Definition 4.1.1
An ordered triple of sets
where
is
called a binary relation between the elements of and the
elements of . The set is called the domain of the
relation, is the range and is the graph of
the relation.

If
, the relation is called the empty relation.
For example, let be the set of children who learn in a given institution; suppose that the set
is made of all the pairs
of children where is a friend of . If
, 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
being displayed. For example, let
and
. If
, 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
to for every pair in
, as in
Figure 1(b).

Notation 2
A binary relation
is often denoted by a single letter (or symbol) like
,
, etc. With this notation we write
instead of
.