Cartesian Product of sets?

Question

I am try to find the cartesian product of 5 small sets, if one of these sets is the empty set, does this make the cartesian product the empty set too?

Answer 1

Yes. The Cartesian product of n sets is simply the set of n-tuples where the kth element is an element of the kth set. Suppose then that, say, the first set is empty. The first element of any element of the cartesian product would have to be an element of the first set, which is impossible, because there are no elements in the first set. It follows that there are no elements in the cartesian product.

It is useful to remember that the number of elements of the product of n sets is simply the product of the number of elements in each set. If one of those numbers is 0, then the entire product will be 0, so there will be no elements in the cartesian product.