# Cartesian product proof

I need help constructing a proof for the

Cartesian product of finitely many countable sets is countable.

https://brainmass.com/math/discrete-math/cartesian-product-proof-13338

#### Solution Preview

First let's state the counting theorem:

"The Counting Theorem":

Let S be a set. The following are equivalent.

(i) S is either finite or a countably infinite set.

(ii) Either S = Null or there is a surjective map alpha: N ---> S.

(iii) There is an injective map beta: S ---> N.

If any of these three equivalent conditions hold, then we say

that S is a countable set.

This theorem has a ...

#### Solution Summary

This is a proof regarding the Cartesian product of countable sets.

$2.19