# Relations (Sets)

##### Updated: October 21, 2020

## Definition #

A *relation* \(R\) from the elements of set \(A\) to the elements of set \(B\) **is a subset** of \(A \times B\).

…alternatively…

Let \(A\) and \(B\) be two non-empty sets, then every subset of \(A \times B\) **defines** a relation from \(A\) to \(B\) and ever relation from \(A\) to \(B\) is a subset of \(A \times B\).

Let \(R \subseteq A \times B\) and \((a, b) \in R\). Then we say that \(a\) is related to \(b\) by the relation \(R\) and write it as \(a R b\). If \((a, b) \in R\), we write it as \(a R b\)