Prove or disprove: If there is a injection f : A →B and a surjection g : A → B, then there is a bijection h : A → B.
This is true. Here is an outline of a proof. Define a function g' : B → A as follows. For each b ∈ B, choose an element xb ∈ g⁻¹({x}). (That is, choose an element xb ∈ A for which g(xb) = b). Now let g' : B → A be the function defined as g’(b) = xb. Chech that g’ is injective and apply the Cantor-Bernstein- Schroeder theorem