If A,B denumerable prove AUB denumerable

[jsmith0476]

Homework Statement

Prove that if A and B are denumerable sets, not necessarily disjoint, then AUB is denumerable.

Homework Equations

The Attempt at a Solution

So I started my saying that since A and B are denumerable then, f:N-->A and g:N-->B are bijections. The I defined h(n) in terms of f and g as such:
h(n) = f(n/2) if n is even
h(n) = g((n+1)/2) if n is odd
This is all I have so far. I am not sure where to go or what this tells me.

 

[Focus]

Try mapping the sets to the naturals, the maps have to be injective. A hint is that n->2n is an injection and composition of injective functions is injective.