x ∈ (A ∪B) ∩ (A ∪ C)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 25 / 1
![Page 136: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/136.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
x ∈ A ∩ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∧ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C)
↔ x ∈ A ∩B ∨ x ∈ A ∩ C
↔ x ∈ (A ∩B) ∪ (A ∩ C)
2. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)
x ∈ A ∪ (B ∩ C) ↔ x ∈ A ∨ x ∈ B ∩ C
↔ x ∈ A ∨ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ x ∈ C)
↔ x ∈ A ∪B ∧ x ∈ A ∪ C
↔ x ∈ (A ∪B) ∩ (A ∪ C)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 25 / 1
![Page 137: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/137.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
x ∈ A ∩ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∧ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C)
↔ x ∈ A ∩B ∨ x ∈ A ∩ C
↔ x ∈ (A ∩B) ∪ (A ∩ C)
2. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)
x ∈ A ∪ (B ∩ C) ↔ x ∈ A ∨ x ∈ B ∩ C
↔ x ∈ A ∨ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ x ∈ C)
↔ x ∈ A ∪B ∧ x ∈ A ∪ C
↔ x ∈ (A ∪B) ∩ (A ∪ C)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 25 / 1
![Page 138: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/138.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
x ∈ A ∩ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∧ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C)
↔ x ∈ A ∩B ∨ x ∈ A ∩ C
↔ x ∈ (A ∩B) ∪ (A ∩ C)
2. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)
x ∈ A ∪ (B ∩ C) ↔ x ∈ A ∨ x ∈ B ∩ C
↔ x ∈ A ∨ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ x ∈ C)
↔ x ∈ A ∪B ∧ x ∈ A ∪ C
↔ x ∈ (A ∪B) ∩ (A ∪ C)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 25 / 1
![Page 139: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/139.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof .
Let A,B and C be sets.
1. A ∩ (B ∪ C) = (A ∩B) ∪ (A ∩ C)
x ∈ A ∩ (B ∪ C) ↔ x ∈ A ∧ x ∈ B ∪ C
↔ x ∈ A ∧ (x ∈ B ∨ x ∈ C)
↔ (x ∈ A ∧ x ∈ B) ∨ (x ∈ A ∧ x ∈ C)
↔ x ∈ A ∩B ∨ x ∈ A ∩ C
↔ x ∈ (A ∩B) ∪ (A ∩ C)
2. A ∪ (B ∩ C) = (A ∪B) ∩ (A ∪ C)
x ∈ A ∪ (B ∩ C) ↔ x ∈ A ∨ x ∈ B ∩ C
↔ x ∈ A ∨ (x ∈ B ∧ x ∈ C)
↔ (x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ x ∈ C)
↔ x ∈ A ∪B ∧ x ∈ A ∪ C
↔ x ∈ (A ∪B) ∩ (A ∪ C)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 25 / 1
![Page 140: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/140.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.12)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x /∈ B. By the axiom of specification , there is a setC such that
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x /∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 26 / 1
![Page 141: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/141.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.12)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Proof .
Let A and B be sets.
Let p(x) be statement x /∈ B. By the axiom of specification , there is a setC such that
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x /∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 26 / 1
![Page 142: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/142.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.12)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x /∈ B.
By the axiom of specification , there is a setC such that
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x /∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 26 / 1
![Page 143: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/143.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.12)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x /∈ B. By the axiom of specification , there is a setC such that
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x /∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 26 / 1
![Page 144: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/144.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.12)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x /∈ B. By the axiom of specification , there is a setC such that
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x /∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 26 / 1
![Page 145: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/145.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.12)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x /∈ B. By the axiom of specification , there is a setC such that
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x /∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 26 / 1
![Page 146: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/146.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.12)
Let A and B be sets. There is a unique set C satisfying
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Proof .
Let A and B be sets. Let p(x) be statement x /∈ B. By the axiom of specification , there is a setC such that
x ∈ C ↔ (x ∈ A ∧ x /∈ B).
Let C2 be a set satisfyingx ∈ C2 ↔ (x ∈ A ∧ x /∈ B).
Thenx ∈ C ↔ x ∈ C2 for all x.
Thus, C = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 26 / 1
![Page 147: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/147.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Definition (2.2.3)
The set C in theorem 2.2.12 is different of A and B, denoted A−B , i.e.,
A−B = {x : x ∈ A ∧ x /∈ B} or x ∈ A−B ↔ (x ∈ A ∧ x /∈ B).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 27 / 1
![Page 148: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/148.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.13)
Let A and B be sets. Then
1. A−A = ∅
2. A−∅ = A
3. ∅−A = ∅
4. A−B ⊆ A
Proof.
Let A and B be sets.
1. A−A = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A−A ↔ x ∈ A ∧ x /∈ A,
x ∈ A−A is also false. Hence, A−A has no element or A−A = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 28 / 1
![Page 149: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/149.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.13)
Let A and B be sets. Then
1. A−A = ∅
2. A−∅ = A
3. ∅−A = ∅
4. A−B ⊆ A
Proof.
Let A and B be sets.
1. A−A = ∅
Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A−A ↔ x ∈ A ∧ x /∈ A,
x ∈ A−A is also false. Hence, A−A has no element or A−A = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 28 / 1
![Page 150: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/150.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.13)
Let A and B be sets. Then
1. A−A = ∅
2. A−∅ = A
3. ∅−A = ∅
4. A−B ⊆ A
Proof.
Let A and B be sets.
1. A−A = ∅ Since x ∈ A ∧ x /∈ A is false for any x
and
x ∈ A−A ↔ x ∈ A ∧ x /∈ A,
x ∈ A−A is also false. Hence, A−A has no element or A−A = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 28 / 1
![Page 151: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/151.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.13)
Let A and B be sets. Then
1. A−A = ∅
2. A−∅ = A
3. ∅−A = ∅
4. A−B ⊆ A
Proof.
Let A and B be sets.
1. A−A = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A−A ↔ x ∈ A ∧ x /∈ A,
x ∈ A−A is also false. Hence, A−A has no element or A−A = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 28 / 1
![Page 152: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/152.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.13)
Let A and B be sets. Then
1. A−A = ∅
2. A−∅ = A
3. ∅−A = ∅
4. A−B ⊆ A
Proof.
Let A and B be sets.
1. A−A = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A−A ↔ x ∈ A ∧ x /∈ A,
x ∈ A−A is also false.
Hence, A−A has no element or A−A = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 28 / 1
![Page 153: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/153.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.13)
Let A and B be sets. Then
1. A−A = ∅
2. A−∅ = A
3. ∅−A = ∅
4. A−B ⊆ A
Proof.
Let A and B be sets.
1. A−A = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A−A ↔ x ∈ A ∧ x /∈ A,
x ∈ A−A is also false. Hence, A−A has no element or A−A = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 28 / 1
![Page 154: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/154.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A
For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 155: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/155.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 156: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/156.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 157: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/157.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.
3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 158: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/158.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅
Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 159: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/159.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x
and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 160: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/160.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 161: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/161.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false.
Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 162: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/162.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.
