Re: Rational Numbers/Irrational Numbers

On Sat, 27 Jan 2007 01:28:41 -0500, David T. Ashley wrote:
<davidmarcus@xxxxxxxxxxxx> wrote in message
news:1169874202.106385.326130@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Jan 26, 11:36 pm, "David T. Ashley" <d...@xxxxxxxx> wrote:
"Leo" <newsdon...@xxxxxxxxxxx> wrote in
messagenews:1169780763.086460.114690@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

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.

Consider one of the standard proofs that the rationals are countable. We
define a mapping f: N -> Q+ by listing the positive rationals in the
following order:

1/1, 1/2, 2/1, 3/1, 2/2, 1/3, 1/4, 2/3, ...

Notice that each rational number appears many times in the list. For
example, 1/1, 2/2, 3/3, ... are all the same number, and likewise 1/2,
2/4, 3/6, ....

This obviously proves that there are infinitely many times more natural
numbers than there are rationals. Or does it?


--
Dave Seaman
U.S. Court of Appeals to review three issues
concerning case of Mumia Abu-Jamal.
<http://www.mumia2000.org/>
.

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