Q.Check whether the following statement is true or not.
If x, y $\in \mathbb{Z}$ are such that x and y are odd, then xy is odd.

The Intuition: A Promise

Imagine you tell a friend: "If it rains tomorrow, then I will carry an umbrella."

That's an if-then statement. In logic, we call it an implication. The whole point is that you've made a promise. Now, let's see when that promise is kept and when it's broken.

There are exactly four things that can happen tomorrow:

1. It rains, and you carry an umbrella. — The promise is kept. The statement is true.
2. It rains, and you do NOT carry an umbrella. — The promise is broken. The statement is false.
3. It does NOT rain, and you carry an umbrella anyway. — Did you break the promise? No. The promise only said what happens if it rains. Since it didn't rain, you're free to do whatever you want. The statement is true.
4. It does NOT rain, and you do NOT carry an umbrella. — Again, the promise wasn't tested. The statement is true.

That last part — the fact that the implication is true when the "if" part is false — is the single most confusing thing for new students. But it makes perfect sense: a promise is only broken when the condition happens and you fail to deliver. If the condition never happens, the promise is intact.

The Precise Statement

Let's give things names.

Let $p$ stand for "it rains" (the hypothesis or antecedent).

Let $q$ stand for "I carry an umbrella" (the conclusion or consequent).

The implication is written as:

Read as: "if $p$, then $q$" or "$p$ implies $q$".

The truth table tells the whole story:

The only row that gives false is the second one: $p$ true and $q$ false. Every other combination yields true.

The Formal Definition

In mathematical logic, the implication $p \implies q$ is defined as equivalent to:

That is, "either $p$ is false, or $q$ is true" (or both). Check it against the truth table:

• If $p$ is true, then $\neg p$ is false, so for $\neg p \lor q$ to be true, $q$ must be true — exactly the condition for the implication to hold.
• If $p$ is false, $\neg p$ is true, so $\neg p \lor q$ is true regardless of $q$ — matching the last two rows.

This is the clean, algebraic way to remember the truth table.

Common Language vs. Logical Implication

In everyday speech, "if…then" often carries extra baggage:

• Causality: "If you study, you will pass" suggests studying causes passing. Logic doesn't require any causal link.
• Time: "If you eat dinner, then you may have dessert" suggests a sequence. Logic doesn't care about time.
• Relevance: "If $2+2=5$, then I am the Pope" sounds absurd, but in logic it's true (because the hypothesis is false). This is called a vacuous truth.

The Terminology You Need for Exams

• Hypothesis / Antecedent: The part after "if" ($p$).
• Conclusion / Consequent: The part after "then" ($q$).
• Implication: The whole statement $p \implies q$.
• Vacuous truth: An implication that is true because the hypothesis is false.

When you see a statement like "If a number is even, then it is divisible by 2", you're looking at an implication. The hypothesis is "a number is even", the conclusion is "it is divisible by 2". The statement is true because whenever the hypothesis holds, the conclusion follows — and if a number isn't even, the statement doesn't care.

One Final Check

Ask yourself: is the following statement true or false?

If $1 = 0$, then $2 = 2$.

The hypothesis ($1=0$) is false. Therefore, the implication is true — it's a vacuous truth. The conclusion ($2=2$) happens to be true, but that doesn't matter; even if the conclusion were false, the implication would still be true because the hypothesis is false.

That's the whole idea. Start with the promise, remember the one forbidden case, and you'll never get lost.