Biconditional

Definition

Let $p$ and $q$ be propositions. The biconditional of $p$ and $q$ is the proposition $p \leftrightarrow q$, read "$p$ if and only if $q$". The proposition $p \leftrightarrow q$ is true when $p$ and $q$ are both true or both false. Otherwise, it is false.

$p$	$q$	$p \leftrightarrow q$
T	T	T
T	F	F
F	T	F
F	F	T

Remarks

Other notations are $p \Leftrightarrow q$ and "$p$ iff $q$". The proposition $p \leftrightarrow q$ has the same truth value as $(p \to q) \land (q \to p)$.

Explanations

L1 human

"A triangle has three equal sides if and only if it has three equal angles" is a biconditional. It says two things: each property gives the other. To prove a biconditional, you usually prove the two implications separately.