The cartesian product is the product of two sets. The resulting product is a set of pairs, one element belonging to and one element belonging to .
This can expressed as:
So what happens if we try to take the cartesian product of a set and the empty set? ?
We end up with: . This is because we try to build a set like this: , but is empty.
It is impossible to create this resulting set of pairs because one of the sets is the empty set. As such, not a single element can be produced from , so our resulting set is the empty set.