Surjektiver

Surjektiver is the plural form of surjektiv in Norwegian, Danish and Swedish, used to describe onto mappings in mathematics. In English, the corresponding term is surjection (or onto function). The concept is named from the French surjectif.

A function f: X → Y is surjective if every element y in Y has at least one preimage x in X such that f(x) = y. Equivalently, the image of f equals the entire codomain Y. Surjectivity concerns the coverage of Y by the outputs of f, not the distinctness of those outputs.

Examples help illustrate the concept. The function f(x) = x^3 from the real numbers to the real numbers is surjective, since every y has a real cube root. The exponential function f(x) = e^x from R to (0, ∞) is surjective onto (0, ∞) but not onto R. A function f: {1, 2} → {a, b} with f(1) = a and f(2) = a is not surjective, because b is not attained.

In finite settings, surjectivity implies a larger or equal domain than codomain: |X| ≥ |Y|, and equality |X| = |Y| holds precisely when the surjection is a bijection (a one-to-one correspondence). In linear algebra, a linear map T: V → W is surjective if its image equals W; for finite-dimensional spaces this means rank(T) = dim(W).

Surjectivity is one of the two main properties paired with injectivity to form bijections. It is central to questions of solvability, coverage of codomains, and the structure of mappings in various areas of mathematics. See also: injective, bijective, function, image, preimage.