Elements of Set Theory

Chapter 1-2: Introduction, Axioms and Operations

January 14 &16, 2014

Teacher: Fan Yang

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Chapter 1: Introduction

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A set is a collection into a whole of definite, distinct objects of our intuition orour thought. The objects are called elements (or members) of the set.

—- Georg Cantor, the founder of set theory

For example, the following collections can be considered sets:

{ , , }

{ , , , }

N = {0,1,2,3, . . . }, A =

{√2,

27, sin 20◦

}, X = { {a},b }, etc.

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Let A be a set, and x some object.If x is a element of A, then we write x ∈ AIf x is not a element of A, then we write ¬(x ∈ A) or x /∈ A

For example:x ∈ {x , y , z}.−5 /∈ {0,1,2,3}.

{ , , , }∈

{ , b }∈

{ {a} , b }{a} ∈

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Principle of ExtensionalityIf two sets A and B have exactly the same elements, then they areequal, denoted by A = B. Otherwise, we write ¬(A = B) or A 6= B.

That is, if for every object x ,

x ∈ A ⇐⇒ x ∈ B,

then A = B.

From this principle, we know the following:

If A = {√

2,−√

2} and B is the set of all solutions to the equationx2 = 2,

then A = B.A set cannot have duplicate elements, that is, e.g.

{a,a,b} = {a,b}.

Elements of a set do not have any order, that is, e.g.{a,b} = {b,a}.

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Usually, we represent a set in the following two ways:

1 List the elements of the set. For example,

A = {a,b, c} is the set consisting of three elements a, b, c.

K = {0,2,4,6,8, . . . } is the set of all even natural numbers.

2 Present the precise condition of an object being an element of theset. For example, the above set K can also be represented as

K = {n | n is an even natural number}or

K = {n ∈ N : n is even}.

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• X = {x ∈ Z | x + 1 = 0}= {−1}.

We call a set which has exactly one element a singleton.

For example, {a,b, c} is a set having 3 elements; {{a,b, c}} is asingleton.

• Let Y = {x ∈ N | x + 1 = 0}. Clearly, Y has no elements at all.

A set which has no elements is called an empty set, denoted by ∅ or ∅.

By Principle of Extensionality, the empty set is unique.

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The empty set may seem to be useless, but in fact, it is very important.Using set-theoretic operations, many sets can be constructed from ∅.

{∅} is a singleton consisting of the unique element ∅. Note that{∅} 6= ∅, since ∅ ∈ {∅} and ∅ /∈ ∅.

{{∅}} is a singleton consisting of the unique element {∅}, and{{∅}} 6= {∅}.

The following are distinct singletons:

{ {{∅}} }, { {{{∅}}} }, { {{{{∅}}}} }, . . .

The set {∅, {∅} } has two elements, namely ∅ and {∅}.

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For any sets A and B, the set whose elements are those belongingeither to A or to B (or both) is called the union of A and B, denoted byA ∪ B. That is

A ∪ B = {x | x ∈ A ∨ x ∈ B}.

A BA ∪ B Venn Diagram

For example:If A = {a,b} and B = {b, c,d}, then A ∪ B = {a,b, c,d}.{∅} ∪ {{∅}} = {∅, {∅}}.∅ ∪ A = A, A ∪ A = A, for any set A.

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We say that a set B is a subset of a set A (or B is included in A),denoted by B ⊆ A, if every element of B is an element of A. That is

B ⊆ A ⇐⇒ ∀x(x ∈ B → x ∈ A).

If B is not a subset of A, then we write B * A.

ABB ⊆ A

AB B 6⊆ A AB

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For example:Let A = {a,b, c,d}, B = {a,d}, C = {c,d}. Then B ⊆ A, B * C.{∅} ⊆ {∅, {∅} }, ∅ ⊆ {∅, {∅} }.For any sets A and B,

A ⊆ A, ∅ ⊆ A, A ⊆ A ∪ B.

If A ⊆ B and B ⊆ C, then A ⊆ C. (Transitivity)

By Principle of Extensionality,

A = B iff A ⊆ B and B ⊆ A.

If B ⊆ A and B 6= A, then we call B a proper subset of A and writeB ⊂ A or B ( A.

For example,if A = {∅, {∅}}, B = {∅}, then B ⊂ A.∅ ⊂ A for any A 6= ∅.

