Animated Solution for Mathematics - Sets and Relations: If the truth value of the statement $P\to (\sim p\vee r)$ is false(F), then the truth values of the statements p, q, r are respectively :

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a problem; we are peeling back the layers of a logical structure. Logic is the bedrock of all mathematics, and understanding it is like learning the secret language of the universe.

Let us dive into the statement $p\to (\sim q\vee r)\equiv F$.

Imagine you are standing before a grand gate. The statement $p\to (\sim q\vee r)$ is a logical gate, and we are told it is False.

In the world of propositional logic, an implication $A\to B$ is a promise. It says, "If $A$ is true, then $B$ must be true." The only way this promise is shattered—the only way the entire statement becomes False—is if the premise $A$ is True, but the conclusion $B$ is False.

In our specific case, $A$ is $p$ and $B$ is $(\sim q\vee r)$. Therefore, for the entire expression to be False, we must have:

This is our first major breakthrough. We have already determined the truth value of $p$! It must be True.

Now, let us turn our attention to the second part: $(\sim q\vee r)\equiv F$. This is a disjunction, an "OR" statement.

Think of an OR statement as a team effort. For the team to fail (be False), every single member must fail. If even one member succeeds, the team succeeds.

Thus, for the disjunction $(\sim q\vee r)$ to be False, both components must be False:

Look at the elegance of this! We have immediately found the value of $r$. It is False. We are now left with the final piece of our puzzle: $\sim q\equiv F$.

We are at the finish line. We have $\sim q\equiv F$. This is the negation of $q$.

If the negation of $q$ is False, then $q$ itself must be the opposite. It must be True. Let us collect our findings like precious gems:

When we put them together, we get the sequence $T,T,F$. It is a beautiful, clean result.

You see, logic isn't about memorizing tables; it's about visualizing the flow of truth. You have successfully navigated the implication, dismantled the disjunction, and arrived at the truth.