Thread: one to one correspondence

Can anyone explain how to find a one to one correspondence between two sets? I need help on the following:

a) the positive integers and the integers
(b) the interval (1,2) and the interval (1,6)
(c) the set of real numbers and the interval [0,1]
(d) the interval (0,1) and the interval [0,1]

Thanks!

Originally Posted by friday616

Can anyone explain how to find a one to one correspondence between two sets? I need help on the following:

a) the positive integers and the integers
(b) the interval (1,2) and the interval (1,6)
(c) the set of real numbers and the interval [0,1]
(d) the interval (0,1) and the interval [0,1]

Thanks!

I'll try to give some hints rather than the functions themselves.
a) Map even positive integers into positive integers and odd positive integers in to negative integers. Don't forget to map one number to 0.

b) Stretch the interval. What do multiply by? Notice that a(x- 1)+ 1 will map 1 to 1 no matter what a is.

c) Stretch again! Only now you have to find a way to map 0 to negative infinity and 1 to positive infinity.

(d) A little bit tricky. Map the irrationals to themselves. The remaining rational numbers in (0, 1) can be put into a list: $\displaystyle r_1$, $\displaystyle r_2$, etc. Map the first rational, $\displaystyle r_1$, to 0, the second, $\displaystyle r_2$, to 1 and I'll let you decide what to do with the rest of them.