4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 163: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/163.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A
For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 164: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/164.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 165: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/165.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 166: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/166.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A− ∅ = A For any x
A− ∅ ↔ x ∈ A ∧ x /∈ A
↔ x ∈ A (x /∈ ∅ is true )
So, A− ∅ = A.3. ∅−A = ∅ Since x ∈ ∅ ∧ x ∈ A is false for any x and
∅−A ↔ x ∈ ∅ ∧ x /∈ A
x ∈ ∅−A is also false. Hence, ∅−A has no element or ∅−A = ∅.4. A−B ⊆ A For any x
x ∈ A−B ↔ x ∈ A ∧ x /∈ B
→ x ∈ A
Thus, A−B ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 29 / 1
![Page 167: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/167.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.14)
Let A,B and C be sets. Then
1. A ⊆ B if and only if A−B = ∅
2. if A ⊆ B, then (A− C) ⊆ (B − C)
3. if A = B, then (A− C) = (B − C)
Proof.
Let A,B and C be sets.
1. A ⊆ B ↔ A−B = ∅
A ⊆ B ↔ ∀x (x ∈ A→ x ∈ B)
A * B ↔ ¬∀x (x ∈ A→ x ∈ B)
↔ ∃x (x ∈ A ∧ x /∈ B)
↔ ∃x (x ∈ A−B)
↔ A−B ̸= ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 30 / 1
![Page 168: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/168.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.14)
Let A,B and C be sets. Then
1. A ⊆ B if and only if A−B = ∅
2. if A ⊆ B, then (A− C) ⊆ (B − C)
3. if A = B, then (A− C) = (B − C)
Proof.
Let A,B and C be sets.
1. A ⊆ B ↔ A−B = ∅
A ⊆ B ↔ ∀x (x ∈ A→ x ∈ B)
A * B ↔ ¬∀x (x ∈ A→ x ∈ B)
↔ ∃x (x ∈ A ∧ x /∈ B)
↔ ∃x (x ∈ A−B)
↔ A−B ̸= ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 30 / 1
![Page 169: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/169.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.14)
Let A,B and C be sets. Then
1. A ⊆ B if and only if A−B = ∅
2. if A ⊆ B, then (A− C) ⊆ (B − C)
3. if A = B, then (A− C) = (B − C)
Proof.
Let A,B and C be sets.
1. A ⊆ B ↔ A−B = ∅
A ⊆ B ↔ ∀x (x ∈ A→ x ∈ B)
A * B ↔ ¬∀x (x ∈ A→ x ∈ B)
↔ ∃x (x ∈ A ∧ x /∈ B)
↔ ∃x (x ∈ A−B)
↔ A−B ̸= ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 30 / 1
![Page 170: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/170.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.14)
Let A,B and C be sets. Then
1. A ⊆ B if and only if A−B = ∅
2. if A ⊆ B, then (A− C) ⊆ (B − C)
3. if A = B, then (A− C) = (B − C)
Proof.
Let A,B and C be sets.
1. A ⊆ B ↔ A−B = ∅
A ⊆ B ↔ ∀x (x ∈ A→ x ∈ B)
A * B ↔ ¬∀x (x ∈ A→ x ∈ B)
↔ ∃x (x ∈ A ∧ x /∈ B)
↔ ∃x (x ∈ A−B)
↔ A−B ̸= ∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 30 / 1
![Page 171: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/171.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C)
Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 172: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/172.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B.
Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 173: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/173.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 174: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/174.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 175: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/175.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 176: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/176.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C)
Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 177: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/177.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B.
Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 178: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/178.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 179: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/179.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 180: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/180.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. A ⊆ B → (A− C) ⊆ (B − C) Assume A ⊆ B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
→ x ∈ B ∧ x /∈ C (∵ A ⊆ B)
↔ x ∈ B − C
Thus, (A− C) ⊆ (B − C).
3. A = B → (A− C) = (B − C) Assume A = B. Then, for any x,
x ∈ A− C ↔ x ∈ A ∧ x /∈ C
↔ x ∈ B ∧ x /∈ C (∵ A = B)
↔ x ∈ B − C
Thus, (A− C) = (B − C).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 31 / 1
![Page 181: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/181.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Take a BreakFor 10 Minutes
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 32 / 1
![Page 182: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/182.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Definition (2.2.4)
A universal set is the set of all elements under consideration, denoted by
U .
Definition (2.2.5)
Let A be a set with a universal set U . The complement of A, denoted Ac ,is
Ac = U −A = {x : x ∈ U ∧ x /∈ A} or x ∈ Ac ↔ (x ∈ U ∧ x /∈ A).
Note that
x /∈ Ac ↔ ¬(x ∈ U ∧ x /∈ A)
↔ (x /∈ U ∨ x ∈ A) (∵ x /∈ U is impossible )
↔ x ∈ A
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 33 / 1
![Page 183: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/183.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Definition (2.2.4)
A universal set is the set of all elements under consideration, denoted by
U .
Definition (2.2.5)
Let A be a set with a universal set U . The complement of A, denoted Ac ,is
Ac = U −A = {x : x ∈ U ∧ x /∈ A} or x ∈ Ac ↔ (x ∈ U ∧ x /∈ A).
Note that
x /∈ Ac ↔ ¬(x ∈ U ∧ x /∈ A)
↔ (x /∈ U ∨ x ∈ A) (∵ x /∈ U is impossible )
↔ x ∈ A
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 33 / 1
![Page 184: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/184.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Definition (2.2.4)
A universal set is the set of all elements under consideration, denoted by
U .
Definition (2.2.5)
Let A be a set with a universal set U . The complement of A, denoted Ac ,is
Ac = U −A = {x : x ∈ U ∧ x /∈ A} or x ∈ Ac ↔ (x ∈ U ∧ x /∈ A).
Note that
x /∈ Ac ↔ ¬(x ∈ U ∧ x /∈ A)
↔ (x /∈ U ∨ x ∈ A) (∵ x /∈ U is impossible )
↔ x ∈ A
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 33 / 1
![Page 185: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/185.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.15)
Let A be a set with a universal set U . Then1 (Ac)c = A
2 ∅c = U3 Uc = ∅4 A ∩Ac = ∅5 A ∪Ac = U
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 34 / 1
![Page 186: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/186.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A be a set with a universal set U .
1. (Ac)c = A
x ∈ (Ac)c ↔ x ∈ U ∧ x /∈ Ac
↔ x ∈ U ∧ x ∈ A
↔ x ∈ A (∵ A ⊆ U)
Thus, (Ac)c = A.
2. ∅c = U
x ∈ ∅c ↔ x ∈ U ∧ x /∈ ∅↔ x ∈ U (∵ x /∈ ∅ is true )
So, ∅c = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 35 / 1
![Page 187: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/187.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A be a set with a universal set U .1. (Ac)c = A
x ∈ (Ac)c ↔ x ∈ U ∧ x /∈ Ac
↔ x ∈ U ∧ x ∈ A
↔ x ∈ A (∵ A ⊆ U)
Thus, (Ac)c = A.
