subsets in the cartesian product Let $A,B,C,D$ be sets. Consider $A\times B$ and $X\subseteq A\times B$. Is it true that $X$ has the form $A'\times B'$ where $A'\subseteq A$ and $B'\subseteq B$ ?
At the same time is it true that $(A\times B)\cup (C\times D)=(A\cup C)\times (B\cup D)$?
For me are both false but I don't know how to prove them.
thank you

For the first, let $A=\{1,2\}, B=\{a,b\}, X=\{(1,a),(2,b)\}$  For the second, note that the pairs on the left have a first element that might come from $B$ while the pairs on the right have a first element that might come from $C$.  Maybe, as in my first example, they are different things.
