Relation

Relation

## Definition / Concept

A relation from $A$ to $B$ is any subset of the Cartesian product $A\times B$.

## Formula

$$R\subseteq A\times B$$

If:

$$(a,b)\in R$$

then $a$ is related to $b$ under $R$.

## Terminologies

- Domain of relation → Set of all first components of ordered pairs in $R$.

- Codomain → The set $B$ to which the relation is defined.

- Range → Set of all second components that actually occur in $R$.

## Formula

$$\text{Domain}(R)=\{a\in A:\exists b\in B,\ (a,b)\in R\}$$

$$\text{Range}(R)=\{b\in B:\exists a\in A,\ (a,b)\in R\}$$

$$\text{Range}(R)\subseteq B$$

## Usage

- Forms the foundation for the study of functions.