Fundamentals of Mathematics: Construct a bijection

[icysruf110]

So here is the question i have been working like a week on and have gotten almost no where. Let f map from X to Y and be one to one and let g map B to A and also be one to one, however neither a necessarily onto. Construct by means of f and g a bijection from X to Y.

 

[pka]

This is a very famous and difficult theorem known as Cantor-Bernstein-Schroeder theorem.
Here is a fairly good proof. There are many such proofs.
http://en.wikipedia.org/wiki/Cantor–Bernstein–Schroeder_theorem