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Given a set A. The set of all subsets of A is called the power set of A,denoted by ℘(A) or ℘A. That is,

℘(A) = {X | X ⊆ A}.

For example,If A = {a,b}, then

℘(A) = {∅, {a}, {b}, {a,b}}.

℘(∅) = {X : X ⊆ ∅}= {∅}.℘({∅}) =

{X : X ⊆ {∅}

}= {∅, {∅}}.

℘({∅, {∅}}) ={

X : X ⊆ {∅, {∅}}}={∅, {∅}, {{∅}}, {∅, {∅}}

}.

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Constructing sets

V0 = {a,b, c,d , . . . }, the set A of atomsV1 = V0 ∪ ℘(V0)= A ∪ ℘(A)V2 = V1 ∪ ℘(V1)...Vn+1 = Vn ∪ ℘(Vn)...Vω = V1 ∪ V2 ∪ · · ·Vω+1 = Vω ∪ ℘(Vω)...

V0 = A

.

.

.

α

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.

.

ω + 1ω

.

.

.

12

0

V0 ⊆ V1 ⊆ V2 ⊆ . . .

0,1,2, . . . , ω, ω + 1, ω + 2, . . . , α, . . .ordinal numbers

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Constructing sets

V0 = ∅V1 = V0 ∪ ℘(V0)= ℘(∅) = {∅}V2 = V1 ∪ ℘(V1)= {∅} ∪ ℘({∅}) = {∅, {∅}}...Vn+1 = Vn ∪ ℘(Vn)...Vω = V1 ∪ V2 ∪ · · ·Vω+1 = Vω ∪ ℘(Vω)...

V0 = ∅

.

.

.

α

.

.

.

ω

.

.

.

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Is there a “set of all sets”?

We will prove later that such a collection is too large to be called a set.Informally, any collection of sets is called a class, but a class A iscalled a set only if it is included in some level Vα of the hierarchy ofsets; otherwise A is called a proper class. The “collection of all sets” isa proper class.

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Consider the class C = {x | x /∈ x}.Question: Is C ∈ C?

If YES, then C /∈ C. Hence C ∈ C =⇒ C /∈ C.

If NO, then C ∈ C. Hence C /∈ C =⇒ C ∈ C.

—- Russell’s Paradox

To avoid “set of all sets" and paradoxes, we will re-examine thedefinition of a set.

Since early 1900’s, mathematicians took the axiomatic approach tosets and developed the so-called “axiomatic set theory”. One of thecommon axiom systems is the Zermelo-Fraenkel system with Axiom ofChoice (ZFC).

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Chapter 2: Axioms and Operations

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An axiom or postulate is a premise so evident as to be accepted astrue without controversy. Axioms serve as a starting point ofreasoning. For example, Euclidean plane geometry has five evidentaxioms or postulates, all theorems and results of lines, triangles,circles, etc. can be derived from these five axioms.

In this course, we will discuss the axioms of ZFC system. From theseaxioms, the theorems of set theory will follow.

It is sometimes said that “mathematics can be embedded in settheory”, meaning that mathematical objects (such as numbers,differentiable functions) can be defined to be certain sets, and thetheorems of mathematics then can be viewed as statements aboutsets. Therefore, these theorems will be provable from the axioms ofset theory.

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In order to state the axioms and give proofs rigorously, we will useformulas.

In our context, the simplest formulas are expressions of the forms:

a ∈ A and a = b.

More complicated formulas can be built up from these using theexpressions

∀x , ∃x , ¬, ∧, ∨, →, ↔together with parentheses to avoid ambiguity. That is, from formulas φand ψ, we can construct longer formulas

∀xφ, ∃xφ, ¬φ, φ ∧ ψ, φ ∨ ψ, φ→ ψ, φ↔ ψ, etc.

For example, the following are formulas:x ∈ A∨ x ∈ B∀x(x /∈ B)

∀x(x ∈ A↔ x ∈ B)

A ⊆ B is not a formula, but it can be viewed as a shorthand for∀x(x ∈ A→ x ∈ B).

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Extensionality AxiomFor any two sets A,B, if they have exactly the same elements, thenthey are equal:

∀A∀B(∀x(x ∈ A↔ x ∈ B)→ A = B

).

SinceB ⊆ A iff ∀x(x ∈ B → x ∈ A),

the Extensionality Axiom can be rephrased as

A = B if A ⊆ B and B ⊆ A.

