Math Is Fun Forum

I need to find a function that is one-to-one and onto between the set A:{(x,y); x,y∈Z+} another words, the set of ordered pairs (x,y) such that x and y are positive integers, and the set of positive integers.
So, I have to come up with f:A to Z+  such that f(x,y) is one-to-one and onto.

Please someone help, I have been trying to come up with one for a while.  I was trying to maybe come up with another set that I know has a one-to-one and onto function g with respect to the positive integers that would be easier to come up with a one-to-one correspondence between A and that set, then I could conclude that f o g is one-to-one and onto.  I don't know if that even makes any sense, but if not, I just need a one-to-one and onto function from A to Z+.

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.

Hi guys, thanks you were a huge help last week. I have a couple more homework questions that are troubling:

Some of these may seem very simple to you guys, but I am just getting started in this, and it is very difficult for me.

Thanks

Prove that  m^2 - 25m > 0     for all m>25

Proof by induction:
Basic step:
Assume P(m): m^2 - 25m > 0  for all m>25
Prove P(m+1): (m+1)^2 - 25(m+1) > 0

Induction step:  P(m+1):  (m+1)^2 - 25(m+1)= m^2 - 25m + 2m - 24  This is where I am stuck!  I don't know what to do from here.

Could someone please help?

Let  a  be a nonzero real number and let S be the set:  S = {x∈R: |x-a| < |a|/2}

(a)  Show that |x|>|a|/2 for all x∈S.

(b)  Write S as either an interval or a union of intervals.

Please help me on this, thanks!

Let n be a positive integer, and consider U(n), the group of units in Z sub n.  We know that
theta(n) = |U(n)|.

a)  Assume a is an element of U(n).  Prove that if r is equivalent to s mod theta(n), then a^r is equivalent to a^s mod n.

b) Is the converse of a) true, in other words, is the following true:  If a is an element of U(n) and a^r is equivalent to a^s mod n, then r is equivalent to s mod theta(n).

Let A be a set. Prove that there is no injection f: P(A) to A.

Please someone help me on this!

I have started it by supposing there is an injection f: P(A) to A.
I am not sure where to start.  I don't even know what my cases could be.

Prove by induction the following:  Let f be a bijection from [m] to [n].  Prove that m=n.

Please someone help me.  Thank you.