{| class="wikitable floatright" style="text-align:center"

|

! scope="col" | surjective

! scope="col" | non-surjective

|-

! scope="row" | injective

|frameless|alt=The sets X = {1, 2, 3, 4} and Y = {A, B, C, D}, and a function mapping 1 to D, 2 to B, 3 to C, and 4 to A.|150x150px

bijective

|frameless|alt=The sets X = {1, 2, 3} and Y = {A, B, C, D}, and a function mapping 1 to D, 2 to B, and 3 to A.|150x150px

injective-only

|-

! scope="row" | non-

injective

|frameless|alt=The sets X = {1, 2, 3, 4} and Y = {B, C, D}, and a function mapping 1 to D, 2 to B, 3 to C, and 4 to C.|150x150px

surjective-only

|frameless|alt=The sets X = {1, 2, 3} and Y = {A, B, C, D}, and a function mapping 1 to B, 2 to B, and 3 to A.|150x150px

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.

A function maps elements from its domain to elements in its codomain. Given a function <math>f \colon X \to Y</math>:

  • The function is injective, or one-to-one, if each element of the codomain is mapped to by at most one element of the domain, or equivalently, if distinct elements of the domain map to distinct elements in the codomain. An injective function is also called an injection. Notationally:

::<math>\forall x, x' \in X, f(x) = f(x') \implies x = x',</math>

:or, equivalently (using logical transposition),

::<math>\forall x,x' \in X, x \neq x' \implies f(x) \neq f(x').</math>

  • The function is surjective, or onto, if each element of the codomain is mapped to by at least one element of the domain; that is, if the image and the codomain of the function are equal. A surjective function is a surjection.

History

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).

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.

See also

  • Horizontal line test
  • Injective module
  • Permutation

References