Definition:Mapping

Definition
A mapping is a special kind of binary relation which relates any given element of one set to a unique element of another.

A mapping $f$ from $S$ to $T$ (or on $S$ into $T$), denoted $f: S \to T$, is a relation $f \subseteq S \times T$ such that:
 * $\forall x \in S: \forall y_1, y_2 \in T: \left({x, y_1}\right) \in f \land \left({x, y_2}\right) \in f \implies y_1 = y_2$

and
 * $\forall x \in S: \exists y \in T: \left({x, y}\right) \in f$

Thus, a mapping is a relation which is:
 * $(1): \quad$ Many-to-one
 * $(2): \quad$ Left-total, that is, defined for all elements in the domain.

Domain, Codomain, Image, Preimage
As a mapping is also a relation, all the results and definitions concerning relations also apply to mappings.

In particular, the concepts of domain and codomain carry over completely, as do the concepts of image and preimage.

Diagrammatic Presentation
The following diagram illustrates the mapping $f: S \to T$.


 * Mapping.png


 * $\operatorname{Dom} \left({f}\right)$ is the domain of $f$.
 * $\operatorname{Cdm} \left({f}\right)$ is the codomain of $f$.
 * $\operatorname{Im} \left({f}\right)$ is the image of $f$.

Mapping as Unary Operation
It can be noted that a mapping can be considered as a unary operation.

Notation
Let $f \subseteq S \times T$ be a mapping.

This is usually denoted $f: S \to T$, which is interpreted to mean:


 * $f$ is a mapping with domain $S$ and codomain $T$
 * $f$ is a mapping of (or from) $S$ to (or into) $T$
 * $f$ maps $S$ to (or into) $T$.

The notation $S \stackrel f {\longrightarrow} T$ is also seen.

For $x \in S, y \in T$, the usual notation is:


 * $f: S \to T: f \left({x}\right) = y$

where $f \left({x}\right) = y$ is interpreted to mean $\left({x, y}\right) \in f$.

It is read $f$ of $x$ equals $y$.

This is the preferred notation on.

Sometimes the brackets are omitted: $f x = y$, as seen in, for example.

The notation $f: x \mapsto y$ is often seen, read $f$ maps, or sends, $x$ to $y$.

Less common notational forms of $f \left({x}\right) = y$ are:


 * $x f = y$, as seen in and, for example
 * $x^f = y$, as seen in, for example
 * $f_x = y$, as remarked on in, for example.

Also known as
Words which mean the same thing as mapping include:
 * map
 * transformation (particularly in the context of self-maps)
 * operator
 * function (usually in the context of numbers)

Sources defining a mapping (function) to be only a many-to-one relation refer to a mapping (function) as a total mapping (total function).

Also see

 * Definition:Linear Transformation
 * Definition:Complex Transformation