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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.
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