Answer
The truth value for 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\[\sim \left[ \left( \sim p\to r \right)\leftrightarrow \left( p\vee \sim q \right) \right]\].
\[\left( \sim p\to r \right)\]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.On solving the conditional\[\left( \text{T}\to \text{F} \right)\], we get F as truth values.
\[\left( p\vee \sim q \right)\]is a disjunction statement, which is false only when both of the variables \[p\]and \[\sim q\] are false. The disjunction\[\left( \text{F}\vee \text{F} \right)\] results in F.
The given compound statement is a negation of \[\left[ \left( \sim p\to r \right)\leftrightarrow \left( p\vee \sim q \right) \right]\] where \[\left[ \left( \sim p\to r \right)\leftrightarrow \left( p\vee \sim q \right) \right]\]is a biconditional statement whose ingredient variables are \[\left( \sim p\to r \right)\]with\[\left( p\vee \sim q \right)\]. This is true only when they both have same truth values, which is either false or true.
Replacing, with the truth values of the simple statement
\[\begin{align}
& \sim \left[ \left( \sim \text{F}\to \text{F} \right)\leftrightarrow \left( \text{F}\vee \sim \text{T} \right) \right] \\
& \sim \left[ \left( \text{T}\to \text{F} \right)\leftrightarrow \left( \text{F}\vee \text{F} \right) \right] \\
& \sim \left[ \left( \text{F}\leftrightarrow \text{F} \right) \right] \\
& \sim \left[ \text{T} \right] \\
\end{align}\]
