
Re: Cardinality of turning wheel
Posted:
Mar 7, 2013 7:14 PM


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 singlevaluedness 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 midstream (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

