EDB — 2BX

Exercises

1. [2BX]Let \(A,B\) be non-empty sets.

   • Suppose that \(f:A→ B\) is an injective function: there exists a surjective function \(g:B→ A\) such that \(g◦ f=\text{Id}_ A\) (the identity function). (Such \(g\) is a left inverse of \(g\)).

   • Suppose that \(g:B→ A\) is a surjective function: there exists a injective function \(f:A→ B\) such that \(g◦ f=\text{Id}_ A\). (Such \(f\) is a right inverse of \(g\)).

   The proof of the second statement requires the Axiom of Choice (see [2BZ]↺↻).

   Vice versa.

   • If \(f:A→ B\) has a left inverse then it is an injective function.

   • If \(g:B→ A\) has a right inverse, then it is a surjective function.

   Solution 1

   [2BY]↺↻