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Re: Cardinality of turning wheel
Posted:
Mar 7, 2013 12:03 PM
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Don Kuenz wrote:
>
> K_h <KHolmes@sx729.com> wrote:
> >
> >
> > "Frederick Williams" wrote in message
> > news:51336CFB.CDE15F2C@btinternet.com...
> >>
> >> > If we can somehow use sin(x) to represent the number of revolutions,
> >> > here's an argument that the cardinality of sin(x) is c^c.
> >>
> >> It is sets that have a cardinality. What set is sin(x)?
> >
> > Don put forward a bad argument. The sine function is the set of all points
> > (x, sin(x)) where x is any real number. So the cardinality of sin(x) is
> > just c, the cardinality of the real line because there are two values in (x,
> > sin(x)) and 2*c=c in cardinal arithmetic.
>
> Agreed, sin(x) is a bad argument for the OP's revolving wheel question.
> The physics forum seems to argue that the cardinality of the range
> created by feeding the set of rational numbers into sin(x) is c^2.
What physics forum? 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.
Here, range(f) = {y : y = f(x) for some x} and is not to be confused
with graph or codomain.
--
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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