Surjektio

Surjektio, in mathematics often called a surjection or onto function, is a type of function between sets that covers its codomain. If f is a function from a domain A to a codomain B, f is surjective when every element of B is the image of at least one element of A.

Formally, f: A -> B is surjective if for every b in B there exists an a in A with f(a) = b. Equivalently, the image of f equals B. Surjective functions are not required to be injective; different elements of A may map to the same element of B.

Examples:

- The function f: Z -> Z given by f(n) = n is surjective (and in fact bijective).

- The function f: R -> R given by f(x) = x^3 is surjective; every real number has a real cube root.

- The function f: Z -> Z given by f(n) = n^2 is not surjective onto Z (negative integers are not attained), but it is surjective onto the nonnegative integers.

- The function f: R -> {0,1} defined by f(x) = 0 if x < 0 and f(x) = 1 if x ≥ 0 is surjective onto {0,1}.

Properties:

- The surjectivity of a composition: if f: A -> B and g: B -> C are surjective, then g ∘ f: A -> C is surjective.

- In finite sets, a surjective f: A -> B implies |A| ≥ |B|; if f is also injective, then |A| = |B| and f is bijective.

Usage:

Surjectivity is a fundamental notion in algebra and analysis, often used to describe when every target element can be reached by the function, and it plays a key role in solving equations and in constructing quotient structures.