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

Welcome, future engineers! Today, we are not just solving a problem; we are peeling back the layers of the language that powers every computer, smartphone, and logic gate on this planet. Boolean Algebra is the bedrock of digital logic.

When you look at an expression like $((p\wedge q)\vee (p\wedge \sim q))\vee (\sim p\wedge \sim q)$, do not see a jumble of symbols. See a map. See a geometric landscape. Let us embark on this journey to simplify this expression, step by step, with the precision of a mathematician and the intuition of an artist.

Imagine you are standing in front of a whiteboard. We have two sets, $p$ and $q$, represented by two overlapping circles in a Venn diagram. The entire universe of possibilities is the rectangle containing them.

Look at the first part of our expression: $((p\wedge q)\vee (p\wedge \sim q))$.

The term $p\wedge q$ is the intersection—the sweet spot where both $p$ and $q$ are true. Now, look at $p\wedge \sim q$. This is the region strictly inside $p$ but outside $q$. If you shade the intersection and the 'p-only' region, you get the entire circle $p$.

Algebraically, we can prove this using the Distributive Law. We factor out the $p$ to get $p\wedge (q\vee \sim q)$.

Since $q\vee \sim q$ is the Complement Law in action, it covers every single possibility and is always True ($T$). Thus, our expression collapses beautifully:

The entire first massive bracket is just $p$. We have already cut the complexity in half!

Now, let us look at what remains. We have simplified the expression to $p\vee (\sim p\wedge \sim q)$.

This is where many students stumble. They try to force a simplification that isn't there. But we have a powerful tool: the Distributive Law. Remember, in Boolean algebra, the OR operator ($\vee$) distributes over the AND operator ($\wedge$).

Think of it like expanding brackets in a polynomial. We distribute the $p\vee$ across the $(\sim p\wedge \sim q)$. This gives us:

Look at the first term: $(p\vee \sim p)$. This is the Complement Law again! It is a tautology, meaning it is always True ($T$). So, our expression becomes:

We are at the finish line. We have $T\wedge (p\vee \sim q)$.

By the Identity Law, we know that $T\wedge X=X$. It is the logical equivalent of multiplying by 1. It changes nothing.

Therefore, the entire expression simplifies down to the elegant, clean result:

Why did we do this? Why simplify? In the world of circuit design, every operator ($\wedge$, $\vee$, $\sim$) represents a physical gate—a transistor, a piece of silicon, a cost.

By simplifying this expression, we have reduced the number of gates required to perform the same logical operation. We have made the circuit faster, cheaper, and more efficient.

This is the essence of engineering. It is not just about getting the right answer; it is about finding the most elegant path to the truth. You have just performed a logical optimization. Keep this mindset, and you will not just solve JEE problems—you will master the logic that builds the future.