Why is $(\cup S) \times (\cup T)$ only a subset of and not equal to $ \cup \{X \times Y | X \in S, Y \in T \} $ for all sets S, T of sets I'm trying to self study some basic set theory but can't understand why this would only be a subset and not equal to where
$$(\cup S) \times (\cup T) \subseteq \cup \{X \times Y | X \in S, Y \in T \} $$
for all sets S, T of sets. I've been thinking about it for a couple of days but just don't get it. Any help would be greatly appreciated. (Question from The Foundations of Mathematics by Stewart and Tall)
 A: Let $A=(\bigcup S)\times(\bigcup T)$ and $B=\bigcup\{X\times Y:X\in S,Y\in T\}$.
Take $X\in S$ and $Y\in T$. Since $X\subseteq \bigcup S$ and $Y\subseteq\bigcup T$, we have that
$$
X\times Y\subseteq A
$$
Therefore $B\subseteq A$.
Suppose $(u,v)\in A$. Then $u\in\bigcup S$ and therefore $u\in X$ for some $X\in S$; similarly $v\in Y$, for some $Y\in T$. Thus $(u,v)\in X\times Y$ and $X\times Y\subseteq B$. Hence $A\subseteq B$.
I proved that $A=B$, so the book authors would not state they are different.
A: They are equal, even if your books claims they are not.
For all $X \in S$ we have $X \subset S^* = \bigcup S$ and For all $Y \in T$ we have $Y \subset T^* = \bigcup T$. Thus all $X \times Y \subset S^* \times T^*$ and therefore
$$\bigcup\{X \times Y \mid X \in S, Y \in T\} \subset S^* \times T^* .$$
Now let $(s,t) \in S^* \times T^*$. Then $s \in X$ for some $X \in S$ and $t \in Y$ for some $Y \in T$. Thus $(s,t) \in X \times Y$ and therefore
$$S^* \times T^*  \subset  \bigcup\{X \times Y \mid X \in S, Y \in T\} .$$
