Q.Rewrite the following statement with “if-then” in five different ways conveying the same meaning.
If a natural number is odd, then its square is also 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. …