Answer
The truth value of the provided compound statement with the provided condition is false.
Work Step by Step
Substitute the truth values for simple statements\[p,q,r\] to determine the truth value for the given compound statement\[\sim p\leftrightarrow \left( \sim q\wedge r \right)\].
\[\left( \sim q\wedge r \right)\]is a conjunction statement, which is true only when both of the variables \[\sim q\]and \[r\] are true. The conjunction\[\left( \text{F}\wedge \text{F} \right)\] results in\[\text{F}\].
The given compound statement is biconditional statement whose ingredient variables are \[\sim p\]with\[\left( \sim q\wedge r \right)\]. This is true only when they both have same truth values, which is either false or either true.
Replace the simple statements present in a compound statement with the truth values of it.
\[\begin{align}
& \sim \text{F}\leftrightarrow \left( \sim \text{T}\wedge \text{F} \right) \\
& \text{T}\leftrightarrow \left( \text{F}\wedge \text{F} \right) \\
& \text{T}\leftrightarrow \text{F} \\
& \text{F} \\
\end{align}\]
