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

Conquering the Tautology

Welcome, future engineer. Today, we are going to dismantle a problem that, at first glance, looks like a tangled mess of arrows and brackets. When you see an expression like $(p\to (q\to p))\to (p\to (p\vee q))$, it is natural to feel a moment of hesitation.

It looks like a wall of symbols designed to confuse you. But here is the secret of JEE Advanced mathematics: complexity is often just simplicity in disguise. We are not going to brute-force this with a truth table; instead, we are going to use the elegance of logical algebra to peel back the layers.

The Divide and Conquer Strategy

Imagine you are standing before a massive fortress. You don't try to knock down the entire wall at once; you find the weak points, the gates, and the sections.

In our expression, we have a major implication arrow connecting two distinct blocks. Let us define the left side as $A$ and the right side as $B$:

Our entire problem now transforms from a terrifying string of symbols into a simple question: What is $A\to B$? By isolating these blocks, we have reduced the cognitive load, allowing us to focus our mental energy on one manageable piece at a time.

The Universal Key

Before we dive into the algebra, we need our primary tool. In propositional logic, the implication arrow ($\to$) is notoriously difficult to manipulate directly, as it does not play well with associative or commutative laws.

However, there is a 'Universal Key' that unlocks every implication, known as the Implication Rule:

This rule is the bridge between the world of implications and the world of OR gates. By converting every implication into a negation and an OR, we enter a domain where we can rearrange, group, and simplify terms with total freedom.

Simplifying Statement A

Let's look at $A=p\to (q\to p)$. We apply our Universal Key to the inner bracket first, where $(q\to p)$ becomes $(\sim q\vee p)$.

Now, our expression for $A$ looks like this:

We apply the rule again to the outer $p$:

Using the Associative Law, we rearrange the terms to group the $p$ variables together:

Here is the moment of realization. The term $(\sim p\vee p)$ is the Law of Excluded Middle, which is always True ($T$). Thus, $A$ becomes $T\vee \sim q$, which simplifies to $T$. We have proven that $A$ is a tautology.

Simplifying Statement B

Now, let's turn our attention to $B=p\to (p\vee q)$. Applying the same logic, the implication becomes:

Again, we use the Associative Law to group the $p$ terms:

Just like before, $(\sim p\vee p)$ is $T$. Consequently, $B$ simplifies to $T\vee q$, which is always $T$. Block $B$ is also a tautology.

We have arrived at the finish line. Our original expression was $A\to B$. We have rigorously proven that $A\equiv T$ and $B\equiv T$.

Substituting these back into our original structure, we get:

By the definition of the implication operator, a True statement implying a True statement is, itself, True. The entire expression is a tautology. You didn't just solve a problem; you mastered the logic behind it. Keep this mindset—divide, simplify, and conquer—and there is no problem in the JEE Advanced paper that can stand against you.