Suppose we have sets and and a function (that is, ’s domain is and its codomain is ). Suppose there is another function such that for every . Is necessarily a bijection? That is, does necessarily match up each element of with a unique element of and vice versa? Or put yet another way, is necessarily invertible?
- If yes, prove it!
- If no, provide a counterexample! For bonus points, what additional assumptions could we impose to make it true?