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: 3

Answer

a. Met does not have an MP3 player b. There is pollution in New Jersey c. 2+1 ≠ 3 d. "The summer in Maine is not both hot and sunny" or “The summer in Maine is not hot or not sunny”

Work Step by Step

The negation of a proposition can be obtained by prefixing the proposition with "It is not the case that". As a further, optional step, the resulting sentence can be reconstructed to a more natural form to make it simpler or it's effective meaning clearer. a. “It is not the case that Met does not have an MP3 player” or the less verbose “Met does not have an MP3 player” b. “It is not the case that there is no pollution in New Jersey” or “There is pollution in New Jersey” c. “It is not the case that 2 + 1 = 3” or “2 + 1 ≠ 3” d. “It is not the case that the summer in Maine is hot and sunny”. To express it in simple English, it should be noted that the proposition “The summer in Maine is hot and sunny” is a compound proposition which is a conjunction of two propositions. The negation of the propositional statement “p AND q” i.e., “NOT (p AND q)” is logically equivalent to “NOT p OR NOT q”. Therefore, the negated proposition would be reconstructed in simple English as “The summer in Maine is not hot or not sunny” or “The summer in Maine is not both hot and sunny”.
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