# Definition:Dual Relation

## Definition

### Inverse of Complement

Let $\RR \subseteq S \times T$ be a binary relation.

Then the dual of $\RR$ is denoted $\RR^d$ and is defined as:

$\RR^d := \paren {\overline \RR}^{-1}$

where:

$\overline \RR$ denotes the complement of $\RR$
$\paren {\overline \RR}^{-1}$ denotes the inverse of the complement of $\RR$.

### Complement of Inverse

Let $\RR \subseteq S \times T$ be a binary relation.

Then the dual of $\RR$ is denoted $\RR^d$ and is defined as:

$\RR^d := \overline {\paren {\RR^{-1} } }$

where:

$\RR^{-1}$ denotes the inverse of $\RR$
$\overline {\paren {\RR^{-1} } }$ denotes the complement of the inverse of $\RR$.