Animated Solution for Mathematics - Sets and Relations: The converse of $((\sim p)\wedge q)\Rightarrow r$ is

Unveiling the Converse

Welcome, fellow traveler on the path to JEE excellence! Today, we are not just solving a problem; we are peeling back the layers of formal logic.

Logic is the bedrock of mathematics, the silent language that governs every theorem you have ever studied. When we look at an implication like $((\sim p)\wedge q)\Rightarrow r$, we are looking at a logical 'contract.'

But what happens when we flip that contract? That is the essence of the converse.

Defining the Foundation

In the world of logic, every implication $A\Rightarrow B$ has a mirror image. The converse is defined as $B\Rightarrow A$.

It is the simplest transformation, yet it is where many students stumble because they forget to treat the entire compound statement as a single block. Here, our $A$ is $((\sim p)\wedge q)$ and our $B$ is $r$.

Therefore, the converse is simply $r\Rightarrow ((\sim p)\wedge q)$. This is our starting point, our North Star.

The Art of Logical Manipulation

Now, you might look at your options and feel a sudden spike of anxiety—your result isn't there! But take a deep breath. In mathematics, there are many ways to say the same thing.

We need to use the Implication Law, which states that $X\Rightarrow Y\equiv (\sim X)\vee Y$. This law is a bridge; it allows us to move from the rigid 'if-then' structure into the flexible world of 'OR' statements.

Applying this to our converse, we get:

The Elegance of De Morgan's Law

We are getting closer, but we need to match the structure of the options. This is where De Morgan's Law shines.

We know that $((\sim p)\wedge q)$ is logically equivalent to $\sim (p\vee (\sim q))$. Think of this as 'factoring out' the negation.

By substituting this back into our expression, we arrive at:

The Final Synthesis

We are almost at the finish line. We have an expression in the form $(\sim X)\vee Y$.

We know that this is equivalent to $X\Rightarrow Y$. By setting $X=(p\vee (\sim q))$ and $Y=(\sim r)$, we can reverse the Implication Law one last time.

The result is:

Look at that! It is beautiful, isn't it? We started with a simple definition, navigated through the forest of logical identities, and arrived at a result that perfectly matches Option 4.

This journey teaches us that in logic, as in life, if the direct path is blocked, there is always a series of logical steps that will lead you to the truth. Keep practicing, keep questioning, and most importantly, keep falling in love with the process!