Definition:Biconditional

Definition
The biconditional is a binary connective:
 * $p \iff q$

defined as:
 * If $p$ is true, then $q$ is true, and if $q$ is true, then $p$ is true.

Thus, $p \iff q$ means:
 * $p$ is true if and only if $q$ is true


 * $p$ is (logically) equivalent to $q$


 * $p$ is true iff $q$ is true

$p \iff q$ can be voiced:
 * $p$ if and only if $q$.

It can be formulated as follows:
 * $\left({p \implies q}\right) \land \left({q \implies p}\right)$

It can be written:
 * $\displaystyle {\left({p \implies q}\right) \quad \left({q \implies p}\right) \over p \iff q} \qquad \qquad {p \iff q \over p \implies q} \qquad \qquad {p \iff q \over q \implies p}$

Boolean Interpretation
From the above, we see that the boolean interpretations for $\mathbf A \iff \mathbf B$ under the model $\mathcal M$ are:


 * $\left({\mathbf A \iff \mathbf B}\right)_{\mathcal M} = \begin{cases}

T & : \mathbf A_{\mathcal M} = \mathbf B_{\mathcal M} \\ F & : \text {otherwise} \end{cases}$

Complement
The complement of $\iff$ is the exclusive or operator.

Truth Function
The biconditional connective defines the truth function $f^\leftrightarrow$ as follows:

Necessary and Sufficient
If $p \iff q$, we can say that $p$ is necessary and sufficient for $q$.

This is a consequence of the definitions of necessary and sufficient conditions.

Also known as
Other names for this operator include:
 * Material Equivalence


 * Logical Equivalence


 * Logical Equality

Notational Variants
Various symbols are encountered that denote the concept of biconditionality:

It is usual in mathematics to use $\iff$, as there are other uses for the other symbols.

Also see

 * Therefore
 * Because
 * Interderivable (Logical Equivalence)
 * Conditional