If $\displaystyle \kappa, \lambda, \mu$ be three cardinals. Prove that $\displaystyle \kappa(\lambda + \mu) = \kappa \lambda + \kappa \mu$.

Thoughts...

My basic problem is that I have never really understood how to construct in/sur/bijections so all this cardinal number stuff is proving to be a bit hard...

But my thoughts on this are that...

Let $\displaystyle \kappa, \lambda, \mu$ be the cardinal numbers of the sets A,B and C respectively such that B and C are disjoint. Then $\displaystyle A \times B$ and $\displaystyle A \times C$ are disjoint and $\displaystyle A \times (B \cup C) = A \times B \cup A \times C$.

I'm not sure whether I'm allowed to assume the disjoint part and how to construct a bijection between the two...