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

You prove the first statement is false with a counter-example.  Consider the diagonal line in $\Bbb R\times\Bbb R$.  It is not of the form $A'\times B'$ because every point is of the form $(x,x)$.
The second is also false.  Consider $A=[0,1]$, $B=[1,2]$ and $C=B$ and $D=A$.  Then the LHS is a union of two squares of side length $1$ while the RHS is a square of side length $2$.
