Surjektif

Surjektif, or surjective, is a property of a function in mathematics. Let f: A → B be a function from a set A (the domain) to a set B (the codomain). f is surjektif if every element of B is the image of at least one element of A. In other words, the image of f equals the codomain B.

This concept contrasts with injective (one-to-one) functions, where distinct elements of A map to distinct elements of B. A function can be surjektif without being injective; several elements of A may share the same output in B.

Examples:

- The function f: R → R defined by f(x) = x^3 is surjektif, since for any y in R there exists x with f(x) = y.

- The function g: R → R defined by g(x) = e^x is not surjektif onto R (its image is (0, ∞)); if the codomain is (0, ∞), then g is surjektif.

- A finite example: f: {1, 2, 3} → {a, b} with f(1) = a, f(2) = b, f(3) = b is surjektif because both a and b occur as outputs.

Properties:

- Surjectivity is equivalent to the existence of a right inverse: there exists h: B → A such that f(h(b)) = b for all b ∈ B.

- If A and B are finite and f is surjektif, then |A| ≥ |B|; if |A| = |B|, surjectivity implies bijectivity (both surjective and injective).

Applications:

- Surjective mappings are described as onto, and they model situations where every possible result in the codomain is attained by some input.