Q.Which number is the odd man out in the numbers 3, 5, 11, 14, 17, 21?
a. 21
b. 17
c. 14
d. 3

The Idea: When Two Things Are Locked Together

Imagine you're told: "You can enter the club if and only if you are wearing a blue badge."

What does that mean? Two things, and both are true:

1. If you have a blue badge, you can enter. (Blue badge → entry allowed.)
2. If you enter, you must have a blue badge. (Entry allowed → blue badge.)

The two statements — "you have a blue badge" and "you can enter" — are equivalent. One cannot happen without the other. They are locked together: whenever one is true, the other is true; whenever one is false, the other is false.

That's the intuition. Now let's make it precise.

The Precise Statement

In logic, "P if and only if Q" (written $P \iff Q$) means:

$P$ is true exactly when $Q$ is true.

It is a two-way implication. It breaks into two separate implications:

• If $P$ then $Q$ ($P \implies Q$) — the "only if" part.
• If $Q$ then $P$ ($Q \implies P$) — the "if" part.

Both must hold. If either direction fails, the "if and only if" is false.

Truth Table

Notice: the statement is true only when $P$ and $Q$ have the same truth value — both true or both false.

Why It Matters

In mathematics, "if and only if" is the language of equivalence. When you prove a theorem that says "A triangle is equilateral if and only if all its angles are $60^\circ$", you are saying:

• If the triangle is equilateral, then each angle is $60^\circ$.
• If each angle is $60^\circ$, then the triangle is equilateral.

These are two separate facts. Proving both is harder than proving just one direction — but the payoff is a complete, reversible description.

A Simple Example

Let $P$ be "An integer is even" and $Q$ be "The integer is divisible by 2".

• If an integer is even, then it is divisible by 2. ($P \implies Q$)
• If an integer is divisible by 2, then it is even. ($Q \implies P$)

Both are true. So: "An integer is even if and only if it is divisible by 2." This is a definition — it tells you that "even" and "divisible by 2" are the same property.

A Non-Example …