Prove that the Cartesian product of a finite number of countable sets is countable?

Problem

We have to prove that the cartesian product of a finite number of countable sets is countable.

Solution

Let the X1, X2 ,…….. Xn be the countable sets.

Yk= X1 * X2 * …….* Xk when k =1……. N). Thus,

Yn := X1 * X2 * · · · * Xn

Proof

Using the induction −

In case k = 1 then Y1 = X1 is countable.

Assuming that Yk (k ∈ n, 1 ≤ k < n) is countable;

Then Yk+1 = ( X1 * X2 * …….* Xk) * Xk+1 = Yk * Xk+1 where the Yk and the Xk+1 can be called countable. Hence the cartesian product of the countable set is always countable. So, Yk+1 is countable.

In the same way, let's prove that the cartesian product of a finite number of countable infinite sets is countably infinite.

Proof

Let the X1, X2 ,…….. Xn be the countable infinite sets.

Define Yk= X1 * X2 * …….* Xk when k =1……. N). Thus

Thus, Yn := X1 * X2 * · · · * Xn

First we need to prove that Yn is countable.

By induction method

If k=1, then the set Y1=X1 is countable infinite.

Assume that Yk( K☐N, 1<=K<N) is countable infinite.

Then,

Yk+1=(X1 * X2 *....Xk) * Xk+1

Yk+1=Yk * Xk+1

Where, Yk and Xk+1 are both countable infinite, we know that the cartesian product of countable sets is countable.

Therefore, Yk+1 is countable infinite.

We concluded that X1 * X2 *.....Xn is countable infinite.

Therefore, the cartesian product of a finite number of countable infinite sets is countable infinite.