JEE 2022 — Mathematics PYQ

To find the negation of the given statement, we will use the logical identity for implication and De Morgan's Laws.

Step 1: Use the Implication Identity

The implication $A\Rightarrow B$ is logically equivalent to $(\sim A)\vee B$.

Let $A=(p\vee q)$ and $B=((\sim r)\vee p)$.

The expression becomes:

Step 2: Apply Negation to the Statement

The question asks for the negation of the entire statement:

Since the negation of an implication $\sim (A\Rightarrow B)$ is equivalent to $A\wedge (\sim B)$, we can write:

Step 3: Simplify using De Morgan's Law

Now, apply the negation to the second bracket $\sim ((\sim r)\vee p)$:

• The negation of $\sim r$ is $r$.

• The negation of $\vee$ is $\wedge$.

• The negation of $p$ is $\sim p$.

So, $\sim ((\sim r)\vee p)\equiv (r\wedge \sim p)$.

Substituting this back into the expression:

Step 4: Use Distributive Law

Rearrange and distribute the terms:

Since $(p\wedge \sim p)$ is always False (F):

Step 5: Final Formatting

By rearranging the terms to match the options:

Correct Option: C