Proposition: Let [tex] \{A_n\}_{n\in I}[/tex] be a family of countable sets. Prove that(adsbygoogle = window.adsbygoogle || []).push({});

[tex] \bigotimes_{i=1}^n A_i[/tex]

is a countable set.

Proof:

Since [tex] \{A_n\}_{n\in I}[/tex]

are countable, there are 1-1 functions

[tex] f_n:A_n->J[/tex]

(J, the set of positive integers)

Now let us define a function

[tex] h:\bigotimes_{i=1}^n A_i->J[/tex]

such that

[tex] h(a_1,a_2,...,a_n)=(p_1)^{f_1(a_1)}*(p_2)^{f_2(a_2)}*...*(p_n)^{f_n(a_n)}[/tex]

Where p_i are prime numbers such that

[tex] 2\leq p_1<p_2<...<p_n[/tex]

Now from the Fundamental Theorem of Arithmetic,it is clear that h is a 1-1 function. So, based on some previous theorems, we conclude that

[tex]\bigotimes_{i=1}^n A_i[/tex]

is a countable set.

Is this correct?

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# Cartesian product of a family of countable sets is countable!

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