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Empty Set AxiomThere is a set having no elements:

∃B∀x(x /∈ B).

Definition 2.1A set which has no elements is called an Empty Set, denoted by ∅ or∅.

By Empty Set Axiom, the empty set exists.By Extensionality Axiom, the empty set is unique.∅ = {x | x 6= x}

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Pairing AxiomFor any sets u and v, there is a set having just u and v as elements:

∀u∀v∃B∀x(x ∈ B ↔ (x = u ∨ x = v)

).

Definition 2.2For any sets u and v, the set whose elements are u and v is called apair set, denoted by {u, v}. That is,

{u, v} = {x | x = u ∨ x = v}.

Given any set x , by Pairing Axiom the set {x , x} exists. We write{x} for {x , x}, and call it a singleton.Since ∅ is a set, by Pairing Axiom, {∅, ∅} = {∅} is a set.Since ∅ and {∅} are sets, {∅, {∅}} is a set.

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Union Axiom (Preliminary Form)For any sets A and B, there is a set whose elements are those setsbelonging either to A or to B (or both):

∀A∀B∃C∀x(x ∈ C ↔ (x ∈ A ∨ x ∈ B)

).

Definition 2.3For any sets A and B, the set whose elements are those setsbelonging either to A or to B (or both) is called the union of A and B,denoted by A ∪ B. That is,

A ∪ B = {x | x ∈ A ∨ x ∈ B}

For example, {∅} ∪ {{∅}} = {∅, {∅}} is a set.

Given sets x1, x2, x3, by Pairing Axiom, {x1, x2} and {x3} are sets. ByUnion Axiom, {x1, x2} ∪ {x3} is a set, and we define

{x1, x2, x3} = {x1, x2} ∪ {x3}.

Similarly, we can define {x1, x2, x3, x4}, and so forth.22/40

Power Set AxiomFor any set A, there is a set whose elements are exactly all of thesubsets of A:

∀A∃B∀x(x ∈ B ↔ x ⊆ A).

Definition 2.4For any set A, the set of all subsets of A is called the power set of A,denoted by ℘(A) or ℘A. That is,

℘(A) = {x | x ⊆ A}.

Note: x ∈ ℘(A)⇐⇒ x ⊆ A.

Example 2.1: For any sets A and B, if ℘(A) = ℘(B), then A = B.

Proof. If ℘(A) = ℘(B), then

A ⊆ A =⇒ A ∈ ℘(A) = ℘(B) =⇒ A ∈ ℘(B) =⇒ A ⊆ B.

Similarly, we get B ⊆ A. Hence A = B.23/40

∅ = {x | x 6= x}{u, v} = {x | x = u ∨ x = v}A ∪ B = {x | x ∈ A ∨ x ∈ B}℘(A) = {x | x ⊆ A}

The above sets are all of the form {x | φ(x)}.

For any formula φ(x), is

A = {x | φ(x)}

always a set?

No. For example, V = {x | x = x} is not a set, that is, the set of all setsdoes not exist.

In order for the class A to be called a set, it must be included in someVα of the hierarchy. In fact, it is enough for A to be included in any setc, for then

A ⊆ c ⊆ Vαfor some α.

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Separation Axioms (or Subset Axioms)For any formula φ(x , y1, . . . , yn) not containing B, any sets c,a1, . . . ,an,

B = {x ∈ c | φ(x ,a1, . . . ,an)}is a set.

Clearly, B ⊆ c.

For example:Let c = {{a}, {a,b}, {b,d ,e}} be a set. Then the following aresets:

B0 = {x ∈ c | a ∈ x} = {{a}, {a,b}},B1 = {x ∈ c | a /∈ x} = {{b,d ,e}}.

Let c = {a,b} be a set. Then

B = {x ∈ ℘(c) | x is a singleton} = { {a}, {b} }is a set, where “x is a singleton” is a shorthand for the formula

∀y(y ⊆ x → (y = x ∨ y = ∅)),or

∀y(∀z(z ∈ y → z ∈ x)→ (y = x ∨ y = ∅)

).

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Theorem 2AThere is no set to which every set belongs.

Proof. Let A be an arbitrary set. We construct a set B such that B /∈ A.

By Separation Axiom, B defined as follows is a set:

B = {x ∈ A | x /∈ x}.

