surjekive

Surjective, also known as onto, is a property of functions in mathematics. A function f from a set A to a set B is surjective if for every element y in the codomain B, there exists at least one element x in the domain A such that f(x) = y. In simpler terms, every element in the target set B is mapped to by at least one element from the source set A. This means that the range of the function is equal to its codomain.

To illustrate, consider a function f mapping integers to integers. If f(x) = x^2, this function is not surjective because there is no integer x for which f(x) = -1. However, if we consider a function g mapping integers to non-negative integers, where g(x) = |x|, then g is surjective because for every non-negative integer y, there exists an integer x (either x = y or x = -y) such that g(x) = y.

A function that is both injective (one-to-one) and surjective is called bijective. Bijective functions have a unique inverse function. The concept of surjectivity is fundamental in set theory and various branches of mathematics, including abstract algebra and topology. It helps classify functions based on how they distribute elements from their domain to their codomain.