 2012 Pearson Education, Inc. Slide Chapter 3 Introduction to Logic

 2012 Pearson Education, Inc. Slide Chapter 3: Introduction to Logic 3.1 Statements and Quantifiers 3.2 Truth Tables and Equivalent Statements 3.3 The Conditional and Circuits 3.4 More on the Conditional 3.5Analyzing Arguments with Euler Diagrams 3.6Analyzing Arguments with Truth Tables

 2012 Pearson Education, Inc. Slide Section 3-2 Truth Tables and Equivalent Statements

 2012 Pearson Education, Inc. Slide Conjunctions Disjunctions Negations Mathematical Statements Truth Tables Alternative Method for Constructing Truth Tables Equivalent Statements and De Morgan’s Laws Truth Tables and Equivalent Statements

 2012 Pearson Education, Inc. Slide 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.” The truth values of component statements are used to find the truth values of compound statements. Conjunctions

 2012 Pearson Education, Inc. Slide p q T TT T FF F TF F FF p and q Conjunction Truth Table

 2012 Pearson Education, Inc. Slide Let p represent the statement 4 > 1, q represent the statement 12 < 9 find the truth of Solution False, since q is false. Example: Finding the Truth Value of a Conjunction

 2012 Pearson Education, Inc. Slide 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.” Disjunctions

 2012 Pearson Education, Inc. Slide p q T TT T FT F TT F FF p or q Disjunctions

 2012 Pearson Education, Inc. Slide Let p represent the statement 4 > 1, q represent the statement 12 < 9 find the truth of Solution True, since p is true. Example: Finding the Truth Value of a Disjunction

 2012 Pearson Education, Inc. Slide The truth values of the negation of p, symbolized are given in the truth table below. p TF FT not p Negation

 2012 Pearson Education, Inc. Slide 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. Solution a) False, since ~ p is false. b) True Example: Mathematical Statements

 2012 Pearson Education, Inc. Slide 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 Truth Tables

 2012 Pearson Education, Inc. Slide p q ~ p~ q T TFFFF T FFTTT F TTFTF F FTTTF Construct the truth table for Solution Example: Constructing a Truth Table

 2012 Pearson Education, Inc. Slide A logical statement having n component statements will have 2 n rows in its truth table. Number of Rows in a Truth Table

 2012 Pearson Education, Inc. Slide After making several truth tables, some people prefer a shortcut method where not every step is written out. Alternative Method for Constructing Truth Tables

 2012 Pearson Education, Inc. Slide Two statements are equivalent if they have the same truth value in every possible situation. Equivalent Statements

 2012 Pearson Education, Inc. Slide Are the following statements equivalent? p q T TFF T FFF F TFF F FTT Solution Yes, see the tables below. Example: Equivalent Statements

 2012 Pearson Education, Inc. Slide For any statements p and q, De Morgan’s Laws

 2012 Pearson Education, Inc. Slide Find 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. Example: Applying De Morgan’s Laws