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Re: Cardinality of turning wheel
Posted:
Mar 7, 2013 7:14 PM
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Don Kuenz wrote:
>
> Frederick Williams <freddywilliams@btinternet.com> wrote:
>
> > What physics forum?
>
> http://www.physicsforums.com/showpost.php?p=1987697
>
> Given A a subset of R... we can map [0,1) onto R, so we
> can map [0,1) onto A, say by a function f. Then extend
> f to a function F by F(x) = f([x]) where [x] is the
> rational part of x. F is periodic and has as its range
> A. Hence for each subset of R, we have a distinct
> periodic function, and then just use the fact that the
> set of periodic functions is a subset of the set of all
> functions R->R and the cardinality of the latter is c^c
>
> The bit about "where [x] is the rational part of x" makes little, if
> any, sense to anyone (including me).
>
> > The cardinality of the range of a function cannot
> > exceed that of its domain. Here I take "function" to mean "single
> > valued function" as is, I think, usual when real functions are
> > discussed. Writing Q for the set of rational numbers and R for the set
> > of real numbers, the function
> >
> > f: Q -> R
> >
> > defined by
> >
> > f(x) = sin(x)
> >
> > is single valued and has a range of cardinality aleph_0.
>
> So, the function
>
> f: R
>
> defined by
>
> f(x) = sin(x)
>
> is single valued and has a range of cardinality c.
>
> Why do we care about single valuedness?
Ha ha, I wrote:
> The cardinality of the range of a function cannot
> exceed that of its domain. Here I take "function" to mean "single
> valued function" as is, I think, usual when real functions are
> discussed.
If the functions _are_ real then of course single-valuedness is
irrelevant. The first sentence there considers other functions (f, say)
in which the cardinality of the codomain exceeds that of the domain.
Say card domain = aleph_0 and card codomain = c, then if f takes on c
values at each point in its domain, then ran f = c. So it seems I
switched horses mid-stream (or whatever the saying is).
> > Here, range(f) = {y : y = f(x) for some x} and is not to be confused
> > with graph or codomain.
>
> This is new territory for me. :)
--
When a true genius appears in the world, you may know him by
this sign, that the dunces are all in confederacy against him.
Jonathan Swift: Thoughts on Various Subjects, Moral and Diverting
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