October 18th, 2010, 04:30 PM  #1 
Member Joined: Apr 2010 Posts: 33 Thanks: 0  Cardinality
Is it true that the cardinality of natural numbers (0, 1 ,2, 3, ...) is equal to the cardinality of integers (..., 3, 2, 1, 0, 1, 2, 3, ...). I don't understand. How can: (..., 3, 2, 1, 0, 1, 2, 3, ...) = (0, 1 ,2, 3, ...) in terms of countable objects. To me it seems like the set of integers contains twice as much countable objects than the set of natural numbers. Does this also apply to the set of rationals too. Is the cardinality of rationals equal to the cardinality of naturals/Integers. 
October 18th, 2010, 05:28 PM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Cardinality
The cardinality of a set A is denoted A. It is true that ? = ? = ?. Two sets are said to have the same cardinality if there is a bijective map from one to the other. The map f > ? f(0) = 0 f(1) = 1 f(2) = 1 f(3) = 2 f(x) = n/2 if n is even, (n + 1)/2 if n is odd is bijective. Conceptually, it can be baffling. It is also true that the cardinality of the prime numbers = P = ?. This seems crazy because there are so many natural numbers in between all the primes, but it's true! The next highest cardinality is that of the real numbers. ? = ? (alephnaught) 
October 18th, 2010, 05:53 PM  #3  
Senior Member Joined: May 2008 From: York, UK Posts: 1,300 Thanks: 0  Re: Cardinality Quote:
 
October 18th, 2010, 05:59 PM  #4  
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Cardinality Quote:
 
October 18th, 2010, 08:52 PM  #5 
Global Moderator Joined: Dec 2006 Posts: 18,142 Thanks: 1417 
The characters you used for N, Q, Z and the alephs don't show up properly in some browsers, e.g. IE6.

October 19th, 2010, 09:46 AM  #6 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  Re: Cardinality
Bummer... 
October 19th, 2010, 11:08 AM  #7 
Member Joined: Apr 2010 Posts: 33 Thanks: 0  Re: Cardinality
So, A set has a cardinality of Alephnaught if and only if it is countably infinite, which is the case if and only if it can be put into a direct bijection, or onetoone correspondence, with the natural numbers. So, any set of numbers that can be paired off with the set of naturals is said to have a cardinality of Alephnaught, which makes it countably infinite. Is this right? 
October 19th, 2010, 11:21 AM  #8 
Member Joined: Apr 2010 Posts: 33 Thanks: 0  Re: Cardinality
If that's true, then the rationals, primes, even, and odd numbers all have a cardinality of Alephnaught.

October 19th, 2010, 11:46 AM  #9 
Senior Member Joined: Jul 2010 From: St. Augustine, FL., U.S.A.'s oldest city Posts: 12,155 Thanks: 463 Math Focus: Calculus/ODEs  Re: Cardinality
You are correct. The cardinality of the reals (the continuum) is 

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