A function is surjective (onto) if every possible image is mapped to by at least one argument. In other words, every element in the codomain has non-empty preimage. Equivalently, a function is surjective if its image is equal to its codomain. A surjective function is a surjection. The formal definition is the following.
- The function is surjective iff for all , there is such that f(a) = b.
- A function f : A → B is surjective if and only if it is right-invertible, that is, if and only if there is a function g: B → A such that f o g = identity function on B. (This statement is equivalent to the axiom of choice.)
- By collapsing all arguments mapping to a given fixed image, every surjection induces a bijection defined on a quotient of its domain. More precisely, every surjection f : A → B can be factored as a projection followed by a bijection as follows. Let A/~ be the equivalence classes of A under the following equivalence relation: x ~ y if and only if f(x) = f(y). Equivalently, A/~ is the set of all preimages under f. Let P(~) : A → A/~ be the projection map which sends each x in A to its equivalence class [x]~, and let fP : A/~ → B be the well-defined function given by fP([x]~) = f(x). Then f = fP o P(~). A dual factorisation is given for injections above.
- The composition of two surjections is again a surjection, but if g o f is surjective, then it can only be concluded that g is surjective. See the figure at right*.
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