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# Mappings

Definition 5.1.1   Let and be two sets. A mapping from to 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 .

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.

Example 5.1.2
• 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 .
• Let 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 to .

Definition 5.1.3 (Composition of mappings)   Let , and be three sets and let and be two mappings. The composition of by is the mapping denoted by such that

Example 5.1.4   Let and be two mappings from to given respectiveley by and . Then we have:

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 and be two mappings from to itself. 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 , hence .

Next: Injections Up: Mappings Previous: Mappings   Contents
root 2002-06-10