AMSCO'S Geometry. New York - Rye High School

78 Logic(1) Eliminate the second row of truth values in whichthe conditional is false.(2) Eliminate the first row of truth values in which pis true. The second row in which p is true hasalready been eliminated.(3) Two rows remain, one in which q is true and theother in which q is false.p q p → qT T TT F FF T TF F TAnswer“I will go to the library” could be either true or false.EXAMPLE 4Draw a conclusion or conclusions base on the following true statements.If I am tired, then I will rest.I do not rest.SolutionA conditional and its contrapositive are logically equivalent. Therefore, “If Ido not rest, then I am not tired” is true.By the Law of Detachment, when the hypothesis of a true conditional is true,the conclusion must be true. Therefore, since “I do not rest” is true, “I am nottired” must be true.AlternativeSolutionLet p represent “I am tired.”Let q represent “I will rest.”Make a truth table for the disjunction p → q and eliminate the rows that donot apply.We know that p → q is true. We also know that since q is true, q is false.(1) Eliminate the second row of truth values in whichthe conditional is false.(2) Eliminate the first and third rows of truth valuesin which q is true.(3) Only one case remains: p is false.pTTFqTFTp → qTFTSince p is false, p is true. “I am not tired” is true.F F TAnswerI am not tired.