1 Introduction to Logic © 2008 Pearson Addison-Wesley.All rights reserved

2 Chapter 3: Introduction to Logic3.1 Statements and Quantifiers 3.2 Truth Tables and Equivalent Statements 3.3 The Conditional and Circuits 3.4 More on the Conditional 3.5 Analyzing Arguments with Euler Diagrams 3.6 Analyzing Arguments with Truth Tables © 2008 Pearson Addison-Wesley. All rights reserved

3 Truth Tables and Equivalent StatementsChapter 1 Section 3-2 Truth Tables and Equivalent Statements © 2008 Pearson Addison-Wesley. All rights reserved

4 Truth Tables and Equivalent StatementsConjunctions 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

5 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 are given in the truth table on the next slide. The connective and implies “both.” © 2008 Pearson Addison-Wesley. All rights reserved

6 Conjunction Truth Tablep and q p q T T T T F F F T F F © 2008 Pearson Addison-Wesley. All rights reserved

7 Example: Finding the Truth Value of a ConjunctionLet p represent the statement 4 > 1, q represent the statement 12 < 9 find the truth of Solution False, since q is false. © 2008 Pearson Addison-Wesley. All rights reserved

8 Disjunctions The truth values of the disjunction p or q, symbolized are given in the truth table on the next slide. The connective or implies “either.” © 2008 Pearson Addison-Wesley. All rights reserved

9 Disjunctions p q p or q T T T T F F T F F F© 2008 Pearson Addison-Wesley. All rights reserved

10 Example: Finding the Truth Value of a DisjunctionLet p represent the statement 4 > 1, q represent the statement 12 < 9 find the truth of Solution True, since p is true. © 2008 Pearson Addison-Wesley. All rights reserved

11 Negation The truth values of the negation of p, symbolized are given in the truth table below. not p p T F © 2008 Pearson Addison-Wesley. All rights reserved

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

13 Truth Tables Use the following standard format for listing the possible truth values in compound statements involving two component statements. p q Compound Statement T T T F F T F F © 2008 Pearson Addison-Wesley. All rights reserved

14 Example: Constructing a Truth TableConstruct the truth table for Solution p q ~ p ~ q T T F T F T F T F F © 2008 Pearson Addison-Wesley. All rights reserved

15 Number of Rows in a Truth TableA logical statement having n component statements will have 2n rows in its truth table. © 2008 Pearson Addison-Wesley. All rights reserved

16 Alternative Method for Constructing Truth TablesAfter making several truth tables, some people prefer a shortcut method where not every step is written out. © 2008 Pearson Addison-Wesley. All rights reserved

17 Equivalent StatementsTwo statements are equivalent if they have the same truth value in every possible situation. © 2008 Pearson Addison-Wesley. All rights reserved

18 Example: Equivalent StatementsAre the following statements equivalent? Solution Yes, see the tables below. p q T T F T F F T F F T © 2008 Pearson Addison-Wesley. All rights reserved

19 De Morgan’s Laws For any statements p and q,© 2008 Pearson Addison-Wesley. All rights reserved

20 Example: Applying De Morgan’s LawsFind a negation of each statement by applying De Morgan’s Law. a) I made an A or I made a B. b) She won’t try and he will succeed. Solution a) I didn’t make an A and I didn’t make a B. b) She will try or he won’t succeed. © 2008 Pearson Addison-Wesley. All rights reserved