The general case

Definition 4.1.1   An ordered triple of sets $(A,B,\mathcal{G})$ where $\mathcal{G} \subset A \times B$ is called a binary relation between the elements of $A$ and the elements of $B$. The set $A$ is called the domain of the relation, $B$ is the range and $G$ is the graph of the relation.

If $\mathcal{G}=\emptyset$, the relation is called the empty relation. For example, let $A$ be the set of children who learn in a given institution; suppose that the set $\mathcal{G} \subset A \times A$ is made of all the pairs $(a,b)$ of children where $a$ is a friend of $b$. If $\mathcal{G}=\emptyset$, the social situation in this institution is very serious and needs help. The same kind of graphical representation we used for the cartesian product of two sets can be used here too, no point other than those of $\mathcal{G}$ being displayed. For example, let $A==\{ a,b,c,d,e \}$ and $B= \{ 1,2,3,4 \}$. If $\mathcal{G}= \{ (a,2),(a,4),(b,1),(c,3),(d,3),(d,4) \}$, the corresponding diagram is displayed on Figure 1(a).

Figure 1: Diagrams of a relation.

Sometimes another kind of diagram is useful: we represent the domain and the range of the relation by Venn diagrams, then we draw an arrow from $a$ to $b$ for every pair $(a,b)$ in $\mathcal{G}$, as in Figure 1(b).

Notation 2   A binary relation $(A,B,\mathcal{G})$ is often denoted by a single letter (or symbol) like $\mathcal{R}$, $\mathcal{S}$, etc. With this notation we write $a \mathcal{R} b$ instead of $(a,b) \in \mathcal{G}$.