An application is a relation of a set E toward a set F so that all elements in set E have one and only one image in set F.

An application of a set E in itself is called a

**transformation**of set E.Synonym for function.

### Properties

**Bijective application**

Application that is both*injective*and*surjective*.

Synonym for**bijection**.**Injective application**

Application*f*of a set E toward a set F in which the distinct elements of the domain have distinct images.

Synonym for**injection**.**Surjective application**

Application*f*of a set E toward a set F in which the image is equal to the set of arrival F.

Synonym for**surjection**.

### Examples

- The relation of \(\mathbb {N}\) in \(\mathbb {N}\) under which every whole number
*x*is made to correspond to its double 2*x*is an injective application of \(\mathbb {N}\) in \(\mathbb {N}\), also known as an**injection**. - The relation of \(\mathbb {Q}\) in \(\mathbb {Q}\) under which every rational number
*x*is made to correspond to its half 0.5*x*is a bijective application of \(\mathbb {Q}\) in \(\mathbb {Q}\), also known as a**bijection**, because on the one hand, each rational number corresponds to its half, and on the other hand, each rational number is the half of another rational number. - The relation of \(\mathbb {Z}\) in \(\mathbb {N}\) under which every integer
*x*is made to correspond to its absolute value |x| is a surjective application of \(\mathbb {Z}\) in \(\mathbb {N}\), also known as a**surjection**, because every whole number is the image (the absolute value) of at least one integer.