I came across the following question while studying.
Let $A,B,C,D$ be pairwise disjoint sets. Prove that if $|A| = |B|$ and $|C| = |D|$ then $|A \cup C| = |B \cup D|$.
I thought of the fact that they intersections are obviously empty but this doesn't help with the progression to a solution. I also tried to find any properties of the pairwise disjoint union that might help but I am stuck. Can anyone offer some suggestions/solutions?