If B ∈ A, then by the definition of B,

B /∈ B ⇐⇒ B ∈ B,

which is impossible. Hence B /∈ A, as desired.

No set is an element of itself: ∀x(x /∈ x). [ Follows from Regularity Axiom(Chapter 7) ]

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Definition 2.5For any sets A and B, the intersection of A and B, denoted by A ∩ B, isdefined as

A ∩ B = {x | x ∈ A ∧ x ∈ B}.

A ∩ B A B

By Separation Axiom, we know that the intersection

A ∩ B = {x | x ∈ A ∧ x ∈ B}= {x ∈ A | x ∈ B}

is a set.

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For example:If A = {a,b, c} and B = {b, c,d}, then A ∩ B = {b, c}.{∅} ∩ {∅, {∅}} = {∅}∅ ∩ A = ∅, A ∩ A = A for any set A.A ∩ B ⊆ A for any sets A and B

Definition 2.6If A ∩ B = ∅, then we say that A and B are disjoint, or A does notintersect B.

BA

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Definition 2.7For any sets A and B, the relative complement of B in A or thedifference of A and B, denoted by A− B or A \ B, is defined as

A− B = {x | x ∈ A ∧ x /∈ B}.

A BA− B

By Separation Axiom, we know that the difference

A− B = {x | x ∈ A ∧ x /∈ B}= {x ∈ A | x /∈ B}.

is a set.

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Definition 2.8Let A be a subset of S. The relative complement of S and A is calledthe complement of A in S, denoted by A or Ac or −A.

S

AA

For example, if S = {a,b, c,d}, A = {a,b, c} and B = {b, c}, then

A− B = {a}, B − A = ∅, A = {d}, B = {a,d}.

Some properties of difference and complement:

A = A∅ = S, S = ∅A ∩ A = ∅, A ∪ A = SA− B = A ∩ B

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Properties of operations on sets

Let A,B,C be sets.

Commutative laws A ∪ B = B ∪ A and A ∩ B = B ∩ A

Associative laws A ∪ (B ∪ C) = (A ∪ B) ∪ C

A ∩ (B ∩ C) = (A ∩ B) ∩ C

Distributive laws A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)

If A ⊆ B, thenA ∪ C ⊆ B ∪ C, A ∩ C ⊆ B ∩ C.

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Example 2.2: Show that A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).

Proof. For any x , x ∈ A ∪ (B ∩ C) =⇒x ∈ A =⇒ x ∈ A ∪ B and x ∈ A ∪ C =⇒ x ∈ (A ∪ B) ∩ (A ∪ C)or ⇑x ∈ B ∩ C =⇒ x ∈ B and x ∈ C

Hence, we conclude that A ∪ (B ∩ C) ⊆ (A ∪ B) ∩ (A ∪ C).

Conversely, for any x , if x ∈ A, then x ∈ A ∪ (B ∩ C); if x /∈ A, then

x ∈ (A ∪ B) ∩ (A ∪ C) =⇒ x ∈ A ∪ B and x ∈ A ∪ C

x /∈A=⇒ x ∈ B and x ∈ C =⇒ x ∈ B ∩ C =⇒ x ∈ A ∪ (B ∩ C).

Hence (A ∪ B) ∩ (A ∪ C) ⊆ A ∪ (B ∩ C).

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Let A,B,C be subsets of a set S.

De Morgan’s laws C − (A ∪ B) = (C − A) ∩ (C − B)

C − (A ∩ B) = (C − A) ∪ (C − B)

Substituting S for C yields

(A ∪ B) = A ∩ B

(A ∩ B) = A ∪ B

Example 2.3: Show that C − (A ∪ B) = (C − A) ∩ (C − B).

Proof. For any element x , we have thatx ∈ C − (A ∪ B) ⇐⇒ x ∈ C and x /∈ A ∪ B

⇐⇒ x ∈ C and (x /∈ A and x /∈ B)

⇐⇒ (x ∈ C and x /∈ A) and (x ∈ C and x /∈ B)

⇐⇒ x ∈ C − A and x ∈ C − B⇐⇒ x ∈ (C − A) ∩ (C − B).

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Example 2.4: Let A,B,C be subsets of a set S. Show that

A− (B − C) = (A− B) ∪ (A ∩ C).

