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TOMERDR
science forum beginner

Joined: 09 May 2006
Posts: 26

Posted: Mon May 15, 2006 6:50 pm    Post subject: Re: Is the set of infinite series of relational numers are countable?

Yes i know,i am using the following method
if q=m/n

then for (m+n) i do:

1: 0/1
2: 1/1
3: 1/2,2/1

etc...

now since each series is countable
i get a countable union of countable sets which is countable
Rick Decker
science forum Guru Wannabe

Joined: 28 Apr 2005
Posts: 210

Posted: Mon May 15, 2006 6:18 pm    Post subject: Re: Is the set of infinite series of relational numers are countable?

TOMERDR wrote:
 Quote: Sorry it should be:is the set of FINITE sequences of rational numbers countable? Sounds suspiciously like homework. The answer is yes. Do you know

of a proof that the rationals are countable? You can modify it
slightly to show that the set of finite sequences of rationals
is countable.

Regards,

Rick
Arturo Magidin
science forum Guru

Joined: 25 Mar 2005
Posts: 1838

Posted: Mon May 15, 2006 6:04 pm    Post subject: Re: Is the set of infinite series of relational numers are countable?

TOMERDR <tomerdr@hotmail.com> wrote:
 Quote: Sorry it should be:is the set of FINITE sequences of rational numbers countable?

Yes. For each positive integer k there are only countably many
sequences of length k all of whose terms are rational numbers; the set
is thus the countable union of countable sets, hence countable.

That the countable union of countable sets is countable usually
requires an invokation of the Axiom of Choice, but I think it need not
be done in this case. We have explicit well-orderings of the
rationals, which give explicit well-orderings of the set of sequences
of length exactly k (by taking the lexicographic order); then one can
obtain an explicit bijection with N by taking the first 1-term
sequence, then the second 1-term sequence, then the first 2-term
sequence, then the third 1-term, the second 2-term, and the first
3-term, etc. (same numbering technique as is used to show there is a
surjection form N to the positive rationals).
--
======================================================================
"It's not denial. I'm just very selective about
what I accept as reality."
--- Calvin ("Calvin and Hobbes")
======================================================================

Arturo Magidin
magidin@math.berkeley.edu
TOMERDR
science forum beginner

Joined: 09 May 2006
Posts: 26

Posted: Mon May 15, 2006 5:26 pm    Post subject: Re: Is the set of infinite series of relational numers are countable?

Sorry it should be:is the set of FINITE sequences of rational numbers
countable?

Thanks
The Qurqirish Dragon
science forum Guru Wannabe

Joined: 30 Apr 2005
Posts: 104

Posted: Mon May 15, 2006 4:28 pm    Post subject: Re: Is the set of infinite series of relational numers are countable?

TOMERDR wrote:
 Quote: Is the set of infinite series or relational numers are countable? If so hwow can i show it? Thanks.

Assuming you mean "Is the set of infinite sequences of rational numbers
countable?" then the answer is no. Something similar to the diagonal
argument for the uncounability of the real numbers wil show this.
TOMERDR
science forum beginner

Joined: 09 May 2006
Posts: 26

 Posted: Mon May 15, 2006 4:24 pm    Post subject: Is the set of infinite series of relational numers are countable? Is the set of infinite series or relational numers are countable? If so hwow can i show it? Thanks.

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