Thinking Mathematically (6th Edition)

Published by Pearson
ISBN 10: 0321867327
ISBN 13: 978-0-32186-732-2

Chapter 3 - Logic - 3.6 Negations of Conditional Statements and De Morgan's Law - Exercise Set 3.6 - Page 179: 53


The negation of the statement \[p\wedge \left( r\to \sim s \right)\] using De Morgan’s law is given as \[\sim p\vee \sim \left( r\to \sim s \right)\]. To form the negation of a conditional statement, leave the antecedent (the first part unchanged, change the \[\text{if-then}\] connective to and) and negate the consequent (the second part). Therefore, \[\sim \left( r\to \sim s \right)\equiv r\wedge \sim \left( \sim s \right)\]. As\[\sim \left( \sim s \right)=s\], therefore, \[r\wedge \sim \left( \sim s \right)\equiv r\wedge s\]. Thus, the negation of \[p\wedge \left( r\to \sim s \right)\] is \[\sim p\vee \left( r\wedge s \right)\].
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