Virgil
Posts:
7,760
Registered:
5/14/06
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Re: Rational Numbers/Irrational Numbers
Posted:
Jan 27, 2007 1:44 AM
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In article <spOdnWCnC5KKbCfYnZ2dnUVZ_uGjnZ2d@giganews.com>,
"David T. Ashley" <dta@e3ft.com> wrote:
> Anonymous wrote in message
> news:1169874202.106385.326130@v45g2000cwv.googlegroups.com...
> > On Jan 26, 11:36 pm, "David T. Ashley" <d...@e3ft.com> wrote:
> >> "Leo" <newsdon...@hotmail.com> wrote in
> >> messagenews:1169780763.086460.114690@s48g2000cws.googlegroups.com...
> >>
> >> > Which set has more numbers, the set of rational numbers or the set of
> >> > irrational numbers?
> >> Well, the set of irrational numbers has at least twice as many elements
> >> as
> >> the set of rational numbers.
> >>
> >> Think about the following functions:
> >>
> >> f(x) = PI + x
> >> g(x) = PI + PI + x
> >>
> >> Every rational number x can be paired with at least two irrationals.
> >>
> >> So, I'm going to go with "irrational" as being bigger.
> >
> > Right answer. Wrong reason. The rationals are countable. The
> > irrationals are uncountable. The rationals have Lebesgue measure zero.
> > The irrationals in [0,1] have Lebesgue measure 1.
>
> I was just clowning around ... but OK, let's explore your logic.
>
> In order for me to have stated the "wrong" reason, there has to be a
> counterexample where my test fits but where there are not "more".
>
> Please provide a counterexample.
It is also the case that every rational can be "paired" with an infinite
set of rationals, but that does not make the set of rationals larger
than the set of rationals.
Are you aware of Cantor's two proofs that there are more reals than
rationals? With slight modification they also prove more irrationals
than rationals.
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