Proving the Countable Union Theorem for Sets with a Prime Mapping Approach

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SUMMARY

The Countable Union Theorem states that the union of countable sets A_1, A_2, ..., A_n is countable. The discussion proposes a prime mapping approach where each element of A_1 is assigned to the first prime, with subsequent sets A_2, A_3, ..., A_n mapped to the second, third, and nth primes respectively. By utilizing the properties of natural numbers and modular arithmetic, the union of these sets can be shown to be enumerable, establishing a bijection to the natural numbers.

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Homework Statement


If [itex]A_1,A_2...A_n[/itex] are countable sets. Then the union
[itex]A_1 \cup A_2\cup ...\cup A_n[/itex] is countable.

The Attempt at a Solution


Since we know there are an infinite amount primes I will assign each element in
[itex]A_1[/itex] to the first prime. I will take every element in [itex]A_1[/itex]
and raise this element to [itex]2^x[/itex] where x is the ith element of the first set.
then I will map all the elements in the second set the second prime.
so the nth set will go to the nth prime.
since these will all be natural numbers, the union of these sets is countable.
 
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It's easier than this, for A_1, align them to the set that of numbers 1 mod n, A_2 to the set of numbers 2 mod n, ..., A_n to sets of numbers of 0 mod n.
each such set is enumarable and this is self evident a bijection from the union of A_i to N.

Cheers!
 
ok, you that's a good way.
 

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