Discrete Mathematics and Its Applications, Seventh Edition

Published by McGraw-Hill Education
ISBN 10: 0073383090
ISBN 13: 978-0-07338-309-5

Chapter 1 - Section 1.1 - Propositional Logic - Exercises: 27

Answer

a) Converse: I will ski tomorrow only if it snows today. Contrapositive: If I do not ski tomorrow, then it will not have snowed today. Inverse: If it does not snow today, then I will not ski tomorrow. ________________________________________________ b) Converse: If I come to class, then there will be a quiz. Contrapositive: If I do not come to class, then there will not be a quiz. Inverse: If there is not going to be a quiz, then I don’t come to class. ______________________________________________________ c) Converse: A positive integer is a prime if it has no divisors other than 1 and itself. Contrapositive: If a positive integer has a divisor other than 1 and itself, then it is not prime. Inverse: If a positive integer is not prime, then it has a divisor other than 1 and itself.

Work Step by Step

For each step, we first come up with the contrapositive, ¬q → ¬p, of a conditional statement p → q that has the same truth value as p → q. Knowing that the contrapositive is false when ¬p is false, and ¬q is true, that is, only when p is true, and q is false. We are currently sure that neither the converse, q → p, nor the inverse, ¬p → ¬q, has the same truth value as p → q for overall possible truth values of p and q. Understanding that when p is true, and q is false, the original conditional statement is false, but the converse and the inverse are both true. ################################# For (a) : Because “q whenever p” can be considered either way to show the conditional statement, we consider that the conditional statement, p → q, where p is “it snows today” and for q is “I will ski tomorrow”: The original statement can be rewritten as “If it snows today, I will ski tomorrow.” The contrapositive of this conditional statement is “If I do not ski tomorrow, then it will not have snowed today. The converse is “I will ski tomorrow only if it snows today.” The inverse is “If it does not snow today, then I will not ski tomorrow.” ################################################# For (b): Same as (a), Let p to be “There is going to be a quiz” and q to be “I come to class” The original statement can be rewritten either as “If there is going to be a quiz, then I come to class.” Or “I come to class whenever there is going to be a quiz.” For Converse, “If I come to class, then there will be a quiz.” For Contrapositive, “If I do not come to class, then there will not be a quiz.” For Inverse, “If there is not going to be a quiz, then I don’t come to class.” ############################ For (c): Same as (a) and (b), p as “A positive integer is a prime” and q as “Has no divisor other than 1 and itself.” The original statement can be rewritten either as “A positive integer is a prime only if it has no divisors other than 1 and itself.” For Converse, “A positive integer is a prime if it has no divisors other than 1 and itself.” For Contrapositive, “If a positive integer has a divisor other than 1 and itself, then it is not prime.” For Inverse, “If a positive integer is not prime, then it has a divisor other than 1 and itself.”
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