November 9th, 2013, 07:43 AM  #1 
Member Joined: May 2012 Posts: 78 Thanks: 0  Countable set
Hello, I have two questions: 1. Determine whether the set is countable set:[*]The set of series of natural number when there are even numbers in even places and odd numbers in odd places (eg, (3,8,5,2,1,0,...)).[*]The set of series of prime numbers (eg, (2,3,5,...) or (13,17,23,31,...)) 2. Let be infinite set of positive numbers. Let that for every : . Prove that A is a countable set. Clue: look at the sets . Thanks! 
November 9th, 2013, 11:50 AM  #2  
Senior Member Joined: Aug 2012 Posts: 2,259 Thanks: 686  Re: Countable set Quote:
Quote:
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November 9th, 2013, 12:17 PM  #3 
Senior Member Joined: Jun 2013 From: London, England Posts: 1,316 Thanks: 116  Re: Countable set
There's a solution and discussion on question 2 here: http://math.stackexchange.com/questions ... venumbers Including an answer to the question of how a sum over an uncountable set B would be defined! 
November 9th, 2013, 12:55 PM  #4  
Member Joined: May 2012 Posts: 78 Thanks: 0  Re: Countable set Quote:
May you help with more clues or detailed solution? Pero, thank you for the link, Iv'e found it very helpful.  
November 9th, 2013, 01:18 PM  #5  
Senior Member Joined: Aug 2012 Posts: 2,259 Thanks: 686  Re: Countable set Quote:
Do you see that if A is uncountable that at least one A_n must be uncountable? If you get that far, then you just need to convince yourself that the sum of an uncountable collection of numbers each of which is greater than some fixed constant 1/n must be infinite. Remember n is fixed here!  
November 9th, 2013, 01:28 PM  #6 
Member Joined: May 2012 Posts: 78 Thanks: 0  Re: Countable set
I understood the answer for my second question. I have problems with my first question... Unfortunately I didn't understand your answer.

November 9th, 2013, 01:43 PM  #7  
Senior Member Joined: Aug 2012 Posts: 2,259 Thanks: 686  Re: Countable set Quote:
 
November 9th, 2013, 06:23 PM  #8 
Member Joined: May 2012 Posts: 78 Thanks: 0  Re: Countable set
You used the same logic for both series, so I have to say both... :\ It's not clear to me why . 
November 9th, 2013, 06:27 PM  #9  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Re: Countable set Quote:
 
November 9th, 2013, 06:32 PM  #10 
Member Joined: May 2012 Posts: 78 Thanks: 0  Re: Countable set
If it's not, may you help with the question or explain Maschke's answer? Thanks. 

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