What is the cartesian product of a non empty set and a set containing an empty set (or $X_1 \times \{ \emptyset \}$ )? Let $X_1$ be a set. What is $X_1 \times \{ \emptyset \}$? I know that a the product of a set and an empty set is an empty set, but what is the product of a set and an empty set WITHIN a set?
The empty set, considered as an element of another set, is nothing special!  It is just an element.  You could do the problem by finding
$$X_1\times\{\,a\,\}$$
and then replacing $a$ by $\varnothing$.
Still empty. Cartesian product does not care about where your set lives. It's the set of all ordered pairs taken from the two sets, and when one of them is empty, there are no such pairs.