AMSCO'S Geometry. New York - Rye High School

64 LogicOn Friday, p is true and q is true. Therefore, both (p → q) and (q → p)are true.On any other day of the week, p is false and q is false. Therefore, both(p → q) and (q → p) are true.A conditional and its converse may have the same truth value.These three examples allow us to make the following conclusion: A conditional (p → q) and its converse (q → p) may or may not have thesame truth value.This conclusion can be shown inthe truth table. Note that the conditional(p → q) and its converse(q → p) have the same truth valuewhen p and q have the same truthvalue. The conditional (p → q) andits converse (q → p) have differenttruth values when p and q have differenttruth values.Conditional Conversep q p → q q → pT T T TT F F TF T T FF F T TThe ContrapositiveWe form the inverse of a conditional by negating both the hypothesis and theconclusion. We form the converse of a conditional by interchanging the hypothesisand the conclusion. We form the contrapositive of a conditional by doingboth of these operations: we negate and interchange the hypothesis and conclusion.In symbols, the contrapositive of (p → q) is (q → p).1. A true conditional can have a true contrapositive.Let p represent “Gary arrives late to class.”Let q represent “Gary is marked tardy.”Conditional If Gary arrives late to class, then Gary is marked tardy.(p → q):gContrapositive If Gary is not marked tardy,(q → p):piqthen Gary does not arrive late to class.If Gary arrives late to class, p is true and q is true. Therefore, (p → q) is true.Also, if p and q are true, p is false and q is false, so (q → p) is true.If p is false, that is, if Gary does not arrive late to class, q is false and(p → q) is true. Similarly, if p and q are false, p is true and q is true, so(q → p) is true.ipfq