Animated Solution for Mathematics - Basic Mathematics: Consider the following three statements: (A) If $3+3=7$ then $4+3=8$. (B) If $5+3=8$ then earth is flat. (C) If both (A) and (B) are true then $5+6=17$. Then, which of the following statements is correct?

Mastering Logical Implication

Welcome, student. Today, we are not just solving a problem; we are peeling back the curtain on the very language of mathematics.

You might look at the statements provided—statements about flat earths and impossible arithmetic—and feel a sense of cognitive dissonance. You might want to scream, "But this makes no sense!" And that is exactly where the JEE wants you.

Logic is not about common sense; it is about formal structure. It is the bedrock upon which calculus, analysis, and set theory are built. Let us embark on this journey to master the truth table, the most powerful tool in your logical arsenal.

The Anatomy of a Promise

In mathematics, the statement "If $p$, then $q$" is denoted as $p\Rightarrow q$. This is an implication. Think of it as a contract.

I, the educator, make a promise to you: "If you study hard ($p$), then you will succeed ($q$)." When have I broken my promise? Only in one scenario: if you study hard ($p$ is True) but you do not succeed ($q$ is False).

If you don't study ($p$ is False), I haven't broken my promise, regardless of whether you succeed or not. This is the fundamental truth table for implication:

Notice the last two cases. These are "vacuous truths." They feel wrong to our human intuition, but they are logically bulletproof. If the premise is false, the implication is true by default. This is the trap that catches thousands of students every year.

Deconstructing Statement (A)

Let us look at Statement (A): "If $3+3=7$ then $4+3=8$."

Your brain immediately flags this as nonsense. $3+3$ is $6$, not $7$, and $4+3$ is $7$, not $8$. But let us apply the formal logic.

The premise $p$ is "$3+3=7$". This is False. The conclusion $q$ is "$4+3=8$". This is also False.

We are looking at the structure $F\Rightarrow F$. According to our truth table, a false premise leading to a false conclusion is a True statement. It is a vacuous truth. Statement (A) is logically True.

Deconstructing Statement (B)

Now, Statement (B): "If $5+3=8$ then earth is flat."

Here, the premise $p$ is "$5+3=8$". This is mathematically correct, so $p$ is True. The conclusion $q$ is "Earth is flat". We know this is scientifically false, so $q$ is False.

We have the structure $T\Rightarrow F$. This is the only scenario where the implication fails. The promise is broken. Therefore, Statement (B) is logically False.

The Compound Challenge of (C)

Finally, Statement (C): "If both (A) and (B) are true then $5+6=17$."

This is where we combine our findings. The premise $p$ is a compound statement: "$A\wedge B$". We know $A$ is True and $B$ is False.

In logic, the conjunction $T\wedge F$ results in False. So, our premise $p$ is False. The conclusion $q$ is "$5+6=17$". This is clearly False.

Again, we find ourselves in the $F\Rightarrow F$ scenario. Just like in Statement (A), this results in a True statement. Statement (C) is logically True.

We have systematically dismantled these statements. We found (A) to be True, (B) to be False, and (C) to be True.

The logic holds firm, independent of the absurdity of the content. This is the power of mathematics. It provides a framework to determine truth even when our intuition is screaming that something is wrong.

Remember this: when you face logic problems in the JEE, do not rely on your gut feeling. Rely on the truth table. Trust the structure. Trust the math. You have now mastered the art of the vacuous truth, and that is a weapon you will carry with you through your entire engineering career.