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.

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