surjectivity

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. Equivalently, the image of f equals the codomain B.

A surjection can have more than one preimage for a given element of B; surjectivity does not require injectivity.

Examples include f(x) = x^3 from R to R, which is surjective; and f(x) = e^x from R to (0, ∞), which is surjective onto its codomain but not onto R. The map g: Z -> Z given by g(n) = n + 1 is also surjective.

Finite sets: If A and B are finite and f: A -> B is surjective, then |A| ≥ |B|; if f is also injective, then |A| = |B| and f is bijective. In general, a surjection f: A -> B has a right inverse g: B -> A with f(g(b)) = b for all b in B, provided one can select a preimage for each b (axiom of choice).

In linear algebra, a linear map T: V -> W is surjective when its image equals W. By rank-nullity, this occurs when rank(T) = dim W and dim V ≥ dim W.

Composition: If f: A -> B and g: B -> C are surjective, then the composition g(f(x)) from A to C is surjective. Surjectivity is a property preserved under composition, and together with injectivity characterizes bijections.