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

Welcome, future engineer. Today, we are peeling back the curtain on the language of mathematics. Logic is the foundation upon which all of calculus, physics, and engineering are built.

When you look at a statement like $p\to (q\to p)$, it might look like a jumble of symbols, but I want you to see it as a structural blueprint. Let us embark on this journey to simplify the complex.

In our toolkit, we have a master key: the conditional identity. Whenever you see an implication $A\to B$, your mind should immediately translate it into the language of OR ($\vee$) and NOT ($ eg$).

The identity is simple yet profound:

Why does this matter? Because OR operations are much easier to manipulate algebraically. Let us apply this to our inner bracket, $(q\to p)$.

By applying our identity, we transform it into $ eg q \vee p$. Now, our original expression $p\to (q\to p)$ becomes $p \rightarrow ( eg q \vee p)$.

Now, look at the expression $p \rightarrow ( eg q \vee p)$. We apply the identity one more time. Let $A=p$ and $B = ( eg q \vee p)$.

The expression becomes:

Here is where the magic happens. Because the OR operator is associative, we can regroup these terms however we like. We can shift the parentheses to bring $ eg p$ and $p$ together:

Do you see the beauty here? We have created a pair of opposites. According to the Complement Law, $ eg p \vee p$ is always True ($T$).

So, our expression simplifies to $T \vee eg q$. In the world of logic, if you have a True statement OR-ed with anything else, the result is always True. We have just proven that our original expression is a tautology—it is universally true, regardless of the values of $p$ or $q$.

Now, we must find which option is equivalent to our tautology. We test Option 2: $p\to (p\vee q)$.

Applying our identity again, we get:

Again, we use the associative property to group $( eg p \vee p) \vee q$. Just as before, the complement law gives us $T\vee q$, which is simply $T$.

We have successfully navigated the logic. We started with a complex implication, broke it down into its fundamental components, and discovered that it was a tautology.

By comparing this to our options, we found that $p\to (p\vee q)$ shares this same logical DNA.

Remember, in the JEE, you are not just looking for the answer; you are looking for the most efficient path. By mastering these identities, you stop guessing and start knowing. You have transformed a daunting logical string into a clear, undeniable truth.