2. ∅c = U
x ∈ ∅c ↔ x ∈ U ∧ x /∈ ∅↔ x ∈ U (∵ x /∈ ∅ is true )
So, ∅c = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 35 / 1
![Page 188: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/188.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A be a set with a universal set U .1. (Ac)c = A
x ∈ (Ac)c ↔ x ∈ U ∧ x /∈ Ac
↔ x ∈ U ∧ x ∈ A
↔ x ∈ A (∵ A ⊆ U)
Thus, (Ac)c = A.
2. ∅c = U
x ∈ ∅c ↔ x ∈ U ∧ x /∈ ∅↔ x ∈ U (∵ x /∈ ∅ is true )
So, ∅c = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 35 / 1
![Page 189: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/189.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A be a set with a universal set U .1. (Ac)c = A
x ∈ (Ac)c ↔ x ∈ U ∧ x /∈ Ac
↔ x ∈ U ∧ x ∈ A
↔ x ∈ A (∵ A ⊆ U)
Thus, (Ac)c = A.
2. ∅c = U
x ∈ ∅c ↔ x ∈ U ∧ x /∈ ∅↔ x ∈ U (∵ x /∈ ∅ is true )
So, ∅c = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 35 / 1
![Page 190: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/190.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A be a set with a universal set U .1. (Ac)c = A
x ∈ (Ac)c ↔ x ∈ U ∧ x /∈ Ac
↔ x ∈ U ∧ x ∈ A
↔ x ∈ A (∵ A ⊆ U)
Thus, (Ac)c = A.
2. ∅c = U
x ∈ ∅c ↔ x ∈ U ∧ x /∈ ∅↔ x ∈ U (∵ x /∈ ∅ is true )
So, ∅c = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 35 / 1
![Page 191: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/191.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A be a set with a universal set U .1. (Ac)c = A
x ∈ (Ac)c ↔ x ∈ U ∧ x /∈ Ac
↔ x ∈ U ∧ x ∈ A
↔ x ∈ A (∵ A ⊆ U)
Thus, (Ac)c = A.
2. ∅c = U
x ∈ ∅c ↔ x ∈ U ∧ x /∈ ∅↔ x ∈ U (∵ x /∈ ∅ is true )
So, ∅c = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 35 / 1
![Page 192: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/192.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅
Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 193: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/193.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x
and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 194: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/194.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 195: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/195.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false.
So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 196: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/196.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.
4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 197: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/197.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅
Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 198: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/198.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x
and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 199: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/199.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 200: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/200.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false.
So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 201: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/201.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
3. Uc = ∅ Since x ∈ U ∧ x /∈ U is false for any x and
x ∈ Uc ↔ x ∈ U ∧ x /∈ U ,
x ∈ Uc is also false. So, Uc has no element or Uc = ∅.4. A ∩Ac = ∅ Since x ∈ A ∧ x /∈ A is false for any x and
x ∈ A ∩Ac ↔ x ∈ A ∧ x ∈ Ac
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∧ x /∈ A) ∧ x ∈ U
x ∈ A ∩Ac is also false. So, A ∩Ac has no element or A ∩Ac = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 36 / 1
![Page 202: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/202.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
5. A ∪Ac = U
Since x ∈ A ∨ x /∈ A is true for any x and
x ∈ A ∪Ac ↔ x ∈ A ∨ x ∈ Ac
↔ x ∈ A ∨ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∨ x ∈ U) ∧ (x ∈ A ∨ x /∈ A)
↔ x ∈ A ∨ x ∈ U↔ x ∈ U (∵ A ⊆ U)
Hence, A ∪Ac = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 37 / 1
![Page 203: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/203.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
5. A ∪Ac = U Since x ∈ A ∨ x /∈ A is true for any x
and
x ∈ A ∪Ac ↔ x ∈ A ∨ x ∈ Ac
↔ x ∈ A ∨ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∨ x ∈ U) ∧ (x ∈ A ∨ x /∈ A)
↔ x ∈ A ∨ x ∈ U↔ x ∈ U (∵ A ⊆ U)
Hence, A ∪Ac = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 37 / 1
![Page 204: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/204.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
5. A ∪Ac = U Since x ∈ A ∨ x /∈ A is true for any x and
x ∈ A ∪Ac ↔ x ∈ A ∨ x ∈ Ac
↔ x ∈ A ∨ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∨ x ∈ U) ∧ (x ∈ A ∨ x /∈ A)
↔ x ∈ A ∨ x ∈ U↔ x ∈ U (∵ A ⊆ U)
Hence, A ∪Ac = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 37 / 1
![Page 205: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/205.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
5. A ∪Ac = U Since x ∈ A ∨ x /∈ A is true for any x and
x ∈ A ∪Ac ↔ x ∈ A ∨ x ∈ Ac
↔ x ∈ A ∨ (x ∈ U ∧ x /∈ A)
↔ (x ∈ A ∨ x ∈ U) ∧ (x ∈ A ∨ x /∈ A)
↔ x ∈ A ∨ x ∈ U↔ x ∈ U (∵ A ⊆ U)
Hence, A ∪Ac = U .
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 37 / 1
![Page 206: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/206.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.16)
Let A and B be sets with a universal set U . Then
1. A−B = A ∩Bc
2. A ⊆ B if and only if Bc ⊆ Ac
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 38 / 1
![Page 207: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/207.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .
