SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof

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


Outline

1 Propositional Logic

2 Propositional Equivalences

3 Proof


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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.


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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


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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”


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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


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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


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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


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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)


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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


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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


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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.


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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


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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.


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Quantifiers

A quantifier is ”an operator that limits the variables of aproposition”

Two types:

Universal: ∀Existential: ∃


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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)


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Proof methods

Proof methods

We will discuss three proof methods for prove:

”If P then Q”:

Direct proof

Proof by Contraposition

Proof by Contradiction


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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.


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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.


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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


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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)


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Documents

SWER ENG 2DM3 Tutorial 1 - McMaster Universityoptlab.mcmaster.ca/~jessiel/SWFR2DMTUT/TUT1_online.pdf · 2011. 9. 15. · Outline Propositional Logic Propositional Equivalences Proof