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.
\[\begin{align}
& \sim \left[ \left( \text{F}\to \sim \text{T} \right)\leftrightarrow \left( \text{F}\wedge \sim \text{T} \right) \right] \\
& \sim \left[ \left( \text{F}\to \text{F} \right)\leftrightarrow \left( \text{F}\wedge \text{F} \right) \right] \\
& \sim \left[ \left( \text{T} \right)\leftrightarrow \left( \text{F} \right) \right] \\
& \sim \left[ \text{F} \right] \\
\end{align}\].