1. A−B = A ∩Bc For any x
x ∈ A ∩Bc ↔ x ∈ A ∧ x ∈ Bc
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ B)
↔ (x ∈ A ∧ x ∈ U) ∧ x /∈ B
↔ x ∈ A ∧ x /∈ B (∵ A ⊆ U)↔ x ∈ A−B
So, A−B = A ∩Bc.2. A ⊆ B ↔ Bc ⊆ Ac
Bc ⊆ Ac ↔ ∀x (x ∈ Bc → x ∈ Ac)
↔ ∀x (x /∈ Ac → x /∈ Bc)
↔ ∀x [¬(x ∈ U ∧ x /∈ A)→ ¬(x ∈ U ∧ x /∈ B)]
↔ ∀x [(x /∈ U ∨ x ∈ A)→ (x /∈ U ∨ x ∈ B)]
↔ ∀x [x ∈ A→ x ∈ B] (∵ x /∈ U is false )
↔ A ⊆ B
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 39 / 1
![Page 208: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/208.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. A−B = A ∩Bc
For any x
x ∈ A ∩Bc ↔ x ∈ A ∧ x ∈ Bc
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ B)
↔ (x ∈ A ∧ x ∈ U) ∧ x /∈ B
↔ x ∈ A ∧ x /∈ B (∵ A ⊆ U)↔ x ∈ A−B
So, A−B = A ∩Bc.2. A ⊆ B ↔ Bc ⊆ Ac
Bc ⊆ Ac ↔ ∀x (x ∈ Bc → x ∈ Ac)
↔ ∀x (x /∈ Ac → x /∈ Bc)
↔ ∀x [¬(x ∈ U ∧ x /∈ A)→ ¬(x ∈ U ∧ x /∈ B)]
↔ ∀x [(x /∈ U ∨ x ∈ A)→ (x /∈ U ∨ x ∈ B)]
↔ ∀x [x ∈ A→ x ∈ B] (∵ x /∈ U is false )
↔ A ⊆ B
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 39 / 1
![Page 209: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/209.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. A−B = A ∩Bc For any x
x ∈ A ∩Bc ↔ x ∈ A ∧ x ∈ Bc
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ B)
↔ (x ∈ A ∧ x ∈ U) ∧ x /∈ B
↔ x ∈ A ∧ x /∈ B (∵ A ⊆ U)↔ x ∈ A−B
So, A−B = A ∩Bc.2. A ⊆ B ↔ Bc ⊆ Ac
Bc ⊆ Ac ↔ ∀x (x ∈ Bc → x ∈ Ac)
↔ ∀x (x /∈ Ac → x /∈ Bc)
↔ ∀x [¬(x ∈ U ∧ x /∈ A)→ ¬(x ∈ U ∧ x /∈ B)]
↔ ∀x [(x /∈ U ∨ x ∈ A)→ (x /∈ U ∨ x ∈ B)]
↔ ∀x [x ∈ A→ x ∈ B] (∵ x /∈ U is false )
↔ A ⊆ B
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 39 / 1
![Page 210: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/210.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. A−B = A ∩Bc For any x
x ∈ A ∩Bc ↔ x ∈ A ∧ x ∈ Bc
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ B)
↔ (x ∈ A ∧ x ∈ U) ∧ x /∈ B
↔ x ∈ A ∧ x /∈ B (∵ A ⊆ U)↔ x ∈ A−B
So, A−B = A ∩Bc.2. A ⊆ B ↔ Bc ⊆ Ac
Bc ⊆ Ac ↔ ∀x (x ∈ Bc → x ∈ Ac)
↔ ∀x (x /∈ Ac → x /∈ Bc)
↔ ∀x [¬(x ∈ U ∧ x /∈ A)→ ¬(x ∈ U ∧ x /∈ B)]
↔ ∀x [(x /∈ U ∨ x ∈ A)→ (x /∈ U ∨ x ∈ B)]
↔ ∀x [x ∈ A→ x ∈ B] (∵ x /∈ U is false )
↔ A ⊆ B
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 39 / 1
![Page 211: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/211.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. A−B = A ∩Bc For any x
x ∈ A ∩Bc ↔ x ∈ A ∧ x ∈ Bc
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ B)
↔ (x ∈ A ∧ x ∈ U) ∧ x /∈ B
↔ x ∈ A ∧ x /∈ B (∵ A ⊆ U)↔ x ∈ A−B
So, A−B = A ∩Bc.
2. A ⊆ B ↔ Bc ⊆ Ac
Bc ⊆ Ac ↔ ∀x (x ∈ Bc → x ∈ Ac)
↔ ∀x (x /∈ Ac → x /∈ Bc)
↔ ∀x [¬(x ∈ U ∧ x /∈ A)→ ¬(x ∈ U ∧ x /∈ B)]
↔ ∀x [(x /∈ U ∨ x ∈ A)→ (x /∈ U ∨ x ∈ B)]
↔ ∀x [x ∈ A→ x ∈ B] (∵ x /∈ U is false )
↔ A ⊆ B
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 39 / 1
![Page 212: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/212.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. A−B = A ∩Bc For any x
x ∈ A ∩Bc ↔ x ∈ A ∧ x ∈ Bc
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ B)
↔ (x ∈ A ∧ x ∈ U) ∧ x /∈ B
↔ x ∈ A ∧ x /∈ B (∵ A ⊆ U)↔ x ∈ A−B
So, A−B = A ∩Bc.2. A ⊆ B ↔ Bc ⊆ Ac
Bc ⊆ Ac ↔ ∀x (x ∈ Bc → x ∈ Ac)
↔ ∀x (x /∈ Ac → x /∈ Bc)
↔ ∀x [¬(x ∈ U ∧ x /∈ A)→ ¬(x ∈ U ∧ x /∈ B)]
↔ ∀x [(x /∈ U ∨ x ∈ A)→ (x /∈ U ∨ x ∈ B)]
↔ ∀x [x ∈ A→ x ∈ B] (∵ x /∈ U is false )
↔ A ⊆ B
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 39 / 1
![Page 213: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/213.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. A−B = A ∩Bc For any x
x ∈ A ∩Bc ↔ x ∈ A ∧ x ∈ Bc
↔ x ∈ A ∧ (x ∈ U ∧ x /∈ B)
↔ (x ∈ A ∧ x ∈ U) ∧ x /∈ B
↔ x ∈ A ∧ x /∈ B (∵ A ⊆ U)↔ x ∈ A−B
So, A−B = A ∩Bc.2. A ⊆ B ↔ Bc ⊆ Ac
Bc ⊆ Ac ↔ ∀x (x ∈ Bc → x ∈ Ac)
↔ ∀x (x /∈ Ac → x /∈ Bc)
↔ ∀x [¬(x ∈ U ∧ x /∈ A)→ ¬(x ∈ U ∧ x /∈ B)]
↔ ∀x [(x /∈ U ∨ x ∈ A)→ (x /∈ U ∨ x ∈ B)]
↔ ∀x [x ∈ A→ x ∈ B] (∵ x /∈ U is false )
↔ A ⊆ B
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 39 / 1
![Page 214: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/214.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
De Morgan’s Law
Theorem (2.2.17)
Let A and B be sets with a universal set U . Then1 (A ∩B)c = Ac ∪Bc
2 (A ∪B)c = Ac ∩Bc
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 40 / 1
![Page 215: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/215.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .
1. (A ∩B)c = Ac ∪Bc For any x
x ∈ (A ∩B)c ↔ x ∈ U ∧ x /∈ A ∩B
↔ x ∈ U ∧ ¬(x ∈ A ∩B)
↔ x ∈ U ∧ ¬(x ∈ A ∧ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∨ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∨ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∨ x ∈ Bc
↔ x ∈ Ac ∪Bc
So, (A ∩B)c = Ac ∪Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 41 / 1
![Page 216: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/216.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. (A ∩B)c = Ac ∪Bc
For any x
x ∈ (A ∩B)c ↔ x ∈ U ∧ x /∈ A ∩B
↔ x ∈ U ∧ ¬(x ∈ A ∩B)
↔ x ∈ U ∧ ¬(x ∈ A ∧ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∨ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∨ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∨ x ∈ Bc
↔ x ∈ Ac ∪Bc
So, (A ∩B)c = Ac ∪Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 41 / 1
![Page 217: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/217.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. (A ∩B)c = Ac ∪Bc For any x
x ∈ (A ∩B)c ↔ x ∈ U ∧ x /∈ A ∩B
↔ x ∈ U ∧ ¬(x ∈ A ∩B)
↔ x ∈ U ∧ ¬(x ∈ A ∧ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∨ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∨ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∨ x ∈ Bc
↔ x ∈ Ac ∪Bc
So, (A ∩B)c = Ac ∪Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 41 / 1
![Page 218: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/218.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. (A ∩B)c = Ac ∪Bc For any x
x ∈ (A ∩B)c ↔ x ∈ U ∧ x /∈ A ∩B
↔ x ∈ U ∧ ¬(x ∈ A ∩B)
↔ x ∈ U ∧ ¬(x ∈ A ∧ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∨ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∨ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∨ x ∈ Bc
↔ x ∈ Ac ∪Bc
So, (A ∩B)c = Ac ∪Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 41 / 1
![Page 219: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/219.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Proof.
