surjectie

Surjectie, or surjection, is a concept from mathematics describing a specific kind of function. A function f from a set A to a set B is surjective (onto) if every element of B is the image of at least one element of A. In other words, the range of f is all of B; for every y in B there exists x in A with f(x) = y.

Consequences and terminology: For a surjective function, the image equals the codomain, and the preimage (fiber) of each y in B is nonempty. Surjectivity is sometimes described as “covering” the codomain. A function can be surjective without being injective (one-to-one); a function that is both surjective and injective is bijective and thus has a two-sided inverse.

Examples:

- f: R -> R defined by f(x) = x^3 is surjective, since every real y has a real cube root x with x^3 = y.

- f: R -> R defined by f(x) = e^x is not surjective onto R; its image is (0, ∞).

- 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.

Notes: Surjectivity depends on the choice of codomain. If you restrict the codomain to the actual image, the function becomes surjective by construction. In finite sets, a surjection from A to B is possible only if |A| ≥ |B|; in particular, a surjection cannot exist from a smaller to a larger finite set. In the context of sets, surjections are epimorphisms.