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

[cragar]

Homework Statement

If A_1,A_2...A_n are countable sets. Then the union
A_1 \cup A_2\cup ...\cup A_n is countable.

The Attempt at a Solution

Since we know there are an infinite amount primes I will assign each element in
A_1 to the first prime. I will take every element in A_1
and raise this element to 2^x 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.

 

[MathematicalPhysicist]

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!

 

[cragar]

ok, you that's a good way.