Thinking Mathematically (6th Edition)

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

Chapter 3 - Logic - 3.4 Truth Tables for the Conditional and the Biconditional - Exercise Set 3.4 - Page 160: 73

Answer

The truth value for the provided compound statement with the provided condition is true.

Work Step by Step

Substitute the truth values for simple statements\[p,q,r\] to determine the truth value for the given compound statement\[\sim \left[ \left( p\to \sim r \right)\leftrightarrow \left( r\wedge \sim p \right) \right]\]. \[\left( p\to \sim r \right)\]is a conditional statement which is false only when the antecedent is true and the consequent is false, at rest all other cases, it is true. The conditional statement \[\left( p\to \sim r \right)\]on the substitution of truth value results in\[\left( \text{F}\to \text{T} \right)\], which can be rewritten as T.\[\left( r\wedge \sim p \right)\]is a conjunction statement, which is true only when both of the variables \[r\]and \[\sim p\] are true. The conjunction\[\left( \text{F}\wedge \text{T} \right)\] results in\[\text{F}\]. The given compound statement is a negation of \[\left[ \left( p\to \sim r \right)\leftrightarrow \left( r\wedge \sim p \right) \right]\]whereit is a biconditional statement whose ingredient variables are \[\left( p\to \sim r \right)\]with\[\left( r\wedge \sim p \right)\]. This is true only when they both have same truth values, which is either false or either true. Replace the truth values of the simple statement. \[\begin{align} & \sim \left[ \left( \text{F}\to \sim \text{T} \right)\leftrightarrow \left( \text{F}\wedge \sim \text{T} \right) \right] \\ & \sim \left[ \left( \text{F}\to \text{F} \right)\leftrightarrow \left( \text{F}\wedge \text{F} \right) \right] \\ & \sim \left[ \left( \text{T} \right)\leftrightarrow \left( \text{F} \right) \right] \\ & \sim \left[ \text{F} \right] \\ \end{align}\].
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