Intro

An n-ary relation is a subset of the cartesian product of $ n $ sets. Most common are binary relations. Given a set $ A $ and a set $ B $ we can form a binary relation $ R $ as a subset of $ A \times B $. For example if we have $$ A = \{1,2,3\} $$ and $$ B = \{a,b\} $$ we can form the following relations $$ R_1 = \{ (1,a),(2,a) \} $$ or $$ R_2 = \{ (1,a),(2,b),(3,a) \} $$ However the relation $$ R_{not} = \{ (1,1),(a,1) \} $$ is not a subset of $ A \times B $ since $ (1,1) \not \in A \times B $ and $ (a,1) \not \in A \times B $.

For a relation, $ R $, we tend to describe that two elements, $ a,b $, are related by $ a\ R\ b $ - meaning $ a $ is related to $ b $ by relation $ R $. The order is important. $ a\ R\ b $ doesn't necessarily imply that $ b\ R\ a $.