Animated Solution for Mathematics - Sets and Relations: Negation of the Boolean statement $(p\vee q)\Rightarrow ((\sim r)\vee p)$ is equivalent to:

We are tasked with simplifying the logical statement $S\equiv (p\vee q)\Rightarrow ((\sim r)\vee p)$ and subsequently finding its negation.

The implication operator $\Rightarrow$ is often difficult to manipulate directly. We utilize the fundamental identity:

By substituting $X=(p\vee q)$ and $Y=((\sim r)\vee p)$, we transform the statement into:

Next, we address the term $\sim (p\vee q)$. According to De Morgan's Law, the negation of a disjunction is the conjunction of the negations:

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

To simplify further, we use the Associative Law to group the terms containing $p$:

We now apply the Distributive Law to the term $((\sim p)\wedge (\sim q))\vee p$:

Since $(\sim p)\vee p$ is a Tautology ($T$), and $T\wedge X\equiv X$, the expression simplifies significantly:

The problem requires the negation of $S$, denoted as $\sim S$. We apply the negation operator to our simplified expression:

Applying De Morgan's Law across the three terms, we flip the OR operators to AND operators:

Using the Double Negation Law, where $\sim (\sim X)\equiv X$, we arrive at the final result:

This result corresponds to the logical structure presented in Option (C). By methodically deconstructing the operators, we have reduced a complex implication to its simplest conjunctive form.