Animated Solution for Mathematics - Sets and Relations: Let $p$ and $q$ be two statements. Then $\sim (p\wedge (p\Rightarrow \sim q))$ is equivalent to

We are tasked with simplifying the logical expression:

This expression may appear complex, but by applying fundamental logical laws systematically, we can dismantle it layer by layer.

We begin by addressing the outermost negation acting upon a conjunction. According to De Morgan's Law, the negation of a conjunction is the disjunction of the negations.

Applying this law, we obtain:

Next, we focus on the inner term $\sim (p\Rightarrow \sim q)$. We utilize the logical identity for implication, which states that $A\Rightarrow B\equiv \sim A\vee B$.

Applying this to the term $p\Rightarrow \sim q$, we get:

Substituting this back into our main expression, we have:

We apply De Morgan's Law once more to the second part of the expression. The negation of $\sim p$ becomes $p$, the disjunction $\vee$ flips to a conjunction $\wedge$, and the negation of $\sim q$ becomes $q$.

The expression now reads:

To simplify further, we apply the Distributive Law, treating the logical operators similarly to algebraic expansion:

The term $(\sim p\vee p)$ represents the Law of Excluded Middle, which is a tautology ($T$). Our expression simplifies to:

By the Identity Law, $T\wedge X\equiv X$. Therefore, the final simplified result is: