# Mapping

### From Online Dictionary of Crystallography

Transformation (*Fr*). Abbildung (*Ge*). Trasformazione (*It*). 写像 (*Ja*). Transformación geométrica (*Sp*).

The term **mapping** is often used in mathematics as a synonym of *function*. In crystallography it is particularly used to indicate a *transformation*.

## Domain, image and codomain

A**mapping**

*f*of

*X*to

*Y*(

*f*:

*X*→

*Y*) assigns to each element

*x*in the

**domain**

*X*a value

*y*in the

**codomain**

*Y*.

The set of values *f*(*X*) = { *f*(*x*) : *x* in *X* } is the **image** of the mapping. The image may be the whole codomain or a proper subset of it.

For an element *y* in the image of *f*, the set { *x* in *X* : *f*(*x*) = *y* } of elements mapped to *y* is called the **preimage** of *y*, denoted by *f*^{−1}{*y*}. Also, the single elements in *f*^{−1}{*y*} are
called **preimages** of *x*.

## Surjective, injective and bijective mappings

The mapping*f*is

**surjective**(

**onto**) if the image coincides with the codomain. The mapping may be

**many-to-one**because more than one element of the domain

*X*can be mapped to the same element of the codomain

*Y*, but every element of

*Y*has a preimage in

*X*. A surjective mapping is a

**surjection**.

*f*is

**injective**(

**one-to-one mapping**) if different elements of the domain

*X*are mapped to different elements in the codomain

*Y*. The image does not have to coincide with the codomain and therefore there may be elements of

*Y*that are not mapped to some elements of

*X*. An injective mapping is an

**injection**.

*f*is

**bijective**(

**one-to-one correspondence**) if and only if it is both injective and surjective. Every element of the codomain

*Y*has exactly one preimage in the domain

*X*. The image coincides with the codomain. A bijective mapping is a

**bijection**.

If the codomain of an injective mapping *f* is restricted to the image *f*(*X*), the resulting mapping is a bijection from *X* to *f*(*X*).