Biconditional

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'A if and only if B' means A and B are either both true or both false.

A biconditional statement 'A if and only if B' (often written as 'A iff B') asserts that A and B have the same truth value: they are both true or both false. This is equivalent to saying both 'If A then B' AND 'If B then A'. Biconditionals express necessary and sufficient conditions and are common in definitions and equivalences.

P ↔ Q ≡ (P → Q) ∧ (Q → P)

• A triangle is equilateral if and only if all its angles are 60°. (Having equal sides is both necessary and sufficient for having 60° angles)
• A number is even if and only if it is divisible by 2.
• You will pass the course if and only if you score above 70%. (Scoring above 70% guarantees passing, and passing requires scoring above 70%)

How to recognize it

• Look for 'if and only if', 'iff', 'exactly when', or 'is equivalent to'
• The relationship works in both directions
• Knowing either part tells you the truth of the other

Common mistakes

• Confusing 'if' with 'if and only if': 'If A then B' does NOT mean 'If B then A'
• Biconditional requires BOTH directions to hold
• 'A only if B' means 'If A then B', NOT biconditional

Practice guidance

Logical Deduction