[Math] the cartesian product of a non empty set and a set containing an empty set (or $X_1 \times \{ \emptyset \}$ )

Question

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?

Answer 1

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 $\varnothing$.