TITLE: What is the cross-product of the null set with another set?
QUESTION [9 upvotes]: Supposing that, for example, we have two sets $A$ and $B$ where $\;A = \varnothing \;$ and $\,B = \{a,b\}$.
What is the result of the cross product of those sets?
My first intuition would be to say that the resulting set would be the empty set as well since you can't set up any ordered pairs of $B$.
Am I correct?

REPLY [6 votes]: Yes, you are correct.
The cartesian product is defined as $$A \times B = \{(a,b) \mid a\in A, b\in B\}.$$
In the case that one or both of the sets $A$ and $B$ are empty, there is no single index pair $(a,b)$ such that $a\in A$ and $b\in B$, which means $A\times B = \emptyset$.