 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 3-4 More on the Conditional

 2012 Pearson Education, Inc. Slide Converse, Inverse, and Contrapositive Alternative Forms of “If p, then q” Biconditionals Summary of Truth Tables More on the Conditional

 2012 Pearson Education, Inc. Slide Conditional Statement If p, then q ConverseIf q, then p InverseIf not p, then not q ContrapositiveIf not q, then not p Converse, Inverse, and Contrapositive

 2012 Pearson Education, Inc. Slide Given the conditional statement If I live in Wisconsin, then I shovel snow, determine each of the following: a) the converse b) the inverse c) the contrapositive Solution a) If I shovel snow, then I live in Wisconsin. b) If I don’t live in Wisconsin, then I don’t shovel snow. c) If I don’t shovel snow, then I don’t live in Wisconsin. Example: Determining Related Conditional Statements

 2012 Pearson Education, Inc. Slide A conditional statement and its contrapositive are equivalent, and the converse and inverse are equivalent. Equivalences

 2012 Pearson Education, Inc. Slide The conditional can be translated in any of the following ways. If p, then q. p is sufficient for q. If p, q.q is necessary for p. p implies q.All p are q. p only if q. q if p. Alternative Forms of “If p, then q”

 2012 Pearson Education, Inc. Slide Write each statement in the form “if p, then q.” a)You’ll be sorry if I go. b)Today is Sunday only if yesterday was Saturday. c)All Chemists wear lab coats. Solution a)If I go, then you’ll be sorry. b)If today is Sunday, then yesterday was Saturday. c)If you are a Chemist, then you wear a lab coat. Example: Rewording Conditional Statements

 2012 Pearson Education, Inc. Slide The compound statement p if and only if q (often abbreviated p iff q) is called a biconditional. It is symbolized, and is interpreted as the conjunction of the two conditionals Biconditionals

 2012 Pearson Education, Inc. Slide p if and only if q p q T TT T FF F TF F FT Truth Table for the Biconditional

 2012 Pearson Education, Inc. Slide Determine whether each biconditional statement is true or false. a)5 + 2 = 7 if and only if = 5. b)3 = 7 if and only if 4 = c)7 + 6 = 12 if and only if = 11. Solution a)True (both component statements are true) b)False (one component is true, one false) c)True (both component statements are false) Example: Determining Whether Biconditionals are True or False

 2012 Pearson Education, Inc. Slide The negation of a statement has truth value opposite of the statement. 2. The conjunction is true only when both statements are true. 3. The disjunction is false only when both statements are false. 4. The biconditional is true only when both statements have the same truth value. Summary of Truth Tables