surjectif

Surjectif is a term used in mathematics to describe a particular property of a function. A function f: A -> B is surjective (onto) if every element of the codomain B is the image of at least one element of the domain A. In formal terms, for every y in B there exists x in A such that f(x) = y. If there exists a y in B with no such x, the function is not surjective.

Examples help illustrate the concept. The function f: R -> R defined by f(x) = x^3 is surjective because every real number y has a real cube root x with f(x) = y. By contrast, g: R -> R with g(x) = x^2 is not surjective, since negative numbers in R have no real preimage. A function h: R -> [0, ∞) given by h(x) = x^2 is surjective onto [0, ∞).

Surjectivity is distinct from injectivity (one-to-one) and from bijectivity (both injective and surjective). A surjective function may fail to be injective, and a bijection has a two-sided inverse. In many contexts, a surjective map has a right inverse: a function s: B -> A with f(s(y)) = y for all y in B, a fact that may require the axiom of choice to guarantee in general.

In finite settings, if f: A -> B is surjective, then the cardinality of A is at least that of B (|A| ≥ |B|). Surjections are common in various areas of mathematics, including algebra, topology, and analysis, and they are often described using the term "onto" in English or "surjection" as a noun.