Animated Solution for Mathematics - Sets and Relations: The expression $\sim (p\to q)$ is logically equivalent to :

Welcome, aspiring engineers! Today, we are embarking on a journey into the bedrock of mathematics: Logical Reasoning. Often, students view logic as a dry set of rules, but I want you to see it as the very language of clarity.

We are tackling the logical equivalence of the negation of an implication, specifically $\sim (p\to q)$. This isn't just about truth tables; it's about understanding the anatomy of a statement.

Imagine you make a promise: 'If I study hard ($p$), then I will clear the JEE ($q$).' In logic, this is represented as $p\to q$.

When have you broken this promise? Only in one scenario: you studied hard ($p$ is True), but you didn't clear the exam ($q$ is False).

In every other case—whether you studied and cleared it, or you didn't study and didn't clear it, or you didn't study but cleared it anyway—the promise remains intact. This is why $p\to q$ is only False when $p$ is True and $q$ is False.

Now, we want to find the negation, $\sim (p\to q)$. If the original statement is a promise that is broken only when $p$ is True and $q$ is False, then the negation of that promise must be the exact moment it is broken.

It must be True only when $p$ is True and $q$ is False. This leads us to the expression $p\wedge \sim q$.

Let's look at the truth table. When we evaluate $p\to q$, we get $T,F,T,T$ for the four possible combinations of $p$ and $q$. When we apply the negation $\sim$, we flip these to $F,T,F,F$.

Now, look at $p\wedge \sim q$. For this to be True, both $p$ must be True AND $\sim q$ must be True (which means $q$ must be False). This happens only in the second row, giving us $F,T,F,F$. The columns are identical!

The beauty here is that we have proven the following logical equivalence:

We have proven this without just blindly following a table. We used the physical intuition of a broken promise.

This is the essence of JEE-level thinking: connecting abstract symbols to real-world scenarios. If you ever encounter a question where the official answer key seems off—as can happen in competitive exams—trust your derivation.

You have the tools to verify the truth, and that is the most powerful asset you possess. Keep practicing, keep questioning, and most importantly, keep falling in love with the logic behind the math.