If and only if

“If and only if” states that two propositions imply each other: each is true exactly when the other is true. It is commonly abbreviated “iff” and represented by the biconditional connective.

Connect

Biconditional: A logical connective that is true when its two propositions have the same truth value. It is the formal connective expressed by “if and only if.”
Proposition: A declarative statement that has a definite truth value, true or false. Each side of an iff is a proposition whose truth is being linked.
Mathematical proof: A deductive argument establishing that a mathematical statement follows from stated assumptions. Proving an iff requires separate arguments for each direction.
Implication: A relation in which one proposition’s truth guarantees another proposition’s truth. A one-way implication lacks the reverse direction required by an iff.
Material implication: A truth-functional connective, written P → Q, that is false only when P is true and Q is false. The biconditional combines implications in both directions.
Conjunction: A logical connective, usually written P ∧ Q, that is true when both propositions are true. An iff can be understood as requiring both directional implications together.
Proof by cases: A proof method that establishes a claim by dividing the possibilities into exhaustive cases. Case splits can establish one or both implications in a biconditional proof.
Converse (logic): The statement formed by reversing the antecedent and consequent of a conditional. An iff holds only when the original implication and its converse both hold.
Truth table: A table listing a proposition’s truth value for every assignment of truth values to its components. Its rows make clear that a biconditional is true when both sides match.
Negation: A logical operation that reverses a proposition’s truth value. Negating an iff’s sides helps derive equivalent forms and test its truth conditions.