I haven't taken a math class for a few years now (since calc. II), and I am currently enrolled in a cryptology/math history course that has posed me a few challenges. I am not sure, but I am guessing that one of the reasons to show a one-to-one correspondence between two sets (I'm not sure 'set' is the right term) is to show that both sets are the same size? I think what I need to do is show that each element in the first set corresponds to another element in a second set.
I have read the sticky posts, and I understand that this website does not provide solutions to problems, but any hints or assistance would be greatly appreciated.
OK... so the questions.
1. Find a one-to-one correspondence between the set of natural numbers and the positive whole number powers of 10.
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.
The Attempt at a Solution
1. I was able to figure out 1.
The two sets are 1 2 3 4 5 ... n and 10^1 10^2 10^3 ... 10^n
So the 1-to-1 correspondence is n <----> 10^n
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).
I am not sure how to pair up these numbers and I am also unsure what it means to pair C with itself, so any advice here would also be appreciated.
I apologize for the rather long series of questions. Thank you in advance!