Surjective function
Induced surjection and induced bijection

Any function induces a surjection by restricting its codomain to its range. Any surjective function induces a bijection defined on a quotient of its domain by collapsing all arguments mapping to a given fixed image. 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, and let f : A /~ → B be the well-defined function given by f () = f (x). Then f = f o P (~).
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