Re: Rational Numbers/Irrational Numbers

On Sat, 27 Jan 2007, David T. Ashley wrote:
"William Elliot" <marsh@xxxxxxxxxxxxxxxxxx> wrote in message
On Sat, 27 Jan 2007, Logan Lee wrote:

I thought rational numbers are subset of real numbers. ?:
It must be that:
Rational numbers are proper subset of real numbers.

Is this right?

That's correct, but that doesn't mean there are fewer of them.
All positive reals are a proper subset of all reals, yet there
are just as many positive reals as there are all reals.

Easy example. The set 2N, of positive even integers is a proper
subset of the set N, of positive integers.

The mapping f:N -> 2N, f(n) = 2n is a one to one correspondance
between the two sets. For each n in N, f maps to exactly one
element in 2N and for each element 2k in 2N, there is exactly
one element, viz k in N, such that f maps it to 2k. Thus N and
2N have the same number of elements, ie have the same cardinality.

Assuming that we're still trying to disprove my armchair babblings, note
that your example is also qualitatively different than mine.

Again, the elements of my reasoning were:

a)There are two mutually exclusive (but perhaps not mutually exhaustive)
sets, A and B.

Mutually exclusive? You mean disjoint? Don't know what mutually
exhaustive means.

b)Every element of A maps to two different elements of B (through two
different functions f() and g()).

f,g:A -> B
for all x in A, f(x) /= g(x)

c)The mapping is unique in both directions, and if the domain of f() and g()
is A, the ranges of f() and g() are _disjoint_ subsets of B.

Which mapping, f,g or both? Unique in both directions? You mean
injective, 1-to-1?

f(A) /\ g(A) = nulset

Mutually exhaustive means
f(A) \/ g(A) = B ?

Note: (c) may not be immediately obvious, but given a real number such as
2/3 + PI, it can only be in the range of f(). 2/3 + PI + PI, on the other
hand, can only be in the range of g(). The range of f() and the range of
g() are disjoint.

What's A and B in this example?

The example you gave is qualitatively different.

Ok, the point you're making is?

BTW, you took my reply too personally, I was answering Logan Lee.
.

Relevant Pages

  • Re: Rational Numbers/Irrational Numbers
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    (sci.math)
  • Re: Rational Numbers/Irrational Numbers
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    (sci.math)
  • Re: Uncountable sets in CZF?
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  • Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
    ... >> thinking on the rationals, that one can get arbitrarily close to any real, ... >> that it therefore constitutes an enumeration of the reals. ... > imply, then why are not the naturals, the odd naturals and the odd ... A proper subset IS smaller than the superset, ...
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  • Re: abundance of irrationals!) - rectangles of area 1.bmp [0/1]
    ... >>> A proper subset IS smaller than the superset, ... >> When TO claimed that the sets of rational and reals are of the same ... > that they are certainly NOT an equivalent set to the naturals. ...
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