Animated Solution for Mathematics - Sets and Relations: The compound statement $(\sim (P\wedge Q))\vee ((\sim P)\wedge Q)\Rightarrow ((\sim P)\wedge (\sim Q))$ is equivalent to

When you look at a statement like $(\sim (P\wedge Q))\vee ((\sim P)\wedge Q)\Rightarrow ((\sim P)\wedge (\sim Q))$, it is easy to feel overwhelmed by the symbols. Logic is not about memorizing tables; it is about visualizing the structure of reality.

Let us break this down into a journey of discovery.

First, let us look at the Antecedent, which we will call $L$. It is defined as:

Instead of panicking, let's use the power of Venn diagrams. Imagine two circles, $P$ and $Q$, inside a universal set with four distinct regions: Region 1: $P$ only Region 2: $P$ and $Q$ (Intersection) Region 3: $Q$ only Region 4: Outside both

The first part, $\sim (P\wedge Q)$, is a classic application of De Morgan's Law. It represents everything except the intersection of $P$ and $Q$, covering Regions 1, 3, and 4.

The second part, $(\sim P)\wedge Q$, represents the region inside $Q$ but outside $P$, which is exactly Region 3. When we take the union of these two, we combine Regions 1, 3, 4 with Region 3.

Since Region 3 is already included, the union remains Regions 1, 3, and 4. Thus, our antecedent $L$ simplifies to $\sim (P\wedge Q)$.

Now, let's look at the Consequent, $R=(\sim P)\wedge (\sim Q)$. This represents the region outside both circles, which is Region 4.

We now face the implication: $L\Rightarrow R$. Never try to evaluate an implication directly; always use the identity:

We know $L$ covers Regions 1, 3, and 4. Therefore, $\sim L$ must be the only remaining region: Region 2.

Now, we take the union of $\sim L$ (Region 2) and $R$ (Region 4). We are left with the set of Regions 2 and 4.

Look closely at Regions 2 and 4. Region 2 is where both $P$ and $Q$ are true, and Region 4 is where both $P$ and $Q$ are false. This is the definition of the Biconditional statement, $P\leftrightarrow Q$.

We have successfully reduced a complex logical expression into a beautiful, symmetric statement. We now compare this to the provided options.

Option 1 is $((\sim P)\vee Q)\wedge ((\sim Q)\vee P)$. This is equivalent to:

This is the standard definition of $P\leftrightarrow Q$. We have arrived at the truth.

Remember, in JEE Advanced, the math is just a language. Once you learn to speak it fluently, you don't solve problems; you simply observe the truth unfolding.