surjects

Surjects is the verb form related to surjection in mathematics. A function f from a set A to a set B surjects onto B if every element of B is the image of some element of A. When this holds, f is called surjective or onto; equivalently, the image or range of f equals the codomain B. If some element of B has no preimage in A, f is not surjective and its image is a proper subset of B.

A concrete way to think about surjectivity is through preimages: for each b in B, the set f^{-1}(b) is nonempty. Surjectivity is thus about covering the entire codomain with outputs from A. Important special cases include finite sets: if A and B are finite and f is surjective, then |A| ≥ |B|, and if |A| = |B| and f is surjective, then f is bijective.

Examples clarify the concept. The function f: R -> R given by f(x) = x^3 is surjective, since every real number y has a real x with x^3 = y. The function f: R -> (0, ∞) given by f(x) = e^x is not surjective onto R; its image is (0, ∞). A function f: {1,2,3} -> {a,b} with f(1)=a, f(2)=b, f(3)=a is surjective because both a and b are attained.

In broader contexts, surjectivity is a standard property of morphisms in algebra and topology; a map is surjective if and only if it has a right-inverse with respect to composition, and surjective maps are often contrasted with injective (one-to-one) maps.