Animated Solution for Mathematics - Sets and Relations: Consider Statement-1 : $(p\wedge \sim q)\wedge (\sim p\wedge q)$ is a fallacy. Statement-2 : $(p\to q)\leftrightarrow (\sim q\to \sim p)$ is a tautology.

Welcome, future IITians! Today, we are embarking on a journey into the bedrock of mathematics: Mathematical Logic. We are analyzing two statements: Statement 1 claims that the expression $(p\wedge \sim q)\wedge (\sim p\wedge q)$ is a fallacy. Statement 2 claims that $(p\to q)\leftrightarrow (\sim q\to \sim p)$ is a tautology.

Imagine you are standing before Statement 1. It looks intimidating, a tangle of conjunctions and negations. But let us look closer at the expression:

If we use the Associative and Commutative laws of logic, we can rearrange this expression. We can group the $p$ terms and the $q$ terms together:

Now, look at the beauty of this simplification. The Law of Contradiction tells us that $p\wedge \sim p$ is always False. It is impossible for a proposition to be both True and False simultaneously.

Therefore, we are left with $False\wedge False$, which is undeniably False. Statement 1 is a fallacy because it is a logical impossibility. It is a ghost in the machine that can never exist.

Now, let us turn our attention to Statement 2. We are looking at the biconditional:

Instead of drawing a massive truth table, let us use the power of algebraic equivalence. Recall the fundamental identity:

Let us apply this to the left-hand side: $p\to q$ becomes $\sim p\vee q$. Now, let us apply it to the right-hand side: $\sim q\to \sim p$. Applying our identity, we get:

Since the double negation of $q$ is simply $q$, this simplifies to $q\vee \sim p$. By the Commutative Law, $\sim p\vee q$ is identical to $q\vee \sim p$. Since both sides of the biconditional are logically equivalent, the statement is always True. It is a tautology, a fundamental truth of the universe.

We have proven that Statement 1 is a fallacy and Statement 2 is a tautology. Both are true. But here is the final hurdle: is Statement 2 the correct explanation for Statement 1?

This is where many students stumble. A 'correct explanation' requires a causal link. Statement 1 is a fallacy because of the Law of Contradiction. Statement 2 is a tautology because of the Law of Contraposition.

These are two separate, beautiful pillars of logic. One does not cause the other; they are independent truths. Therefore, while both statements are true, the second does not explain the first. You have navigated the trap, mastered the algebra, and understood the underlying philosophy.