Animated Solution for Mathematics - Sets and Relations: Which of the following statements is a tautology?

Decoding Tautologies

Welcome, fellow traveler on the path to JEE mastery. Today, we are not just solving a problem; we are peeling back the curtain on the very fabric of mathematical logic.

We are hunting for a Tautology—a statement so robust, so perfectly constructed, that it remains true in every conceivable universe. It is the 'universal truth' of Boolean algebra.

The Philosophy of the Tautology

In the realm of logic, a tautology is a compound statement that yields a truth value of $T$ (True) regardless of the individual truth values of its components, $p$ and $q$.

Imagine a machine where you can flip the switches of $p$ and $q$ to any combination of True or False, and yet, the output light remains stubbornly, brilliantly lit. That is a tautology. Our mission is to identify which of the given options possesses this inherent, unbreakable truth.

The Toolkit of the Logical Architect

Before we charge into the options, we must sharpen our tools. We cannot rely on brute force; we need the precision of the laws of logic.

1. The Conditional Identity: The implication $A\to B$ is often the most intimidating part of a logical expression. We tame it with the identity $A\to B\equiv \sim A\vee B$. This converts the 'arrow' into a familiar 'OR' ($\vee$) and 'NOT' ($\sim$) structure.

2. The Double Negation Law: Just as in arithmetic where two negatives make a positive, in logic, $\sim (\sim p)\equiv p$. It is a beautiful cancellation that clears the clutter.

3. The Complement Law: This is our secret weapon. A statement $q$ must be either true or false. Therefore, $q\vee \sim q$ covers all possibilities, making it universally true: $q\vee \sim q\equiv T$.

The Deep Dive into Option 4

Let us examine the candidate: $\sim (p\vee \sim q)\to (p\vee q)$.

It looks complex, but let us apply our first tool, the Conditional Identity. We treat the entire left side as $A$ and the right side as $B$. The expression becomes:

Now, witness the power of the Double Negation Law. The two negations at the start, $\sim (\sim \dots )$, effectively vanish, leaving us with:

The Collapse to Truth

We are now left with a string of 'OR' operations. Because the 'OR' operator is associative and commutative, we can rearrange these terms as we please. Let us group the $p$ terms and the $q$ terms:

Since $p\vee p\equiv p$, we simplify this to:

Look closely at the bracketed term $(q\vee \sim q)$. By the Complement Law, this is simply $T$. Our expression has now collapsed into:

Finally, by the Identity Law, any statement 'OR'ed with a guaranteed True statement is itself True. Thus, $p\vee T\equiv T$.

We have arrived! The expression is always true, and therefore, it is a Tautology. You have successfully navigated the logic, simplified the complexity, and found the universal truth.