Let A and B be a set with a universal set U .1. (A ∩B)c = Ac ∪Bc For any x
x ∈ (A ∩B)c ↔ x ∈ U ∧ x /∈ A ∩B
↔ x ∈ U ∧ ¬(x ∈ A ∩B)
↔ x ∈ U ∧ ¬(x ∈ A ∧ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∨ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∨ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∨ x ∈ Bc
↔ x ∈ Ac ∪Bc
So, (A ∩B)c = Ac ∪Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 41 / 1
![Page 220: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/220.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. (A ∪B)c = Ac ∩Bc
For any x
x ∈ (A ∪B)c ↔ x ∈ U ∧ x /∈ A ∪B
↔ x ∈ U ∧ ¬(x ∈ A ∪B)
↔ x ∈ U ∧ ¬(x ∈ A ∨ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∧ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∧ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∧ x ∈ Bc
↔ x ∈ Ac ∩Bc
Thus, (A ∪B)c = Ac ∩Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 42 / 1
![Page 221: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/221.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. (A ∪B)c = Ac ∩Bc For any x
x ∈ (A ∪B)c ↔ x ∈ U ∧ x /∈ A ∪B
↔ x ∈ U ∧ ¬(x ∈ A ∪B)
↔ x ∈ U ∧ ¬(x ∈ A ∨ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∧ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∧ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∧ x ∈ Bc
↔ x ∈ Ac ∩Bc
Thus, (A ∪B)c = Ac ∩Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 42 / 1
![Page 222: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/222.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. (A ∪B)c = Ac ∩Bc For any x
x ∈ (A ∪B)c ↔ x ∈ U ∧ x /∈ A ∪B
↔ x ∈ U ∧ ¬(x ∈ A ∪B)
↔ x ∈ U ∧ ¬(x ∈ A ∨ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∧ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∧ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∧ x ∈ Bc
↔ x ∈ Ac ∩Bc
Thus, (A ∪B)c = Ac ∩Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 42 / 1
![Page 223: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/223.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
2. (A ∪B)c = Ac ∩Bc For any x
x ∈ (A ∪B)c ↔ x ∈ U ∧ x /∈ A ∪B
↔ x ∈ U ∧ ¬(x ∈ A ∪B)
↔ x ∈ U ∧ ¬(x ∈ A ∨ x ∈ B)
↔ x ∈ U ∧ (x /∈ A ∧ x /∈ B)
↔ (x ∈ U ∧ x /∈ A) ∧ (x ∈ U ∧ x /∈ B)
↔ x ∈ Ac ∧ x ∈ Bc
↔ x ∈ Ac ∩Bc
Thus, (A ∪B)c = Ac ∩Bc.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 42 / 1
![Page 224: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/224.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Axiom of Operations
Theorem (2.2.18)
Let A,B and C be sets. Then
1. A− (B ∩ C) = (A−B) ∪ (A− C)
2. A− (B ∪ C) = (A−B) ∩ (A− C)
Proof. Assignment 2
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 43 / 1
![Page 225: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/225.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2.3 The Power Set Axiom
Axiom 2.3.1 (Axiom of Power set)
Let A be a set. There is a set C satisfying
x ∈ C ↔ x ⊆ A.
Theorem (2.3.1)
Let A be a set. There is a unique set C satisfying
x ∈ C ↔ x ⊆ A.
Proof .
Let A be a set and let C1 and C2 be sets satisfying condition of theorem 2.3.1. Then
x ∈ C1 ↔ x ⊆ A
x ∈ C2 ↔ x ⊆ A
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 44 / 1
![Page 226: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/226.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2.3 The Power Set Axiom
Axiom 2.3.1 (Axiom of Power set)
Let A be a set. There is a set C satisfying
x ∈ C ↔ x ⊆ A.
Theorem (2.3.1)
Let A be a set. There is a unique set C satisfying
x ∈ C ↔ x ⊆ A.
Proof .
Let A be a set and let C1 and C2 be sets satisfying condition of theorem 2.3.1. Then
x ∈ C1 ↔ x ⊆ A
x ∈ C2 ↔ x ⊆ A
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 44 / 1
![Page 227: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/227.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2.3 The Power Set Axiom
Axiom 2.3.1 (Axiom of Power set)
Let A be a set. There is a set C satisfying
x ∈ C ↔ x ⊆ A.
Theorem (2.3.1)
Let A be a set. There is a unique set C satisfying
x ∈ C ↔ x ⊆ A.
Proof .
Let A be a set and let C1 and C2 be sets satisfying condition of theorem 2.3.1.
Then
x ∈ C1 ↔ x ⊆ A
x ∈ C2 ↔ x ⊆ A
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 44 / 1
![Page 228: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/228.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2.3 The Power Set Axiom
Axiom 2.3.1 (Axiom of Power set)
Let A be a set. There is a set C satisfying
x ∈ C ↔ x ⊆ A.
Theorem (2.3.1)
Let A be a set. There is a unique set C satisfying
x ∈ C ↔ x ⊆ A.
Proof .
Let A be a set and let C1 and C2 be sets satisfying condition of theorem 2.3.1. Then
x ∈ C1 ↔ x ⊆ A
x ∈ C2 ↔ x ⊆ A
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 44 / 1
![Page 229: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/229.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2.3 The Power Set Axiom
Axiom 2.3.1 (Axiom of Power set)
Let A be a set. There is a set C satisfying
x ∈ C ↔ x ⊆ A.
Theorem (2.3.1)
Let A be a set. There is a unique set C satisfying
x ∈ C ↔ x ⊆ A.
Proof .
Let A be a set and let C1 and C2 be sets satisfying condition of theorem 2.3.1. Then
x ∈ C1 ↔ x ⊆ A
x ∈ C2 ↔ x ⊆ A
∴ x ∈ C1 ↔ x ∈ C2
Therefore, C1 = C2.Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 44 / 1
![Page 230: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/230.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Definition (2.3.1)
The set C in therorem 2.3.1 is a power set of A, denoted P(A) , i.e.,
P(A) = {x : x ⊆ A} or x ∈ P(A) ↔ x ⊆ A.
Theorem (2.3.2)
Let A be a set. Then
1. A ∈ P(A)
2. ∅ ∈ P(A)
3. P (∅) = {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 45 / 1
![Page 231: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/231.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Definition (2.3.1)
The set C in therorem 2.3.1 is a power set of A, denoted P(A) , i.e.,
P(A) = {x : x ⊆ A} or x ∈ P(A) ↔ x ⊆ A.
Theorem (2.3.2)
Let A be a set. Then
1. A ∈ P(A)
2. ∅ ∈ P(A)
3. P (∅) = {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 45 / 1
![Page 232: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/232.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A be a set.
