Definition:Mapping

Definition
Let $S$ and $T$ be sets.

Let $S\times T$ be their cartesian product.

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.

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

Also known as
Words which mean the same thing as mapping include:
 * transformation (particularly in the context of self-maps)
 * operator
 * function (usually in the context of numbers)
 * map (but this term is discouraged, as the term is also used by some writers for planar graph).

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

The wording can vary.

A mapping $f$ from $S$ to $T$ is also described as a mapping on $S$ into $T$.

Also defined as
Some approaches, for example, define a mapping as a many-to-one relation from $S$ to $T$, and then separately specify its requisite left-total nature by restricting $S$ to the domain. However, this approach is sufficiently different from the mainstream approach that it will not be used on and limited to this mention.

Also see

 * Definition:Linear Transformation
 * Definition:Complex Transformation