As for general binary relations, is called the domain of the mapping and is the range of the mapping.
Usually, mappings are denoetd in another way: the mapping is denoted by a single letter
, and the triple is displayed as in the following diagram:
The element of is denoted by and is called the image of by the mapping . If , the element in is called a pre-image of by .
For example, let
such that for any
, . Then is the image of by , has two pre-images by , namely and , and has no pre-image by .
Figure 1 displays diagrams of binary relations. The first diagram determines a mapping, as it fulfills the requirements of Definition 1.1. The second one does not determine a mapping, as there exists at least one element in the domain without an image; the third one does not determine a mapping, as there exist a domain element related to two different elements in the range.
be two sets. A mapping from
is a binary relation, whose graph
verify the following condition:
i.e. a mapping establishes a correspondence between the elements of
and the elements of
such that every element of
appears in a single pair in
Mapping or not.
- Let be the set of students in a classroom; the correspondence which assigns to each student his left neighbour is not a mapping from to , because at the left end of a row, there is a student who has no left neighbour (i.e. there exists at least an element of which can have no image in this correspondence).
- Let be the set of inhabitants of a given town and be the set of all blood types (O+, O-, A+, etc.). When we assign to each element of his/her blood type, we define a mapping
be the set of all the natural numbers. Let us assign to each natural number its double , we define a mapping
- The correspondence
determines a mapping from
(Composition of mappings)
be three sets and let
be two mappings. The composition of
is the mapping denoted by
Composition of two mappings.
In general, we cannot compute both compositions and (if the given sets are all different), but even when both compositions are defined, they are generally different. For example, let
be two mappings from
to itself. Then we have:
be two mappings from
given respectiveley by
. Then we have:
The mappings and have the same domain and the same range, but their graphs are different (i.e. the correspondences they define are different). For example, we have and