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October 18th, 2010, 03:30 PM   #1
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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.
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October 18th, 2010, 04:28 PM   #2
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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.
|?| = ? (aleph-naught)
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October 18th, 2010, 04:53 PM   #3
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Re: Cardinality

Quote:
Originally Posted by The Chaz
The next highest cardinality is that of the real numbers.
Do you have a proof for that?
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October 18th, 2010, 04:59 PM   #4
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Re: Cardinality

Quote:
Originally Posted by mattpi
Quote:
Originally Posted by The Chaz
The next highest cardinality is that of the real numbers.
Do you have a proof for that?
I *would* write it, but it wouldn't fit in the margin of this... forum...
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October 18th, 2010, 07:52 PM   #5
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The characters you used for N, Q, Z and the alephs don't show up properly in some browsers, e.g. IE6.
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October 19th, 2010, 08:46 AM   #6
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Re: Cardinality

Bummer...
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October 19th, 2010, 10:08 AM   #7
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Re: Cardinality

So, A set has a cardinality of Aleph-naught 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 one-to-one 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 Aleph-naught, which makes it countably infinite.

Is this right?
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October 19th, 2010, 10:21 AM   #8
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Re: Cardinality

If that's true, then the rationals, primes, even, and odd numbers all have a cardinality of Aleph-naught.
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October 19th, 2010, 10:46 AM   #9
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Re: Cardinality

You are correct. The cardinality of the reals (the continuum) is
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