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

Mastering Logical Negation

Welcome, future engineers. Today, we are not just solving a problem; we are peeling back the layers of logical reasoning. In the JEE Advanced arena, Mathematical Logic is often seen as a 'scoring' topic, but I want you to see it as something more: it is the language of the digital world.

Every circuit in your smartphone, every algorithm in your computer, operates on these exact principles. Let us dive into the negation of the expression $\sim s\vee (\sim r\wedge s)$.

The Outer Shell

When we are asked for the negation of a complex logical statement, our first instinct should be to treat the entire expression as a single entity. Imagine you are looking at a complex machine; you do not start by unscrewing the smallest bolt. You look at the casing.

We place the entire expression inside a bracket and apply the negation operator $\sim$ to the outside:

This is our starting point. We are essentially asking: "What is the exact opposite of this entire logical state?"

The Great Distributor (De Morgan's Law)

Now, we need to push that negation inside. This is where De Morgan's Law becomes our most trusted ally. Remember the rule: $\sim (A\vee B)\equiv \sim A\wedge \sim B$.

It is a beautiful symmetry. The negation distributes, and the 'or' ($\vee$) flips into an 'and' ($\wedge$). Let $A=\sim s$ and $B=(\sim r\wedge s)$.

Applying the law, we get:

Notice how the central 'or' has transformed into an 'and'. This is the first major victory in our simplification journey.

The Beauty of Cancellation

Look at the first part of our new expression: $\sim (\sim s)$. This is the Double Negation Law. Just as two negatives make a positive in arithmetic, two negations cancel each other out in logic: $\sim (\sim p)\equiv p$.

So, $\sim (\sim s)$ simply becomes $s$. Our expression now stands as:

We are making progress, but we still have that pesky negation outside the second bracket. Let us apply De Morgan's Law again, but this time to the inner conjunction: $\sim (P\wedge Q)\equiv \sim P\vee \sim Q$.

Here, our $P$ is $\sim r$ and our $Q$ is $s$. Pushing the negation inside, we get $\sim (\sim r)\vee \sim s$. Again, the 'and' flips to an 'or'.

The Final Simplification

We are almost there. Inside our bracket, we have $\sim (\sim r)$, which simplifies back to $r$ via the Double Negation Law. Now, our expression looks like this:

This is where many students stumble, but you won't. We have an 'and' outside and an 'or' inside. This is the perfect setup for the Distributive Law: $p\wedge (q\vee r)\equiv (p\wedge q)\vee (p\wedge r)$.

Distributing $s\wedge$ across the bracket, we obtain:

The Contradiction and the Identity

Look closely at the second term: $(s\wedge \sim s)$. Can something be true AND false at the same time? No. This is the Contradiction Law, and it collapses to $F$ (False).

Our expression is now:

Finally, we invoke the Identity Law: any statement $P$ ORed with False is just $P$. Thus, $(s\wedge r)\vee F\equiv s\wedge r$.

We have arrived at the destination. The negation of the original expression is simply $s\wedge r$. You have successfully navigated the logic, simplified the complexity, and found the truth.