Animated Solution for Mathematics - Sets and Relations: The negation of the Boolean expression $x\leftrightarrow \sim y$ is equivalent to :

In the world of Boolean algebra, we deal with the fundamental truth of existence—true or false, on or off. Today, we are going to dissect the negation of a biconditional expression: $\sim (x\leftrightarrow \sim y)$.

This might look like a jumble of symbols, but it is actually a beautiful puzzle of symmetry.

Our first step is to recognize the structure. We are looking for the negation of a biconditional statement.

There is a powerful identity in logic that acts like a key to this door:

This identity tells us that negating a biconditional is equivalent to negating just one of its components. It is an elegant way to shift the complexity.

When we apply this to our expression, where $p=x$ and $q=\sim y$, the expression $\sim (x\leftrightarrow \sim y)$ transforms into $x\leftrightarrow \sim (\sim y)$. Notice how the outer negation has moved inside, attaching itself to the second term.

Now, we encounter the double negation law. We have $\sim (\sim y)$, which is the logical equivalent of saying "not not $y$."

Just as two negatives make a positive in arithmetic, two negations cancel each other out in logic. So, $\sim (\sim y)$ simplifies beautifully to $y$.

Our entire expression has now collapsed into the much simpler form: $x\leftrightarrow y$. This is the moment where the complexity vanishes, leaving us with a clean, fundamental logical relationship.

Finally, we must translate this biconditional into the language of AND and OR. What does $x\leftrightarrow y$ actually mean?

It means that $x$ and $y$ must have the same truth value. They must be in agreement.

This can happen in two ways: either both are true, which we write as $(x\wedge y)$, or both are false, which we write as $(\sim x\wedge \sim y)$.

Since either of these scenarios makes the statement true, we connect them with an OR operator. Thus, we arrive at our final result:

You have just navigated the logical landscape, simplified a complex expression, and arrived at the truth. Keep this mindset—look for the symmetry, simplify the structure, and the answer will reveal itself.