# Equivalence of Definitions of Dual Relation

## Theorem

The following definitions of the concept of Dual Relation are equivalent:

Let $\mathcal R \subseteq S \times T$ be a relation.

### Inverse of Complement

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

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

$\mathcal R^d := \left({\overline{\mathcal R}}\right)^{-1}$

where:

$\overline {\mathcal R}$ denotes the complement of $\mathcal R$
$\left({\overline{\mathcal R}}\right)^{-1}$ denotes the inverse of the complement of $\mathcal R$.

### Complement of Inverse

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

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

$\mathcal R^d := \overline{\left({\mathcal R^{-1}}\right)}$

where:

$\mathcal R^{-1}$ denotes the inverse of $\mathcal R$
$\overline{\left({\mathcal R^{-1}}\right)}$ denotes the complement of the inverse of $\mathcal R$.

## Proof

Let $\left({x, y}\right) \in \left({\overline{\mathcal R}}\right)^{-1}$.

Then:

 $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \overline{\mathcal R}$ $\displaystyle \iff \ \$ $\displaystyle \left({y, x}\right)$ $\in$ $\displaystyle \overline{\mathcal R}$ Definition of Inverse Relation $\displaystyle \iff \ \$ $\displaystyle \left({y, x}\right)$ $\notin$ $\displaystyle \mathcal R$ Definition of Complement of Relation $\displaystyle \iff \ \$ $\displaystyle \left({x, y}\right)$ $\notin$ $\displaystyle \mathcal R^{-1}$ Definition of Inverse Relation $\displaystyle \iff \ \$ $\displaystyle \left({x, y}\right)$ $\in$ $\displaystyle \overline {\left({\mathcal R^{-1} }\right)}$ Definition of Complement of Relation

$\blacksquare$