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.