One-to-one Correspondence Questions
Here are the questions I have been struggling with. Any help would be greatly appreciated!
2. Find a one-to-one correspondence between the set of natural numbers and the whole number powers of 10.
3. Find a one-to-one correspondence between the numbers in the closed interval [0,1] and the numbers in the closed interval [3,8]. A word on notation: the closed interval [a,b] is the set of numbers x such that a <=x <= b; it includes the numbers a and b.
4. Find a one-to-one correspondence between the set A of reciprocals of the positive integers and the set B consisting of 0 and the reciprocals of the positive integers.
5. Find a one-to-one correspondence between the numbers in the closed interval [0,1] and the open interval (0,1). Notation: the open interval (a,b) is the set of numbers x such that a < x < b; it excludes both a and b.
My solution attempts:
2. The two sets here are 1 2 3 4 5 .... n and .... 10^-1 10^0 10^1 10^2 ...
I am not sure how to address the second set (whole number powers of 10) Since it does not have a defined beginning or end...
3. So these two sets are [0,1] and [3,8]
I found that for each x in [0,1] there is a 5x+3 in [3,8], but I am not sure as to the correct way to express this as a 1-to-1 correspondence.
4. So the two sets here are:
A: 1/1 1/2 1/3 1/4 ... 1/n?
B: 0 1/1 1/2 1/3 1/4...
I am not sure how to show these two sets relate. At first I thought set B was 1/(n-1) but that would result in dividing by zero for the first element in the set... So I am not sure where to go from here.
5. Upon discussion with my professor, this is the advice I have so far gathered.
So the numbers in (0,1) come in two flavors: there are those numbers of
the form 1/n where n is a whole number greater than 1 (that is, ½, 1/3,
¼,…) and those numbers not of this form. Let C be the numbers not of
the form ½, 1/3, ¼,..).
Then the numbers in [0,1] also come in two flavors: flavor one are those
of the form 0,1, 1/2 , 1/3, ¼,… and flavor two consists of exactly those
numbers in C.
So I have learned that to get the desired one-to-one correspondence between (0,1) and [0,1], I must first
pair up the numbers ½, 1/3, ¼,1/ 5,… with the numbers 0,1,1/2, 1/3,
¼,1/5,… and then pair up each number in C with itself).
Thank you again for any help!