Relations

Definition: Relation

A relation $R\subseteq A\times B$ between two sets $A$ and $B$ is any subset of the Cartesian product $A\times B$.

NOTATION

For any $a\in A$ and any $b\in B$, if $(a;b)\in R$, then we write

$$aRb$$

INTUITION

The statement $aRb$ translates to “there is a relationship between $a$ and $b$ which is expressed by $R$“.

EXAMPLE

Let $R\subseteq \mathbb{N}\times \mathbb{N}$ be the relation

$$R=\{(a;b)\mid a,b\in \mathbb{N}\text{ and }b\text{ is a divisor of a}\}$$

For any $x,y\in \mathbb{N}$, the statement $xRy$ means that $(x,y)\in R$ which in this case translates to ”$y$ is a divisor of $b$“.