9/10/2010

Chapter 3
Introduction
to Logic

© 2008 Pearson Addison-Wesley.
All rights reserved

Chapter 3: Introduction to Logic
3.1
3.2
3.3
3.4
3.5
3.6

Statements and Quantifiers
Truth Tables and Equivalent Statements
The Conditional and Circuits
More on the Conditional
Analyzing Arguments with Euler Diagrams
Analyzing Arguments with Truth Tables

© 2008 Pearson Addison-Wesley. All rights reserved

3-2-2

Chapter 1

Section 3-2
Truth Tables and Equivalent
Statements
© 2008 Pearson Addison-Wesley. All rights reserved

3-2-3

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9/10/2010

Truth Tables and Equivalent
Statements
•
•
•
•
•
•

Conjunctions
Disjunctions
Negations
Mathematical Statements
Truth Tables
Alternative Method for Constructing Truth
Tables
• Equivalent Statements and De Morgan’s Laws
© 2008 Pearson Addison-Wesley. All rights reserved

3-2-4

Conjunctions
The truth values of component statements are used to find the truth values of compound statements.
The truth values of the conjunction p and q, symbolized p ∧ q are given in the truth table on the next slide. The connective and implies “both.”
© 2008 Pearson Addison-Wesley. All rights reserved

3-2-5

Conjunction Truth Table p and q q p∧q

T

T

T

F

F

F

T

F

F

F

F

p
T

© 2008 Pearson Addison-Wesley. All rights reserved

3-2-6

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Example: Finding the Truth Value of a Conjunction
Let p represent the statement 4 > 1, q represent the statement 12 < 9 find the truth of p ∧ q.

Solution
False, since q is false.

© 2008 Pearson Addison-Wesley. All rights reserved

3-2-7

Disjunctions
The truth values of the disjunction p or q, symbolized p ∨ q are given in the truth table on the next slide. The connective or implies “either.”

© 2008 Pearson Addison-Wesley. All rights reserved

3-2-8

Disjunctions p or q q p∨q

T

T

T

T

F

T

F

T

T

F

F

F

p

© 2008 Pearson Addison-Wesley. All rights reserved

3-2-9

3

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Example: Finding the Truth Value of a Disjunction
Let p represent the statement 4 > 1, q represent the statement 12 < 9 find the truth of p ∨ q.

Solution
True, since p is true.

© 2008 Pearson Addison-Wesley. All rights reserved

3-2-10

Negation
The truth values of the negation of p, symbolized ∼ p , are given in the truth table below. not p p ∼ p
T

F

F

T

© 2008 Pearson Addison-Wesley. All rights reserved

3-2-11

Example: Mathematical Statements
Let p represent the statement 4 > 1, q represent the statement 12 < 9, and r represent 0 < 1.
Decide whether each statement is true or false.
a) ∼ p ∧ ∼ q

b)

(∼ p ∧ r ) ∨ (∼ q ∧ p)

Solution
a) False, since ~ p is false.
b) True
© 2008 Pearson Addison-Wesley. All rights reserved

3-2-12

4

9/10/2010

Truth Tables
Use the following standard format for listing the possible truth values in compound statements involving two component statements. p Compound Statement

q

T

T

T

F

F

T

F

F
3-2-13

© 2008 Pearson Addison-Wesley. All rights reserved

Example: Constructing a Truth Table