# Mapping

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

mapping f of X to Y (f : X → Y assigns  to each element x in the domain X a value yin 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 yis 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.

• The mapping f is injective (one-to-one mapping) if different elements of the domain Xare 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.
• The mapping 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).