Animated Solution for Mathematics - Sets and Relations: The statement among the following that is a tautology is:

A Journey into Tautologies

My dear students, welcome to the world of mathematical reasoning. Today, we are not just solving a problem; we are peeling back the layers of logic itself.

We are hunting for a Tautology—a statement so robust, so inherently true, that it stands unshaken regardless of the truth values of its components. Imagine a fortress that remains standing no matter what storm hits it. That is a tautology.

Many students immediately reach for truth tables. While truth tables are a valid tool, they are the 'brute force' approach.

In the high-stakes environment of the JEE, we value elegance and speed. We want to dismantle these expressions using the laws of logic, much like a master chess player anticipates moves.

Let us look at our options. Options 1 and 3 involve the Absorption Law. For instance, $A\wedge (A\vee B)$ simplifies directly to $A$. Since $A$ can be False, these cannot be tautologies. We have already eliminated half our work!

Analyzing the Implication

Now, we turn our gaze to the final option: $[A\wedge (A\to B)]\to B$. This looks intimidating, but let us break it down.

We start by tackling the inner implication $A\to B$. Remember the golden rule of logic: $P \to Q \equiv eg P \lor Q$.

By applying this, our expression transforms into:

Now, we use the Distributive Law, just as you would in algebra. We distribute $A$ across the parenthesis to get:

Look closely at the term $(A \land eg A)$. This is the Law of Contradiction. A statement cannot be both true and false simultaneously. It is always False ($F$).

Our expression now simplifies to:

By the Identity Law, $F$ does nothing in an 'or' operation, leaving us with $(A\wedge B)\to B$. We are almost there!

We apply the implication rule one last time:

Now, we invoke De Morgan's Law to expand the negation:

By the Associative Law, we regroup this as $ eg A \lor ( eg B \lor B)$. Since $( eg B \lor B)$ is always True ($T$), we are left with $ eg A \lor T$.

In logic, anything 'or' True is always True. We have arrived at the destination! The statement is a tautology.

Keep practicing this art of simplification, and you will find that even the most complex problems bow before the power of logical laws.