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

The expression we are tackling is $(p\wedge (\sim q))\vee ((\sim p)\wedge q)\vee ((\sim p)\wedge (\sim q))$. While it may appear intimidating, we can decompose it systematically.

The Visual Intuition

Imagine you are standing in front of a whiteboard drawing two overlapping circles, $p$ and $q$. The entire area inside these circles represents our universe of discourse.

The first term, $p\wedge (\sim q)$, is the crescent of $p$ that does not touch $q$. The second term, $(\sim p)\wedge q$, is the crescent of $q$ that does not touch $p$. The third term, $(\sim p)\wedge (\sim q)$, is the vast space outside both circles.

If you shade all these regions, you will notice something profound: the only part left unshaded is the intersection, $p\wedge q$. This visual insight serves as our compass for the algebraic proof.

The Algebraic Journey

We start with our expression:

We use the Commutative Law to group the terms containing $\sim p$ together:

Now, we apply the Distributive Law in reverse, effectively factoring out $\sim p$:

Here is where the logic simplifies. The Complement Law states that $q\vee \sim q$ is a Tautology, denoted as $T$. Thus, our expression becomes:

By the Identity Law, $\sim p\wedge T$ simplifies to $\sim p$. We are left with the following expression:

The Final Simplification

We apply the Distributive Law again, this time distributing the OR operator over the AND operator:

Again, the Complement Law strikes, as $\sim p\vee p$ is equivalent to $T$. We now have:

Finally, the Identity Law tells us that $T\wedge A$ is simply $A$. Therefore, our final result is:

This result matches our Venn diagram perfectly. We have successfully navigated the maze of logic, proving that every complex problem is merely a series of simple, elegant steps.