Thinking Mathematically Seventh Edition Chapter 3 Logic Copyright

Thinking Mathematically Seventh Edition Chapter 3 Logic Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 1

Section 3. 3 Truth Tables for Negation, Conjunction, and Disjunction Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 2

Objectives 1. Use the definitions of negation, conjunction, and disjunction. 2. Construct truth tables. 3. Determine the truth value of a compound statement for a specific case. Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 3

Truth Tables (1 of 3) Negation (not): Opposite truth value from the statement. Negation Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 4

Truth Tables (2 of 3) Conjunction (and): Only true when both statements are true. Conjunction Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 5

Truth Tables (3 of 3) Disjunction (or): Only false when both statements are false. Disjunction Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 6

Example 1: Using the Definitions of Negation, Conjunction, and Disjunction (1 of 2) Let p and q represent the following statements: p: 10 > 4 q: 3 < 5 Determine the truth value for each statement: a. Since both are true, the conjunction is true. b. Since p is true, is false, so the conjunction is false. Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 7

Example 1: Using the Definitions of Negation, Conjunction, and Disjunction (2 of 2) Let p and q represent the following statements: p: 10 > 4 q: 3 < 5 Determine the truth value for each statement: c. is false. A disjunction is false only when 10 > 4 is true or both components are false. Only one component is false, this is a true statement d. Since q is true, is false. So, the disjunction is false. Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 8

Example 2: Constructing Truth Tables (1 of 2) Construct a truth table for Step 1: First list the simple statements on top and show all the possible truth values. and fill in the truth values. Step 2: Make a column for Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 9

Example 2: Constructing Truth Tables (2 of 2) Step 3: Construct one more column for The final column tells us that the statement is false only when both p and q are true. For example: p: Harvard is a college. (true) q: Yale is a college. (true) It is not true that Harvard and Yale are colleges. A compound statement that is always true is called a tautology. Is this a tautology? No Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 10

Example 3: Constructing a Truth Table Construct a truth table for Copyright © 2019, 2015, 2011 Pearson Education, Inc. All Rights Reserved Slide - 11