Definition:Dual Relation

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This page is about Dual Relations. For other uses, see Definition:Dual.

Definition

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 $\mathcal R$.


Also see