A function f:AB is surjective [or onto] if for every element b of the set B, there is an element a in the set A such that f(a) = b. Compare with partial function.

The interesting thing about surjectivity is that the range of the function [in the example f:AB, the range is B] is considered to be part of the definition of the function. This is not always immediately obvious. For example, the function sin:ℜ → ℜ [where ℜ is the set of real numbers] is not surjective, but the function sin:ℜ → {x : (x ∈ ℜ) ∧ (x ≥ -1) ∧ (x ≤ 1)} is surjective. In general, for every function f, there is another function g which is identical except the range is limited to the actual results of the function, and so is essentially the same function but is always surjective.

Log in or registerto write something here or to contact authors.