1. A ∈ P(A) Since A ⊆ A, A ∈ P(A).
2. ∅ ∈ P(A) Since ∅ ⊆ A, ∅ ∈ P(A).
3. P(∅) = {∅}
x ∈ P(∅) ↔ x ⊆ ∅↔ x = ∅ ( By theorem 2.1.15 )
↔ x ∈ {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 46 / 1
![Page 233: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/233.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A be a set.1. A ∈ P(A)
Since A ⊆ A, A ∈ P(A).
2. ∅ ∈ P(A) Since ∅ ⊆ A, ∅ ∈ P(A).
3. P(∅) = {∅}
x ∈ P(∅) ↔ x ⊆ ∅↔ x = ∅ ( By theorem 2.1.15 )
↔ x ∈ {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 46 / 1
![Page 234: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/234.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A be a set.1. A ∈ P(A) Since A ⊆ A, A ∈ P(A).
2. ∅ ∈ P(A) Since ∅ ⊆ A, ∅ ∈ P(A).
3. P(∅) = {∅}
x ∈ P(∅) ↔ x ⊆ ∅↔ x = ∅ ( By theorem 2.1.15 )
↔ x ∈ {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 46 / 1
![Page 235: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/235.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A be a set.1. A ∈ P(A) Since A ⊆ A, A ∈ P(A).
2. ∅ ∈ P(A)
Since ∅ ⊆ A, ∅ ∈ P(A).
3. P(∅) = {∅}
x ∈ P(∅) ↔ x ⊆ ∅↔ x = ∅ ( By theorem 2.1.15 )
↔ x ∈ {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 46 / 1
![Page 236: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/236.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A be a set.1. A ∈ P(A) Since A ⊆ A, A ∈ P(A).
2. ∅ ∈ P(A) Since ∅ ⊆ A, ∅ ∈ P(A).
3. P(∅) = {∅}
x ∈ P(∅) ↔ x ⊆ ∅↔ x = ∅ ( By theorem 2.1.15 )
↔ x ∈ {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 46 / 1
![Page 237: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/237.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A be a set.1. A ∈ P(A) Since A ⊆ A, A ∈ P(A).
2. ∅ ∈ P(A) Since ∅ ⊆ A, ∅ ∈ P(A).
3. P(∅) = {∅}
x ∈ P(∅) ↔ x ⊆ ∅↔ x = ∅ ( By theorem 2.1.15 )
↔ x ∈ {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 46 / 1
![Page 238: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/238.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A be a set.1. A ∈ P(A) Since A ⊆ A, A ∈ P(A).
2. ∅ ∈ P(A) Since ∅ ⊆ A, ∅ ∈ P(A).
3. P(∅) = {∅}
x ∈ P(∅) ↔ x ⊆ ∅↔ x = ∅ ( By theorem 2.1.15 )
↔ x ∈ {∅}
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 46 / 1
![Page 239: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/239.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Example (2.3.1)
Draw Hasse diagram of all elements of power set A = {x, y, z}.
Solution
P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
{x, y, z}
{x, y} {y, z} {x, z}
{y} {x} {z}
∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 47 / 1
![Page 240: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/240.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Example (2.3.1)
Draw Hasse diagram of all elements of power set A = {x, y, z}.
Solution
P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
{x, y, z}
{x, y} {y, z} {x, z}
{y} {x} {z}
∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 47 / 1
![Page 241: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/241.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Example (2.3.1)
Draw Hasse diagram of all elements of power set A = {x, y, z}.
Solution
P(A) = {∅, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}}
{x, y, z}
{x, y} {y, z} {x, z}
{y} {x} {z}
∅
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 47 / 1
![Page 242: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/242.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.3)
If a set has n elements, then its power set will contain 2n elements.
Proof .
Let n be the number of elements of A. Then numbers of subset sets of A which have
0, 1, 2, ..., n elements,
respectively, are (n0
),
(n1
),
(n2
), ...,
(nn
).
By binomial theorem, it implies that(n0
)+
(n1
)+
(n2
)+ · · ·+
(nn
)= 2n.
Thus, the number of elements in the power set A is 2n.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 48 / 1
![Page 243: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/243.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.3)
If a set has n elements, then its power set will contain 2n elements.
Proof .
Let n be the number of elements of A.
Then numbers of subset sets of A which have
0, 1, 2, ..., n elements,
respectively, are (n0
),
(n1
),
(n2
), ...,
(nn
).
By binomial theorem, it implies that(n0
)+
(n1
)+
(n2
)+ · · ·+
(nn
)= 2n.
Thus, the number of elements in the power set A is 2n.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 48 / 1
![Page 244: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/244.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.3)
If a set has n elements, then its power set will contain 2n elements.
Proof .
Let n be the number of elements of A. Then numbers of subset sets of A which have
0, 1, 2, ..., n elements,
respectively,
are (n0
),
(n1
),
(n2
), ...,
(nn
).
By binomial theorem, it implies that(n0
)+
(n1
)+
(n2
)+ · · ·+
(nn
)= 2n.
Thus, the number of elements in the power set A is 2n.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 48 / 1
![Page 245: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/245.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.3)
If a set has n elements, then its power set will contain 2n elements.
Proof .
Let n be the number of elements of A. Then numbers of subset sets of A which have
0, 1, 2, ..., n elements,
respectively, are (n0
),
(n1
),
(n2
), ...,
(nn
).
By binomial theorem, it implies that(n0
)+
(n1
)+
(n2
)+ · · ·+
(nn
)= 2n.
Thus, the number of elements in the power set A is 2n.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 48 / 1
![Page 246: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/246.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.3)
If a set has n elements, then its power set will contain 2n elements.
Proof .
Let n be the number of elements of A. Then numbers of subset sets of A which have
0, 1, 2, ..., n elements,
respectively, are (n0
),
(n1
),
(n2
), ...,
(nn
).
By binomial theorem, it implies that
(n0
)+
(n1
)+
(n2
)+ · · ·+
(nn
)= 2n.
Thus, the number of elements in the power set A is 2n.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 48 / 1
![Page 247: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/247.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.3)
If a set has n elements, then its power set will contain 2n elements.
Proof .
Let n be the number of elements of A. Then numbers of subset sets of A which have
0, 1, 2, ..., n elements,
respectively, are (n0
),
(n1
),
(n2
), ...,
(nn
).
By binomial theorem, it implies that(n0
)+
(n1
)+
(n2
)+ · · ·+
(nn
)= 2n.
Thus, the number of elements in the power set A is 2n.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 48 / 1
![Page 248: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/248.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.3)
If a set has n elements, then its power set will contain 2n elements.
Proof .
Let n be the number of elements of A. Then numbers of subset sets of A which have
0, 1, 2, ..., n elements,
respectively, are (n0
),
(n1
),
(n2
), ...,
(nn
).
By binomial theorem, it implies that(n0
)+
(n1
)+
(n2
)+ · · ·+
(nn
)= 2n.
Thus, the number of elements in the power set A is 2n.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 48 / 1
![Page 249: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/249.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Example (2.3.2)
Compute number of elements of a power set
A = {x ∈ Z : |x| < 100 and 3 | x}.
