Animated Solution for Mathematics - Sets and Relations: The negation of the boolean expression $\sim s\vee (\sim r\wedge s)$ is equivalent to :

Unraveling the Boolean Knot

Welcome, future engineers! Today, we are going to dive into the beautiful, structured world of Boolean algebra. Often, students look at a problem like the negation of $\sim s\vee (\sim r\wedge s)$ and see a jumble of symbols.

But I want you to see something else: a puzzle, a sequence of transformations that, when handled with care, reveals a stunningly simple truth. Let us embark on this journey together.

The Gatekeeper of Negation

We start with the expression $\sim s\vee (\sim r\wedge s)$. Our mission is to find its negation.

The first step is the most critical: we must wrap the entire expression in brackets and place the negation operator outside. This is our 'gatekeeper'. We write it as:

Why do we do this? Because the negation operator is not just a symbol; it is an instruction to invert the truth value of the entire logical structure. If you forget the brackets, you are only negating the first term, and the entire logic collapses. Always respect the scope of your operators!

The De Morgan's Dance

Now, we invoke the legendary De Morgan's Laws. Think of these laws as the 'distributive property' of logic.

When we apply the outer negation to the bracket $\sim [\sim s\vee (\sim r\wedge s)]$, the negation must distribute to both terms, and—this is the crucial part—the operator in the middle must flip. The OR ($\vee$) becomes an AND ($\wedge$). So, we get:

Look at the first term: $\sim (\sim s)$. This is the Double Negation Law in action. Just as a double negative in language often creates a positive, in logic, negating a negation brings us back to the original state.

So, $\sim (\sim s)$ simplifies beautifully to $s$. Our expression now looks like $s\wedge \sim (\sim r\wedge s)$. We are making progress!

The Second Transformation

We are not done yet. We still have that pesky negation outside the second bracket: $\sim (\sim r\wedge s)$.

We apply De Morgan's Law again. The negation hits $\sim r$, the AND ($\wedge$) flips to an OR ($\vee$), and the negation hits $s$. This gives us $(\sim (\sim r)\vee \sim s)$.

Again, we see the Double Negation Law: $\sim (\sim r)$ becomes $r$. The expression now stands as $s\wedge (r\vee \sim s)$.

The Distributive Power

Now, we reach the heart of the problem. We have $s\wedge (r\vee \sim s)$. This looks like basic algebra, doesn't it?

We distribute the $s$ across the bracket. We get:

This is the Distributive Law, and it is the bridge that connects our complex expression to the final, elegant answer.

The Final Simplification

Look closely at the second part of our expression: $(s\wedge \sim s)$. This is the Complement Law.

Can a statement $s$ be true AND its negation $\sim s$ be true at the same time? Absolutely not! This is a contradiction, which we represent as False ($F$). So, our expression becomes:

Finally, we use the Identity Law. In logic, any statement $P$ ORed with False ($P\vee F$) is just $P$. It is the logical equivalent of adding zero.

Thus, $(s\wedge r)\vee F$ simplifies to $s\wedge r$.

And there we have it! The entire complex expression boils down to $s\wedge r$. We started with a tangled web of negations and operators, and through systematic application of logical laws, we arrived at a clear, concise result. This is the power of mathematical logic—it turns chaos into order.