Animated Solution for Mathematics - Sets and Relations: Negation of $(p\to q)\to (q\to p)$ is

A Journey Through Implication

Welcome, fellow explorer of the mathematical universe. Today, we are not just solving a problem; we are peeling back the layers of a logical structure.

We are looking at the expression $(p\to q)\to (q\to p)$. At first glance, it might seem like a tangled mess of arrows and variables, but I want you to see it as a puzzle waiting to be solved.

In the world of JEE Advanced, logic is not just about memorizing tables; it is about understanding the flow of truth.

The Master Key

The first step in our journey is to recognize the 'Implication Identity.' In logic, an implication $A\to B$ is not just a statement; it is a promise. It tells us that if $A$ is true, then $B$ must be true.

But how do we translate this into the language of algebra? We use the identity:

This is our master key. It allows us to convert the rigid, directional arrow into a flexible 'OR' (disjunction) statement. By doing this, we strip away the complexity of the implication and reveal the underlying truth values.

The Nested Implication

Now, look at our expression: $(p\to q)\to (q\to p)$. We have two inner implications and one outer one. Let's tackle the inner ones first.

Applying our master key, $(p\to q)$ becomes $(\sim p\vee q)$, and $(q\to p)$ becomes $(\sim q\vee p)$. Now, substitute these back into the original expression:

The arrows are starting to disappear, aren't they? This is the beauty of simplification—we are making the problem speak our language.

The De Morgan's Dance

We still have that main implication arrow in the middle. Let's treat the first bracket as $A$ and the second as $B$.

Applying the identity $A\to B\equiv \sim A\vee B$ again, we get:

Now, we encounter the negation of a disjunction. This is where De Morgan's Law comes into play, which states that $\sim (X\vee Y)\equiv \sim X\wedge \sim Y$.

When we apply this to $\sim (\sim p\vee q)$, the negation travels inside, flipping the 'NOT' to $p$ and the 'OR' to 'AND'. The result is $(p\wedge \sim q)$. Our expression is now $(p\wedge \sim q)\vee (\sim q\vee p)$.

The Absorption Law

We are nearing the finish line. We have $(p\wedge \sim q)\vee (\sim q\vee p)$. Let's rearrange this using the associative property:

Look closely at the first part: $(p\wedge \sim q)\vee \sim q$. This is the perfect setup for the Absorption Law, which states that $(A\wedge B)\vee B\equiv B$.

Here, our $B$ is $\sim q$. So, the entire term $(p\wedge \sim q)\vee \sim q$ collapses into just $\sim q$. We are left with $\sim q\vee p$.

The Final Negation

The question asks for the negation of the entire expression. We have simplified the expression to $(\sim q\vee p)$.

Now, we must negate this: $\sim (\sim q\vee p)$. Applying De Morgan's Law one last time, the negation goes inside: $\sim (\sim q)$ becomes $q$, the 'OR' becomes 'AND', and $p$ becomes $\sim p$.

The final result is $q\wedge \sim p$.

Look at what we have achieved. We started with a complex, nested implication and, through the systematic application of logical laws, arrived at a clean, elegant result: $q\wedge \sim p$.

This matches option B. Remember, in JEE Advanced, the complexity is often just a mask.

If you stay calm, apply your identities, and trust the process, the truth will always reveal itself. Keep practicing, keep questioning, and most importantly, keep falling in love with the logic behind the math.