Equivalence classes and Cardinal number

Question

For groups $A,B,C,D$, if $C\sim D,\ A\sim B$ and $A\cap C=B\cap D=\emptyset$, Prove that $A\cup C\sim B\cup D$.

I don't know how to approach this question. i thought about dividing the problem into cases. Would like some guidance. Thanks

Answer 1

Hint:

Let $f:A\to B$ and $g:C\to D$ be bijections and identify these functions with their graphs.

Then what can be said of $f\cup g$ under the given conditions?