surjektive

Surjektive, or surjective, refers to a property of functions in mathematics. A function f from a domain X to a codomain Y is called surjective if every element of Y is the image of at least one element of X. In formal terms, for every y in Y there exists x in X with f(x) = y. Equivalently, the image of X under f equals the entire codomain Y, i.e., f(X) = Y.

Surjectivity is often discussed together with injectivity. A function can be surjective without being injective, meaning multiple elements of X may map to the same element of Y. Conversely, a function is injective if different inputs produce different outputs, and it is bijective when it is both injective and surjective, establishing a one-to-one correspondence between X and Y.

Common examples illustrate the concept. The function f: R -> R defined by f(x) = x^3 is surjective, since every real y has a real x with x^3 = y. The function f: R -> R defined by f(x) = e^x is not surjective because negative numbers are not in the image. A finite example: f: {1,2,3} -> {a,b} with f(1)=a, f(2)=b, f(3)=b is surjective, as both a and b are hit, but it is not injective.

Surjectivity also has implications in higher mathematics. A surjective function has a right inverse (a function g: Y -> X with f(g(y)) = y for all y in Y) provided a choice function is available. In category theory, surjections in sets correspond to epimorphisms.