# 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?

• It is just the set $\{(a,\emptyset)\mid a\in X_1\}$. This set is of course empty if $X_1$ is. In any case, there is an obvious bijection between it and $X_1$. – Andrés E. Caicedo Mar 24 '14 at 5:43

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 $\varnothing$.
As a clarifying example, take $$\{\emptyset\}\times\{\emptyset\}$$ It's the product of two sets with one element, and it's exactly $$\{(\emptyset, \emptyset)\}$$ with the standard representation of pairs ($(x,y) = \{\{x\},\{x,y\}\}$), it becomes $$\{ \{\{\emptyset\},\{\emptyset,\emptyset\}\} \}$$ which is the same as $$\{ \{\{\emptyset\}\} \}$$
• The second set is $\{\emptyset\}$, which is not empty. – Andrés E. Caicedo Mar 24 '14 at 5:45