Bijection, injection and surjection

bijective injective-only injective surjective-only general In mathematics, injections, surjections, and bijections are classes of functions distinguished by the manner in which arguments (input expressions from the domain) and images (output expressions from the codomain) are related or mapped to each other.

: An injective function need not be surjective (not all elements of the codomain may be associated with arguments), and a surjective function need not be injective (some images may be associated with more than one argument).

The four possible combinations of injective and surjective features are illustrated in the adjacent diagrams.

A function is injective (one-to-one) if each possible element of the codomain is mapped to by at most one argument.

Equivalently, a function is surjective if its image is equal to its codomain.

A function is bijective if and only if every possible image is mapped to by exactly one argument.

[1] This equivalent condition is formally expressed as follows: The following are some facts related to bijections: Suppose that one wants to define what it means for two sets to "have the same number of elements".

In the category of sets, injections, surjections, and bijections correspond precisely to monomorphisms, epimorphisms, and isomorphisms, respectively.

[5] The Oxford English Dictionary records the use of the word injection as a noun by S. Mac Lane in Bulletin of the American Mathematical Society (1950), and injective as an adjective by Eilenberg and Steenrod in Foundations of Algebraic Topology (1952).

[6] However, it was not until the French Bourbaki group coined the injective-surjective-bijective terminology (both as nouns and adjectives) that they achieved widespread adoption.