Remark

The reader should beware of the confusion which may arise from the simultaneous use of the notations $f(x)$ and $f(\mathrm{X})$ (synonymous with $f\langle \mathrm{X} \rangle$) introduced in Definition 3 (cf. Exercise 11).

Let A and B be two sets; a mapping of A into B is a function $f$ whose source (which is equal to its domain) is equal to A and whose target is equal to B; such a function is also said to be defined on A and to take its values in B.

Instead of the phrase "let $f$ be a mapping of A into B", the following phrases are often used : "let $f : \mathrm{A} \to \mathrm{B}$ be a mapping" or even "let $f : \mathrm{A} \to \mathrm{B}$". To simplify the presentation of an argument involving several mappings, we use diagrams such as

in which a group of signs such as $\mathrm{A} \overset{f}{\to} \mathrm{B}$ is to be interpreted as meaning that $f$ is a mapping of A into B.

A function $f$ defined on A is said to transform x into $f(x)$ (for all $x \in \mathrm{A}$); $f(x)$ is called the transform of x by f or (by abuse of language) the image of $x$ under $f$.

¶ Under certain circumstances, a functional graph is called a family; the domain is then called the index set, and the range is called (by abuse of language) the set of elements of the family. It is mainly in this connection that the indicial notation $f_x$ is used to denote the value of $f$ at the element $x$. When the index set is the product of two sets, we often speak of a double family.

¶ Likewise a function whose target is E is sometimes called a family of elements of E. If every element of E is a subset of a set F, we speak of a family of subsets of F.

Throughout this series we shall often use the word “function” in place of “functional graph”.

Examples of functions