Re: How can you prove it?
- From: "[Mr.] Lynn Kurtz" <kurtz@xxxxxxxxxxxxxxx>
- Date: Sun, 30 Sep 2007 20:47:35 GMT
On Sun, 30 Sep 2007 12:38:46 -0700, mike3 <mike4ty4@xxxxxxxxx> wrote:
Hi.
How could one prove that there are "more" functions from reals to
reals than there are reals, anyway? How also can one show that the set
of such functions also has "just as many" items in it as the power set
of the reals?
You have received one answer; here's another way to look at it
directly. Suppose there exists a surjection H from the reals to those
functions. So for each a in R there corresponds a distinct function
H(a) = f_a.
Now consider the function g defined with the condition that for each a
in R, g(a) <> f_a(a). Since H is onto, there is w in R such that H(w)
= g, so g = f_w. Then g(w) = f_w(w), a contradiction. So no such
surjection exists.
--Lynn
.
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