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

The expression $(p\wedge q)\Rightarrow ((r\wedge q)\wedge p)$ may appear as a chaotic jumble of symbols, but it is a puzzle waiting to be decoded. By stripping away the complexity, we can reveal the hidden simplicity beneath.

The Implication Trap

The first hurdle is the implication operator, $\Rightarrow$. We must transform it using the fundamental identity:

Applying this to our expression, where $A=(p\wedge q)$ and $B=((r\wedge q)\wedge p)$, we obtain:

The arrow is now replaced by the familiar operators of NOT, AND, and OR. We have taken the first step toward clarity.

The Art of Rearrangement

Consider the second part of our expression: $((r\wedge q)\wedge p)$. Because the operator is exclusively AND, we are governed by the Associative and Commutative laws.

We can shuffle these variables to find patterns. By grouping $p$ and $q$ together, we rewrite the expression as:

This rearrangement is the "Aha!" moment. We have successfully created a common structure.

The Power of Substitution

To reduce cognitive load, let us use a substitution. Let $X=(p\wedge q)$.

The expression now transforms into:

It is no longer a terrifying string of symbols; it is a simple, elegant logical statement. We have moved from the forest of complexity into the clearing of simplicity.

The Distributive Dance

Next, we apply the Distributive Law. Distributing $\sim X\vee$ over the bracket $(r\wedge X)$ yields:

We have expanded the logic, creating two distinct branches connected by an AND operator.

The Tautology Reveal

Look closely at the second bracket: $(\sim X\vee X)$. This is a Tautology.

Since a statement $X$ is either true or false, $X$ OR NOT $X$ must always be true, represented by $T$. Our expression becomes:

Because anything AND True is simply the thing itself, the $T$ vanishes, leaving us with:

The Final Synthesis

Now, we reverse our substitution by replacing $X$ with $(p\wedge q)$:

Applying the implication rule in reverse, we rewrite this as:

We have successfully distilled the complexity into a concise, elegant implication. By testing the options, we find that the expression simplifies to the equivalent of $(p\wedge q)\Rightarrow r$.