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

Understanding Tautologies

Welcome, future engineers! Today, we are stepping into the elegant world of Mathematical Reasoning. We are going to dissect a problem that often appears in the JEE Advanced syllabus: identifying a tautology.

Imagine you are an architect of logic. A tautology is your strongest structure—a compound statement that remains 'True' under every possible condition. It is the bedrock of mathematical certainty.

Setting the Stage

To determine if an expression is a tautology, we use the Truth Table. Think of this as a stress test.

We have two variables, $p$ and $q$. Since each can be either True ($T$) or False ($F$), we have $2^2=4$ possible scenarios. We list them systematically: $(T,T)$, $(T,F)$, $(F,T)$, and $(F,F)$.

This is our foundation. Without this, we are building on sand.

The Strict Gatekeeper

First, we evaluate the conjunction $p\wedge q$. This is the 'AND' operator.

It is incredibly strict. It demands that both $p$ and $q$ be True simultaneously. If even one of them is False, the whole operation collapses into False.

So, in our table, only the first row $(T,T)$ yields a True. The rest are all False. It is a harsh gatekeeper, but that is its nature.

The Promise

Now, we tackle the conditional statement $p\to q$. This is where most students stumble.

Think of $p\to q$ as a promise: "If $p$ occurs, then $q$ must follow." When is this promise broken? Only when $p$ happens (True) but $q$ fails to happen (False).

That is the only scenario where the implication is False. If $p$ is False, the promise is never tested, so we consider it "vacuously True." This is why $F\to T$ and $F\to F$ are both True. Master this, and you master logic.

Now, we bring it all together to test the expression $(p\wedge q)\to (p\to q)$. We treat $(p\wedge q)$ as our new premise and $(p\to q)$ as our conclusion.

Let us look at the rows:

In the first row, both are True, so the implication is True. In the second, third, and fourth rows, our premise $(p\wedge q)$ is False.

Remember our rule? If the premise is False, the implication is automatically True! Look at the final column—it is a solid wall of 'True' values.

We have found our tautology. This is the beauty of logic: when you break down complex expressions into these simple, fundamental rules, the truth reveals itself clearly. Keep practicing, stay curious, and remember that every complex problem is just a collection of simple truths waiting to be organized.