Existence of Mappings Creating Identity Mappings

Table of Contents


Let’s introduce a fundamental theorem regarding the existence of mappings that create identity mappings.

Theorem

Let f be a mapping from A to B.

  1. If f is surjective, then and only then there exists a mapping s:BA such that fs=IB.
  2. If f is injective, then and only then there exists a mapping r:BA such that rf=IA.

Proof

  1. If fs=IB, then for any bB, f(s(b))=b. Therefore, f is surjective.

    Let f:AB be surjective. Define s(b) for bB such that s(b)f1(b). By the Axiom of Choice, such s(b) can be defined. Then, for any bB, f(s(b))=b.

  2. If rf=IA, then if f(a1)=f(a2) for any elements a1, a2 of A, it follows that a1=r(f(a1))=r(f(a2))=a2. Therefore, f is injective.

    Let f:AB be injective. For bB, if bf(A), there exists an element aA such that b=f(a). Define r using that a as r(b)=a. If bf(A), pick an element a0 from A and define r(b)=a0. Then, for any aA, r(f(a))=a.

Corollary

Let A and B be two sets. The necessary and sufficient condition for the existence of an injective mapping from A to B is the existence of a surjective mapping from B to A.

Proof

If there exists an injective mapping φ:AB, then by Theorem 2, a mapping ψ:BA exists such that ψφ=IA. ψ is surjective according to Theorem 1.

If there exists a surjective mapping ψ:BA, then by Theorem 1, a mapping φ:AB exists such that ψφ=IA. φ is injective according to Theorem 2.