Animated Solution for Mathematics - Sets and Relations: The negation of the Boolean expression $((\sim q)\wedge p)\Rightarrow ((\sim p)\vee q)$ is logically equivalent to

The problem asks us to find the negation of the Boolean expression:

At first glance, this expression appears complex, but we can simplify it systematically by applying the fundamental laws of Boolean algebra.

The primary challenge is the implication operator, $\Rightarrow$. We utilize the standard logical equivalence:

By treating the left side as $X$ and the right side as $Y$, we rewrite the expression $S$ as:

Next, we focus on the first term: $\sim ((\sim q)\wedge p)$. We apply De Morgan's Law, which states that $\sim (A\wedge B)\equiv \sim A\vee \sim B$.

Distributing the negation, the conjunction $\wedge$ becomes a disjunction $\vee$, and the internal terms are negated:

Substituting this back into our expression for $S$, we obtain:

Observe that both components of the disjunction are identical due to the Commutative Law of the OR operator. We are effectively evaluating:

Applying the Idempotent Law, which states that $A\vee A\equiv A$, the expression collapses into its simplest form:

The problem specifically requests the negation of the original expression, which is $\sim S$. Given our simplified result $S\equiv \sim p\vee q$, we calculate:

Applying De Morgan's Law again:

Alternatively, recognizing that $\sim p\vee q$ is equivalent to the implication $p\Rightarrow q$, the negation is simply $\sim (p\Rightarrow q)$. By following these logical steps, we have successfully navigated the complexity to reach the final simplified negation.