Proof. We have that

A− (B − C) = A ∩ (B − C) (since X − Y = X ∩ Y )

= A ∩ (B ∩ C)

= A ∩ (B ∪ C) (by De Morgan’s law)

= A ∩ (B ∪ C)

= (A ∩ B) ∪ (A ∩ C) (by distributive law)= (A− B) ∪ (A ∩ C).

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Suppose we have an infinite collection of sets

A = {A1,A2,A3, . . . ,An, . . . }.

By the preliminary form of Union Axiom, we know that the followingfinite unions are sets:

A1 ∪ A2, (A1 ∪ A2) ∪ A3, . . . A1 ∪ A2 ∪ · · · ∪ An, . . .

Definition 2.9For any set A , the union of A , denoted by

⋃A , is defined as⋃

A = {x | x belongs to some element of A } = {x | ∃A ∈ A (x ∈ A)}.

To guarantee that⋃

A , particularly the infinite union A1 ∪ A2 ∪ · · · , is aset, we need the general form of Union Axiom.

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Union AxiomFor any set A , there exists a set B whose elements are exactly theelements of the elements of A :

∀A ∃B∀x [ x ∈ B ↔ ∃A ∈ A (x ∈ A) ].

For example:If A = {A1,A2, . . . ,An, . . . }, then

⋃A = A1 ∪ A2 ∪ · · · ∪ An ∪ · · · .⋃

{{a,b, c}, {c,d}, {c,e}} = {a,b, c,d ,e}⋃{a,b} = a ∪ b,

⋃{a,b, c} = a ∪ b ∪ c, . . .⋃

{a} = a,⋃∅ = ∅

If A = {Ai | i ∈ I}, where I is referred to as an index set, we alsowrite

⋃A =

⋃i∈I Ai .

If A ∈ A , then x ∈ A =⇒ x ∈⋃

A , thus A ⊆⋃

A .

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⋃℘({a,b}) =

⋃({∅, {a}, {b}, {a,b}})= ∅ ∪ {a} ∪ {b} ∪ {a,b} = {a,b}

Example 2.5: Show that for any set A,⋃℘(A) = A.

Proof. Since A ∈ ℘(A), we have that A ⊆⋃℘(A). Conversely, for any

x ∈⋃℘(A), by definition, there exists X ∈ ℘(A) such that x ∈ X . Since

x ∈ X ⊆ A, we obtain x ∈ A.

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Definition 2.10For any nonempty set A , the intersection of A , denoted by

⋂A , is

defined as⋂A = {x | x belongs to every element of A } = {x | ∀A ∈ A (x ∈ A)}.

If A = {A1,A2, . . . ,An, . . . }, then⋂

A = A1 ∩ A2 ∩ A3 ∩ · · · .

Since A 6= ∅, there exists C ∈ A , thus by Separation Axiom,⋂A = {x | ∀A ∈ A (x ∈ A)}= {x ∈ C | ∀A ∈ A (x ∈ A)}

is a set.

For example⋂{{a,b, c}, {c,d}, {c,e}} = {a,b, c} ∩ {c,d} ∩ {c,e}= {c}⋂{a,b} = a ∩ b,

⋂{a,b, c} = a ∩ b ∩ c, . . .⋂

{a} = a.⋂∅ is undefined or equal to some fixed set.

If A = {Ai | i ∈ I}, we also write⋂

A =⋂

i∈I Ai .

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Example 2.6:⋃⋂{{a}, {a,b}} =

⋃({a} ∩ {a,b}) =

⋃{a} = a.

⋂⋃{{a}, {a,b}} =

⋂({a} ∪ {a,b}) =

⋂{a,b} = a ∩ b.

Example 2.7: If ∅ 6= A ⊆ B, then⋂

B ⊆⋂

A

Proof. For any x ∈⋂

B, we show that x ∈⋂

A , which is equivalent toshowing that x ∈ X for all X ∈ A .

From X ∈ A ⊆ B, it follows that X ∈ B, thereby⋂

B ⊆ X . Hencex ∈

⋂B implies that x ∈ X , as required.

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Distributive laws

A ∩⋃

B =⋃{A ∩ X | X ∈ B}=

⋃X∈B

(A ∩ X )

A ∪⋂

B =⋂{A ∪ X | X ∈ B}=

⋂X∈B

(A ∪ X )

De Morgan’s laws (for A 6= ∅)⋃A =

⋂{X | X ∈ A }=

⋂X∈A

X

⋂A =

⋃{X | X ∈ A }=

⋃X∈A

X

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