Animated Solution for Mathematics - Sets and Relations: $(p\ast r)\iff (p\ast (\sim q))$ is equivalent to $(\sim p)$ when $r$ is

Welcome, future engineer! Today, we are not just solving a problem; we are stepping into the shoes of a logician. Logic is the bedrock of computer science and advanced mathematics.

When you look at an expression like $(p\ast r)\iff (p\ast \sim q)$, it might look like a jumble of symbols, but I want you to see it as a balance scale. The biconditional operator $\iff$ is the fulcrum. For the scale to be balanced, the left side must mirror the right side exactly.

The Power of Truth Values

Our target is to make the entire expression equivalent to $\sim p$. This is our 'North Star.' Whenever you face a logical equivalence problem, do not panic.

Instead, use the most powerful tool in your arsenal: the Case Method. We will test the two possible states of the universe for $p$: when $p$ is True ($T$) and when $p$ is False ($F$).

Let us start with the case where $p=F$. On the right-hand side (RHS), we have $\sim p$. Since $p$ is False, $\sim F$ becomes True.

Now, look at the left-hand side (LHS): $(F\ast r)\iff (F\ast \sim q)$. For the whole statement to be True (to match our RHS), both sides of the $\iff$ must be identical.

If we test the logical AND operator ($\wedge$), we know that $F\wedge r=F$ and $F\wedge \sim q=F$. Thus, $F\iff F$ is indeed True. We have found our operator! The mystery operator $\ast$ is the logical AND, $\wedge$.

The Moment of Truth

Now that we have identified the operator, we move to the second case: $p=T$. On the RHS, $\sim T$ gives us False. Our goal is to make the LHS equal to False.

Substituting our operator, the LHS becomes $(T\wedge r)\iff (T\wedge \sim q)$.

Here, we use the Identity Law of logic. Just as multiplying a number by 1 leaves it unchanged, applying the AND operator with True leaves the other variable unchanged. So, $T\wedge r$ simplifies to $r$, and $T\wedge \sim q$ simplifies to $\sim q$.

Our equation now stands elegantly as:

The Final Deduction

We are at the finish line. We have a biconditional statement that must result in False. Recall the soul of the biconditional: it only yields True if both inputs are identical.

If the result is False, the inputs must be opposites. Therefore, $r$ must be the logical negation of $\sim q$.

What is the negation of a negation? It is the original statement itself! The negation of $\sim q$ is simply $q$.

Look at what we have achieved. We took a complex, abstract logical structure and, by testing the boundaries of its truth values, stripped away the complexity to reveal the simple, elegant truth hidden underneath.

The final result is $r=q$. You didn't just find the answer; you navigated the logic of the universe. Keep this analytical mindset, and no problem in the JEE Advanced will ever be able to stand in your way.