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

When you see a question asking for a 'tautology,' do not panic. Think of it as a logical identity, much like an algebraic identity. Just as $(a+b{)}^2=a^2+2ab+b^2$ is true for all real numbers, a tautology is a statement that remains true regardless of the truth values of its components.

To master this, we use the Truth Table—our ultimate tool for verification. Let us dissect the expression:

We start by peeling the onion from the inside out. The innermost component is the implication $p\to q$. Remember, an implication is only false when a true premise leads to a false conclusion. If the premise is false, the implication is 'vacuously true.'

The implication $p\to q$ is often the most misunderstood operator in the JEE syllabus. Students often try to map it to 'causality' in the real world, but in logic, it is purely about truth values.

Imagine you are a judge in a court of law. You have a rule: 'If the defendant is guilty ($p$), then they must be punished ($q$).' If the defendant is innocent (premise $p$ is False), the rule is not violated regardless of the outcome. This is why $F\to T$ and $F\to F$ are both True.

When you are staring at a complex logical expression, do not let the symbols intimidate you. Break them down. If you see $p\to q$, immediately check the truth table for the case where $p$ is True and $q$ is False. That is the only case that can break your implication.

Why do we insist on the truth table? Because it is the ultimate stress test. In the heat of an exam, your brain might skip a step or make a sign error. The truth table is mechanical, reliable, and systematic.

For two variables $p$ and $q$, we have $2^2=4$ scenarios. For three variables, we have $2^3=8$. It is a finite, manageable universe. By systematically listing $(T,T)$, $(T,F)$, $(F,T)$, and $(F,F)$, you are essentially checking every possible reality.

When you evaluate $(p\wedge (p\to q))\to q$, you are not just doing arithmetic; you are verifying that the logical structure holds up under every possible condition. When you see the final column of your table filled with nothing but 'True' (or 'T'), you are witnessing a logical certainty.

The expression $(p\wedge (p\to q))\to q$ is not just any random collection of symbols. It is the formalization of Modus Ponens, one of the most important rules of inference in human history.

It states: if you know that $p\to q$ is true, and you know that $p$ is true, then you are logically forced to accept that $q$ is true. This is how we build mathematical proofs. Every theorem you have ever studied, from the Pythagorean theorem to the laws of thermodynamics, relies on this chain of reasoning.

By solving this problem, you are not just getting a mark on an exam; you are practicing the very method that scientists and mathematicians use to uncover the secrets of the universe. Look for the structure, look for the Modus Ponens, and you will find the beauty in the truth table.