Equivalence

logic

Definition

Equivalence

The equivalence (or biconditional) is a binary logical connective. For formulas $A$ and $B$, the formula $A\leftrightarrow B$ is true exactly when $A$ and $B$ have the same truth value.

Truth Table

$A$	$B$	$A\leftrightarrow B$
$0$	$0$	$1$
$0$	$1$	$0$
$1$	$0$	$0$
$1$	$1$	$1$

Relation to Equivalence of Formulas

Semantic Equivalence

Two formulas $A$ and $B$ are equivalent if and only if $A\leftrightarrow B$ is valid.

Equivalently, $A\equiv B$ if and only if $\neg (A\leftrightarrow B)$ is unsatisfiable.