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dchilow

Member

Registered: 2007-03-05

Posts: 27

Help with Real analysis.

I am having trouble with a couple of problems on my practice homework, could someone please help me figure these out, thanks.

1)  For each pair of sets, find an explicit one-to-one correspondence between them.  That is, find a one-to-one and onto function that maps one set onto the other.

(a) The even positive integers and the odd positive integers.

(b) The interval (-1,1) and the interval (0,1).

(c) The interval (-∞,∞) and the interval (-1,1).

(d) The interval (0,1) and the interval [0,1].

2)  Let S be a nonempty set of real numbers that is bounded above.  Let B=supS and let E>0 be a constant.

(a) Suppose that B is not an element of S.  Prove that the set A={x∈S:x>B-E} is infinite.

(b) Give an example in which the set A is countably infinite, and another example in which A is uncountably infinite.

Please help, I would greatly appreciate it.

JaneFairfax

Member

Registered: 2007-02-23

Posts: 6,868

Re: Help with Real analysis.

#1
In what follows, f is a function from the first set to the second set. I will give you the examples and leave you to prove that the functions are injective and surjective.

(a)

(b)

(c)

(d)

#2

(a)
Note that B−E < B is not an upper bound of S. Hence

But B ∉ S, so s[sub]1[/sub] ≠ B. So s[sub]1[/sub] < B. Then s[sub]1[/sub] is not an upper bound of S and so

Again s[sub]2[/sub] < B because B ∉ S, so s[sub]2[/sub] is not an upper bound of S and so

Continuing this way we get an infinite subset

of A. Since

is infinite, so is A.

(b)
Take

Then

, and A is uncountably infinite. To make A countably infinite, take

Last edited by JaneFairfax (2008-02-20 14:53:26)