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