# Application

## Application

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 2x 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 0.5x 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.