Animated Solution for Mathematics - Sets and Relations: The statement $(\sim (p\iff \sim q))\wedge q$ is :

We are tasked with simplifying the logical expression:

First, we address the inner expression $\sim (p\iff \sim q)$. We utilize the identity for the negation of a biconditional, which states that $\sim (p\iff r)$ is equivalent to $p\iff \sim r$.

Applying this identity, the term becomes $p\iff \sim (\sim q)$. Since double negation is equivalent to affirmation, this simplifies to:

Substituting this back into our original expression, we obtain:

We expand the biconditional into its constituent implications:

Using the Associative Law, we regroup the terms to isolate the $q$ variables:

Focusing on the inner bracket $(q\Rightarrow p)\wedge q$, we apply the implication identity $q\Rightarrow p\equiv \sim q\vee p$:

Distributing $q$ across the disjunction, we get:

Since $\sim q\wedge q$ is a contradiction (False), the expression simplifies to $F\vee (p\wedge q)$, which is simply:

Our full expression is now $(p\Rightarrow q)\wedge (p\wedge q)$. If $p\wedge q$ is true, then both $p$ and $q$ are true, which makes $p\Rightarrow q$ true as well. Thus, the intersection is simply $p\wedge q$.

To verify the result against the provided options, we test the expression $(p\Rightarrow q)\wedge p$:

This simplifies to $F\vee (q\wedge p)$, which is $p\wedge q$. Therefore, the expression is equivalent to $(p\Rightarrow q)\wedge p$.