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

Welcome, future engineer. Today, we are not just solving a problem; we are peeling back the layers of a logical structure to reveal its core. In the world of JEE Advanced, logic is the bedrock of computer science and complex reasoning.

The expression we are tackling today is $\sim (p⟹(\sim q))$. At first glance, it looks like a tangle of symbols, but I want you to see it as a puzzle waiting to be simplified.

Many students see the implication symbol $⟹$ and feel stuck. Remember this: the implication $A⟹B$ is not a primitive operation. It is a shorthand for the statement $\sim A\vee B$.

Think of it as saying, 'If it rains, I will carry an umbrella.' This is logically equivalent to saying, 'It is not raining OR I am carrying an umbrella.'

In our problem, we have $p⟹(\sim q)$. Here, our antecedent $A$ is $p$, and our consequent $B$ is $\sim q$. By applying our identity, we transform the inner part of the expression into $\sim p\vee \sim q$.

Now, our entire expression looks like this: Suddenly, the 'scary' implication has vanished, replaced by a clean, manageable disjunction.

Now, we stand before the gate of De Morgan's Law. We have a negation sitting outside a pair of parentheses: $\sim (\sim p\vee \sim q)$.

De Morgan's Law tells us that when we distribute a negation over an OR ($\vee$) operator, the operator itself must flip to an AND ($\wedge$).

Imagine the negation as a wave passing through the parentheses. It hits the $\sim p$, it hits the $\sim q$, and as it passes through the $\vee$, it forces it to invert into $\wedge$. Mathematically, this gives us:

We are almost at the finish line. We are left with two double negations. In logic, just as in mathematics where two negatives make a positive, the Law of Double Negation states that $\sim (\sim X)\equiv X$.

Applying this to our terms, $\sim (\sim p)$ becomes $p$, and $\sim (\sim q)$ becomes $q$. The expression collapses beautifully into its simplest form:

The Beauty of Simplicity

Look at what we have achieved. We started with a complex, nested logical statement and, through the systematic application of identity laws, reduced it to the most fundamental conjunction: $p\wedge q$.

This is the essence of mathematical maturity—taking the complex and finding the simple truth hidden within. You have successfully navigated the logic, and the correct option is clearly (4). Keep this clarity of thought, and no problem in the JEE will ever be able to intimidate you again.