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Outline Propositional Logic Propositional Equivalences Proof
SWER ENG 2DM3 Tutorial 1
Min Jing Liu
Department of Computing and SoftwareMcMaster University
Sept 15, 2011
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3 Tutorial 1
![Page 2: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/2.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Outline
1 Propositional Logic
2 Propositional Equivalences
3 Proof
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 3: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/3.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Propositional Logic
Propositional Logic
Example for proposition:
5 is odd.
3 + 4 = 8
A proposition is a declarative sentence is eithertrue or false, but not both.
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 4: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/4.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Propositional Logic
Propositional Logic
Question:
1. x2 + 1 = 3
2. The moon is made of green cheese. (Page 12,Ex 2 f)
3. Answer this question. (Page 12,Ex 1 f)
1 and 3 is not a proposition, 2 is false proposition.
We use letter to denote propositional variable: p, q, r,...The truth value of a proposition is true: TThe truth value of a proposition is false: F
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 5: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/5.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Propositional Logic
Negation
Let p be a proposition: The negation of p denotedby ¬p(also denoted by p̄)
”It is not the case that p”
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 6: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/6.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Propositional Logic
Negation
Example: p: Mei have an Iphone 5.¬p: Mei does not have an Iphone 5.
Truth Table for negation:
p ¬pT FF T
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 7: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/7.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Propositional Logic
And, Or
Let p, q be proposition:
The conjunction of p and q: p ∧ q
The disjunction of p and q: p ∨ q
Truth Table:
p q p ∧ q p ∨ q
T T T TT F F TF T F TF F F F
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 8: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/8.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Propositional Logic
Conditional Statements
Let p, q be proposition:The conditional statement p → q is the proposition
”if p, then q.”
Truth Table:
p q p → q
T T TT F FF T TF F T
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 9: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/9.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Propositional Logic
Biconditional
Let p, q be proposition:The Biconditional statement p ↔ q is the proposition
”p if and only if q.”
Truth Table:
p q p ↔ q
T T TT F FF T FF F T
Remark: p ↔ q has exactly the same truth value as (p → q) ∧ (q → p)
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 10: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/10.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Exercise
Example for Logic
Write the truth table of the proposition:
1. (p ∨ q) ∧ ¬pSolution:
p q p ∨ q ¬p (p ∨ q) ∧ ¬pT T T F FT F T F FF T T T TF F F T F
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 11: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/11.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Exercise
Example for Logic
Write the truth table of the proposition:
2. [¬(p ∧ q)]→ (q ∧ ¬p)
Solution:
p q p ∧ q ¬(p ∧ q) ¬p q ∧ ¬p [¬(p ∧ q)]→ (q ∧ ¬p)
T T T F F F TT F F T F F FF T F T T T TF F F T T F F
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 12: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/12.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Logical Equivalences
Tautology
A compound proposition that is always true, no matter what thetruth value of the propositional variables that occurs in it, is calleda tautology.E.g (p → q) ∨ (p ∧ ¬q)
p q p → q ¬q p ∧ ¬q (p → q) ∨ (p ∧ ¬q)
T T T F F TT F F T T TF T T F F TF F T T F T
This is a tautology.
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 13: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/13.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Logical Equivalences
Logically Equivalent
The compound propositions p and q are called logicallyequivalent if p ↔ q is tautology.
The notation p ≡ q denotes that p and q are logicallyequivalent.
How to determine whether two compound propositions areequivalent:Truth Table
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 14: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/14.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Predicates and Quantifiers
Predicates
Statement involving variables, such as x > 3, x2 = 4, ...
Example:P(x , y) denote the statement ”x + y > 5”. What are the truthvalue of P(2, 4), P(1, 3)?
solution:P(2, 4) is True, P(1, 3) is False.
P denote the predicate ”the sum is greater than 5” and x,y are the
variable.
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 15: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/15.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Predicates and Quantifiers
Quantifiers
A quantifier is ”an operator that limits the variables of aproposition”
Two types:
Universal: ∀Existential: ∃
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 16: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/16.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Predicates and Quantifiers
Exercise
Page 67, Ex 27 (a, c, e, g, i)
a) ∀n∃m(n2 < m)
c) ∀n∃m(n + m = 0)
e) ∃n∃m(n2 + m2 = 5)
g) ∃n∃m(n + m = 4 ∧ n −m = 1)
i) ∀n∀m∃p(p = (m + n)/2)
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 17: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/17.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Proof methods
Proof methods
We will discuss three proof methods for prove:
”If P then Q”:
Direct proof
Proof by Contraposition
Proof by Contradiction
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 18: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/18.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Proof methods
”If P then Q” - Direct Proof
”If P then Q”:
Assume P is true, and show Q must therefor be true.
Example
Proof ”if n is even, then n2 is even”.
solution
Assume n is even:
Thus, n = 2k , for some k (definition of even numbers)
n2 = (2k)2 = 4k2 = 2(2k2)
As n2 is 2 times an integer, n2 is thus even.
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 19: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/19.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Proof methods
”If P then Q” - by Contraposition
”If P then Q”:
Assume Q is false; prove P is false.
Example
Proof ”If n is an integer and n3 + 5 is odd, then n is even”.
Solution
We must prove the contrapositive: If n is odd, then n3 + 5 is even.Assume n is odd:
Thus, n = 2k + 1, for some k (definition of odd numbers)
n3 +5 = (2k +1)3 +5 = 8k3 +12k2 +6k +6 = 2(4k3 +6k2 +3k +2)
n3 + 5 is thus even.
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 20: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/20.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Proof methods
”If P then Q” - by Contradiction
”If P then Q”:Showing that the proposition’s being false would imply acontradiction
Example
Proof ”If n is an integer and n3 + 5 is odd, then n is even”.
Solution
Suppose that n3 + 5 is odd and that n is odd.
Since n is odd, the product of odd number is odd. Thus n3 is odd
5 is odd, the sum of odd number is even. Thus n3 + 5 is even
This is not true. Therefore our supposition was wrong
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3
![Page 21: SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof](https://reader034.vdocuments.us/reader034/viewer/2022051808/600bd7a3e283eb3aac73b6b6/html5/thumbnails/21.jpg)
Outline Propositional Logic Propositional Equivalences Proof
Exercise
Exercise
Every odd integer is the difference of two squares (Directproof) (pg 91, ex 7)
Prove that if m and n are integers and mn is even, then m iseven or n is even.(contradiction) (pg 91, ex 16)
If x + y ≥ 2, where x and y are real numbers, then x ≥ 1 ory ≥ 1 (contraposition) (pg 91, ex 15)
Min Jing Liu Department of Computing and Software McMaster University
SWER ENG 2DM3