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

Unlocking $\sim (p\leftrightarrow \sim q)$

My dear student, welcome to the fascinating world of mathematical reasoning. Today, we are going to peel back the layers of a logical expression that might look intimidating at first glance: $\sim (p\leftrightarrow \sim q)$.

Logic is the bedrock of all mathematics and computer science, and mastering it is like learning the secret language of the universe. Let us embark on this journey together.

The Anatomy of the Expression

Before we dive into the mechanics, let us understand what we are looking at. We have two fundamental propositions, $p$ and $q$. These are our building blocks.

The operator $\leftrightarrow$ is the bi-conditional, often read as 'if and only if'. Think of it as a logical equality checker. It returns True only when both sides are in perfect agreement—either both are True or both are False.

Then, we have the negation operator $\sim$, which acts like a mirror, flipping True to False and False to True. Our expression $\sim (p\leftrightarrow \sim q)$ is essentially asking us to evaluate the bi-conditional between $p$ and the negation of $q$, and then flip the final result.

The Truth Table Strategy

When you face a logical expression in the JEE, the truth table is your most reliable weapon. It is the 'brute force' method that never fails.

Since we have two variables, $p$ and $q$, we have $2^2=4$ possible combinations of truth values. We list them systematically: $(T,T)$, $(T,F)$, $(F,T)$, and $(F,F)$. This structure ensures we do not miss a single possibility.

Step-by-Step Execution

Let us build our table. First, we define the column for $\sim q$. We take the $q$ column and flip it: $T$ becomes $F$, and $F$ becomes $T$.

Next, we evaluate the inner bi-conditional $p\leftrightarrow \sim q$. Remember the rule: it is True if both sides are identical.

In the first row, $p$ is $T$ and $\sim q$ is $F$, so the result is $F$. In the second row, $p$ is $T$ and $\sim q$ is $T$, so the result is $T$.

In the third row, $p$ is $F$ and $\sim q$ is $F$, so the result is $T$. In the final row, $p$ is $F$ and $\sim q$ is $T$, so the result is $F$.

Now, we apply the outer negation $\sim$. We take our column $F,T,T,F$ and flip it to get $T,F,F,T$. This is the truth profile of our original expression.

The Moment of Truth

Finally, we compare this result with our options. Let us look at $p\leftrightarrow q$. This is True when $p$ and $q$ are identical.

For $(T,T)$, it is $T$. For $(T,F)$, it is $F$. For $(F,T)$, it is $F$. For $(F,F)$, it is $T$. The sequence is $T,F,F,T$.

Look at that! The columns are identical. We have proven that:

You have successfully navigated the logic, my friend. Keep this clarity of thought, and no logical problem will ever stand in your way.