Definition:Relation

Let $$S \times T$$ be the cartesian product of two sets $$S$$ and $$T$$.

A relation (in this context, technically speaking, a binary relation) in $$S$$ to $$T$$ is an arbitrary subset $$\mathcal{R} \subseteq S \times T$$.

What this means is that a binary relation "relates" (certain) elements of one set with (certain) elements of another. Not all elements need to be related.

When $$\left({s, t}\right) \in \mathcal{R}$$, we can write: $$s \mathcal{R} t$$.

If $$\left({s, t}\right) \notin \mathcal{R}$$, we can write: $$s \not \mathcal{R} t$$, that is, by drawing a line through the relation symbol. See Complement of Relation.

If $$S = T$$, then $$\mathcal{R} \subseteq S \times S$$, and $$\mathcal{R}$$ is referred to as a relation in $$S$$ or relation on $$S$$.