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?
Best Answer
You know that $$|A \cup C| = |A|+|C|-|A \cap C|.$$ Similarly, $$|B \cup D| = |B|+|D|-|B \cap D|.$$ Also, since $A,B,C,D$ are pairwise disjoint, the intersection of any two of them is the empty set. Hence, $$|A \cap C| = |B \cap D| = 0.$$
Now, do you see how to prove that $|A \cup C| = |B \cup D|$?