Solution
A = {x ∈ Z : |x| < 100 and 3 | x}A = {0,±3,±6, ...,±99}
Hence, A has number of elements to be 33× 2 + 1 = 67.Therefore, number of elements of P(A) is 267.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 49 / 1
![Page 250: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/250.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Example (2.3.2)
Compute number of elements of a power set
A = {x ∈ Z : |x| < 100 and 3 | x}.
Solution
A = {x ∈ Z : |x| < 100 and 3 | x}A = {0,±3,±6, ...,±99}
Hence, A has number of elements to be 33× 2 + 1 = 67.Therefore, number of elements of P(A) is 267.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 49 / 1
![Page 251: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/251.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Example (2.3.2)
Compute number of elements of a power set
A = {x ∈ Z : |x| < 100 and 3 | x}.
Solution
A = {x ∈ Z : |x| < 100 and 3 | x}A = {0,±3,±6, ...,±99}
Hence, A has number of elements to be 33× 2 + 1 = 67.
Therefore, number of elements of P(A) is 267.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 49 / 1
![Page 252: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/252.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Example (2.3.2)
Compute number of elements of a power set
A = {x ∈ Z : |x| < 100 and 3 | x}.
Solution
A = {x ∈ Z : |x| < 100 and 3 | x}A = {0,±3,±6, ...,±99}
Hence, A has number of elements to be 33× 2 + 1 = 67.Therefore, number of elements of P(A) is 267.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 49 / 1
![Page 253: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/253.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.4)
Let A and B be sets. Then
1. A ⊆ B if and only if P(A) ⊆ P(B)
2. P(A ∩B) = P(A) ∩ P(B)
3. P(A ∪B) ⊇ P(A) ∪ P(B)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 50 / 1
![Page 254: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/254.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A and B be sets.
1. A ⊆ B ↔ P(A) ⊆ P(B)
(→) Assume A ⊆ B.
x ∈ P(A) → x ⊆ A
→ x ⊆ B (∵ A ⊆ B)
→ x ∈ P(B)
Thus, P(A) ⊆ P(B).(←) Assume P(A) ⊆ P(B).
x ∈ A → {x} ⊆ A
→ {x} ∈ P(A)
→ {x} ∈ P(B) (∵ P(A) ⊆ P(B))
→ {x} ⊆ B
→ x ∈ B
Thus, A ⊆ B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 51 / 1
![Page 255: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/255.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A and B be sets.1. A ⊆ B ↔ P(A) ⊆ P(B)
(→) Assume A ⊆ B.
x ∈ P(A) → x ⊆ A
→ x ⊆ B (∵ A ⊆ B)
→ x ∈ P(B)
Thus, P(A) ⊆ P(B).(←) Assume P(A) ⊆ P(B).
x ∈ A → {x} ⊆ A
→ {x} ∈ P(A)
→ {x} ∈ P(B) (∵ P(A) ⊆ P(B))
→ {x} ⊆ B
→ x ∈ B
Thus, A ⊆ B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 51 / 1
![Page 256: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/256.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A and B be sets.1. A ⊆ B ↔ P(A) ⊆ P(B)
(→) Assume A ⊆ B.
x ∈ P(A) → x ⊆ A
→ x ⊆ B (∵ A ⊆ B)
→ x ∈ P(B)
Thus, P(A) ⊆ P(B).(←) Assume P(A) ⊆ P(B).
x ∈ A → {x} ⊆ A
→ {x} ∈ P(A)
→ {x} ∈ P(B) (∵ P(A) ⊆ P(B))
→ {x} ⊆ B
→ x ∈ B
Thus, A ⊆ B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 51 / 1
![Page 257: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/257.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A and B be sets.1. A ⊆ B ↔ P(A) ⊆ P(B)
(→) Assume A ⊆ B.
x ∈ P(A) → x ⊆ A
→ x ⊆ B (∵ A ⊆ B)
→ x ∈ P(B)
Thus, P(A) ⊆ P(B).
(←) Assume P(A) ⊆ P(B).
x ∈ A → {x} ⊆ A
→ {x} ∈ P(A)
→ {x} ∈ P(B) (∵ P(A) ⊆ P(B))
→ {x} ⊆ B
→ x ∈ B
Thus, A ⊆ B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 51 / 1
![Page 258: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/258.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A and B be sets.1. A ⊆ B ↔ P(A) ⊆ P(B)
(→) Assume A ⊆ B.
x ∈ P(A) → x ⊆ A
→ x ⊆ B (∵ A ⊆ B)
→ x ∈ P(B)
Thus, P(A) ⊆ P(B).(←) Assume P(A) ⊆ P(B).
x ∈ A → {x} ⊆ A
→ {x} ∈ P(A)
→ {x} ∈ P(B) (∵ P(A) ⊆ P(B))
→ {x} ⊆ B
→ x ∈ B
Thus, A ⊆ B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 51 / 1
![Page 259: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/259.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A and B be sets.1. A ⊆ B ↔ P(A) ⊆ P(B)
(→) Assume A ⊆ B.
x ∈ P(A) → x ⊆ A
→ x ⊆ B (∵ A ⊆ B)
→ x ∈ P(B)
Thus, P(A) ⊆ P(B).(←) Assume P(A) ⊆ P(B).
x ∈ A → {x} ⊆ A
→ {x} ∈ P(A)
→ {x} ∈ P(B) (∵ P(A) ⊆ P(B))
→ {x} ⊆ B
→ x ∈ B
Thus, A ⊆ B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 51 / 1
![Page 260: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/260.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Proof.
Let A and B be sets.1. A ⊆ B ↔ P(A) ⊆ P(B)
(→) Assume A ⊆ B.
x ∈ P(A) → x ⊆ A
→ x ⊆ B (∵ A ⊆ B)
→ x ∈ P(B)
Thus, P(A) ⊆ P(B).(←) Assume P(A) ⊆ P(B).
