TITLE: Is the union of two cartesian products equal to the product of their unions?
QUESTION [5 upvotes]: Can we prove that
$$(A \times B)\cup (C \times D) = (A \cup C) \times (B \cup D)  \;?$$
If my understanding is correct, we cannot prove because we do not know if $A \times B$ and $C \times D$ share common elements or not. Please tell me if I'm wrong.

REPLY [6 votes]: Hint: What do you get for $A=\{1\}$, $B=\{2\}$, $C=\{3\}$, $D=\{4\}$?
(Which is almost the simplest case you can think of.)