Animated Solution for Mathematics - Sets and Relations: Statement-1: $\sim (p\leftrightarrow \sim q)$ is equivalent to $p\leftrightarrow q$. Statement-2: $\sim (p\leftrightarrow \sim q)$ is a tautology

Unveiling the Truth Table

Welcome, future engineers. Today, we are not just solving a problem; we are embarking on a journey into the very bedrock of mathematical reasoning. Logic is the language of the universe, and the problem before us—evaluating the equivalence of logical statements—is a classic test of your precision and patience.

Imagine you are an architect designing a digital circuit. You need to know exactly when a signal passes through and when it is blocked. This is the essence of the biconditional operator, $p\leftrightarrow q$.

The biconditional operator is a 'Same-Value Detector'. It only outputs True if both inputs are identical. If they differ, it shuts down, outputting False. This is the fundamental truth we must carry into our analysis.

Building the Truth Table

When we face a complex expression like $\sim (p\leftrightarrow \sim q)$, the temptation is to guess or rush. But in the JEE Advanced arena, we do not guess. We build.

We construct a truth table, a systematic grid that leaves no room for error. We start with our atomic propositions, $p$ and $q$. With two variables, we have $2^2=4$ possible states: $(T,T),(T,F),(F,T),$ and $(F,F)$.

Now, let us calculate $p\leftrightarrow q$. For the first row, $p=T$ and $q=T$, they match, so we get $T$. For the second and third rows, they differ, so we get $F$. For the fourth row, $p=F$ and $q=F$, they match again, giving us $T$. Our result column is $\{T,F,F,T\}$.

Next, we turn our attention to the more complex expression: $\sim (p\leftrightarrow \sim q)$. We must break this down.

First, we find $\sim q$. This is the simplest step—we just flip the values of $q$. If $q$ is $T$, $\sim q$ is $F$. If $q$ is $F$, $\sim q$ is $T$.

Now, we evaluate $p\leftrightarrow \sim q$. We compare $p$ with our new $\sim q$ column:

In row one, $p=T$ and $\sim q=F$, so they differ, resulting in $F$. In row two, $p=T$ and $\sim q=T$, they match, resulting in $T$. In row three, $p=F$ and $\sim q=F$, they match, resulting in $T$. In row four, $p=F$ and $\sim q=T$, they differ, resulting in $F$.

The final step is to negate this entire result. We flip the column $\{F,T,T,F\}$ to get $\{T,F,F,T\}$.

Truth and Tautology

Now, look at the two columns side-by-side. The column for $p\leftrightarrow q$ is $\{T,F,F,T\}$. The column for $\sim (p\leftrightarrow \sim q)$ is also $\{T,F,F,T\}$.

They are identical! This confirms that Statement-1 is absolutely True.

But what about Statement-2? It claims that $\sim (p\leftrightarrow \sim q)$ is a tautology. A tautology must be True in every single case.

Our result column contains False values in the second and third rows. Therefore, it cannot be a tautology. Statement-2 is False.

You see, logic is not about memorizing formulas; it is about the systematic breakdown of complexity. By taking it step-by-step, you have conquered this problem. Keep this clarity, keep this patience, and you will dominate the JEE.