x ∈ A → {x} ⊆ A
→ {x} ∈ P(A)
→ {x} ∈ P(B) (∵ P(A) ⊆ P(B))
→ {x} ⊆ B
→ x ∈ B
Thus, A ⊆ B.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 51 / 1
![Page 261: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/261.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2. P(A ∩B) = P(A) ∩ P(B)
For any x
x ∈ P(A) ∩ P(B) ↔ x ∈ P(A) ∧ x ∈ P(B)
↔ x ⊆ A ∧ x ⊆ B
↔ x ⊆ A ∩B ( By theorem 2.2.5)↔ x ∈ P(A ∩B)
3. P(A ∪B) ⊇ P(A) ∪ P(B)
x ∈ P(A) ∪ P(B) → x ∈ P(A) ∨ x ∈ P(B)
→ x ⊆ A ∨ x ⊆ B
→ x ⊆ A ∪B ∨ x ⊆ A ∪B (∵ A ⊆ A ∪B and B ⊆ A ∪B)
→ x ∈ P(A ∪B)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 52 / 1
![Page 262: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/262.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2. P(A ∩B) = P(A) ∩ P(B) For any x
x ∈ P(A) ∩ P(B) ↔ x ∈ P(A) ∧ x ∈ P(B)
↔ x ⊆ A ∧ x ⊆ B
↔ x ⊆ A ∩B ( By theorem 2.2.5)↔ x ∈ P(A ∩B)
3. P(A ∪B) ⊇ P(A) ∪ P(B)
x ∈ P(A) ∪ P(B) → x ∈ P(A) ∨ x ∈ P(B)
→ x ⊆ A ∨ x ⊆ B
→ x ⊆ A ∪B ∨ x ⊆ A ∪B (∵ A ⊆ A ∪B and B ⊆ A ∪B)
→ x ∈ P(A ∪B)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 52 / 1
![Page 263: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/263.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2. P(A ∩B) = P(A) ∩ P(B) For any x
x ∈ P(A) ∩ P(B) ↔ x ∈ P(A) ∧ x ∈ P(B)
↔ x ⊆ A ∧ x ⊆ B
↔ x ⊆ A ∩B ( By theorem 2.2.5)↔ x ∈ P(A ∩B)
3. P(A ∪B) ⊇ P(A) ∪ P(B)
x ∈ P(A) ∪ P(B) → x ∈ P(A) ∨ x ∈ P(B)
→ x ⊆ A ∨ x ⊆ B
→ x ⊆ A ∪B ∨ x ⊆ A ∪B (∵ A ⊆ A ∪B and B ⊆ A ∪B)
→ x ∈ P(A ∪B)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 52 / 1
![Page 264: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/264.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2. P(A ∩B) = P(A) ∩ P(B) For any x
x ∈ P(A) ∩ P(B) ↔ x ∈ P(A) ∧ x ∈ P(B)
↔ x ⊆ A ∧ x ⊆ B
↔ x ⊆ A ∩B ( By theorem 2.2.5)↔ x ∈ P(A ∩B)
3. P(A ∪B) ⊇ P(A) ∪ P(B)
x ∈ P(A) ∪ P(B) → x ∈ P(A) ∨ x ∈ P(B)
→ x ⊆ A ∨ x ⊆ B
→ x ⊆ A ∪B ∨ x ⊆ A ∪B (∵ A ⊆ A ∪B and B ⊆ A ∪B)
→ x ∈ P(A ∪B)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 52 / 1
![Page 265: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/265.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
2. P(A ∩B) = P(A) ∩ P(B) For any x
x ∈ P(A) ∩ P(B) ↔ x ∈ P(A) ∧ x ∈ P(B)
↔ x ⊆ A ∧ x ⊆ B
↔ x ⊆ A ∩B ( By theorem 2.2.5)↔ x ∈ P(A ∩B)
3. P(A ∪B) ⊇ P(A) ∪ P(B)
x ∈ P(A) ∪ P(B) → x ∈ P(A) ∨ x ∈ P(B)
→ x ⊆ A ∨ x ⊆ B
→ x ⊆ A ∪B ∨ x ⊆ A ∪B (∵ A ⊆ A ∪B and B ⊆ A ∪B)
→ x ∈ P(A ∪B)
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 52 / 1
![Page 266: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/266.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Axiom 2.3.2 (The Axiom of Regularity)
Every non-empty set x contains a member y such that x and y are disjoint sets.
∀x [∃a (a ∈ x) → ∃y (y ∈ x ∧ ¬∃z (z ∈ x ∧ z ∈ y))]
In other word, for any A ̸= ∅, there exists C ∈ A such that
∀x [x ∈ C → x /∈ A] or C ∩A = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 53 / 1
![Page 267: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/267.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Axiom 2.3.2 (The Axiom of Regularity)
Every non-empty set x contains a member y such that x and y are disjoint sets.
∀x [∃a (a ∈ x) → ∃y (y ∈ x ∧ ¬∃z (z ∈ x ∧ z ∈ y))]
In other word, for any A ̸= ∅, there exists C ∈ A such that
∀x [x ∈ C → x /∈ A] or C ∩A = ∅.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 53 / 1
![Page 268: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/268.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.5)
For any set A, A /∈ A.
Proof .
Let A be a set. We will prove by contradiction. Suppose that A ∈ A.Since A ∈ {A}, A ∈ A ∩ {A}. So,
A ∩ {A} ̸= ∅ (1)
Since {A} ̸= ∅ by the axiom of regularity , there is set C ∈ {A} such that C ∩ {A} = ∅.
But C ∈ {A}means that C = A, so A ∩ {A} = ∅.
It contradics (1).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 54 / 1
![Page 269: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/269.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Theorem (2.3.11)
For set A and B, A /∈ B or B /∈ A.
Proof .
Let A and B be sets. We will prove by contradiction. Suppose that A ∈ B and B ∈ A. SinceA ∈ {A,B} and B ∈ {A,B},
(A ∈ B ∩ {A,B}) ∧ (B ∈ A ∩ {A,B}) (2)
Since {A,B} ̸= ∅ by the axiom of regularity , there is set C ∈ {A,B} such thatC ∩ {A,B} = ∅. But C ∈ {A,B}means that C = A or C = B, so
A ∩ {A,B} = ∅ or B ∩ {A,B} = ∅.
It contradics (2).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 55 / 1
![Page 270: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/270.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Conclusion
The Existential Axiom There is a set at least one.The Axiom of Extensionality Two sets are equal (are the same set) if they have the sameelements.The Axiom of Specification To every set A and to every condition p(x) therecorresponds a set B whose elements are exactly those elements x of A for which p(x)holds.Axiom of Union Let A and B be sets. There is a set C satisfying
x ∈ C ↔ (x ∈ A ∨ x ∈ B).
Axiom of Power set Let A be a set. There is a set C satisfying
x ∈ C ↔ x ⊆ A.
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 56 / 1
![Page 271: MAT1202 SET THEORY :Week2 · Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 4 / 1..... The Aixiomatic Set Theory The Axiom of Operations 2.2 The Axiom of Operations Theorem (2.2.1)](https://reader034.vdocuments.us/reader034/viewer/2022050306/5f6eac2ad51ad95309787ae2/html5/thumbnails/271.jpg)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
The Aixiomatic Set Theory The Power Set Axiom
Assignment 2 (30 Minutes)
1. (EVEN and ODD) Let A1, A2, ..., An, B1, B2, ..., Bn be sets. Then, for any n ∈ N,
A1 = B1 ∧A2 = B2 ∧ ...∧An = Bn → (A1 ∩A2 ∩ ...∩An) = (B1 ∩B2 ∩ ...∩Bn).
2. Let A,B and C be sets. Prove that
(a) (EVEN) A− (B ∩ C) = (A−B) ∪ (A− C)
(b) (ODD) A− (B ∪ C) = (A−B) ∩ (A− C)
3. (EVEN and ODD) Let A = {1, 2, 3, 4}.
(a) Cumpute numbers of subset sets of A which have 0, 1, 2, 3, 4 elements by binomialtheorem.
(b) Write out elements of P(A).(c) Draw Hasse diagram of elements of P(A).
Thanatyod Jampawai, Ph.D. MAT1202 SET THEORY :Week2 57 / 1