The concept of a group is intimately related to the concept of a mapping or, rather, a set of mappings. We shall now introduce this concept (which is basic for much of modern mathematics) by considering some simple examples.

The word “mapping” ordinarily means “making a map of something”. The technical sense in which the word “mapping” is used in mathematics does not wander very far from this everyday meaning, in contrast to the usual situation where a borrowed word is given specialized mathematical meaning far removed from the sense of the original. For example, consider such concepts as group, field, ring.

The mathematical concept of a mapping is abstracted in a natural way from the ordinary notion of the map of a city. Ideally, such a map is a representation of the original object (a city) on a sheet of paper in such a way that every point of the original (city) has as its counterpart one (and only one) point on the paper. In all its ramifications, the mathematical concept of mapping never strays from this basic notion of correspondence between elements of the original and elements of the image, or map.

We begin our study of mapping by considering the simple case where we map a set with finitely many elements. Suppose we have a set X = {a, b, c} consisting of three elements, and a set Y = {r, s, t} consisting of three elements.