# One-to-one Correspondence Questions

by kolssi
Tags: correspondence, onetoone
 P: 2 1. The problem statement, all variables and given/known data 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. 2. Relevant equations 3. 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!
Emeritus
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PF Gold
P: 11,378
 Quote by kolssi 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...
You've confused the whole numbers with the integers. The whole numbers are 0, 1, 2, ....
 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.
You want to give a one-to-one function f:[0,1]->[3,8]; that is, just write it as you normally would write a function.
 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.
You don't have to use a single formula to describe the correspondence. It's perfectly fine to say: f(1/n)=1/(n-1) when n is not equal to 1 and f(1/1)=0.

 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.
Hint: Part of problem 5 is very similar to problem 4.
 P: 2 OK so I made some attempts at solutions. If someone could tell me if I am going in the right direction, that would be great. I have done some reading and I found the Schroder-Bernstein theorem that I think states that for two sets A and B, if injective functions A-->B and B-->A exist, then A and B have one-to-one correspondence (bijection). I hope this can apply to my homework. I just wanted to know if I am going about this in a correct manner. ------------------------------------------------------------------------------------ Find a one-to-one correspondence between the set of natural numbers and the whole number powers of 10. (my professor said that by "whole numbers" he means all integers) Set A: 1,2,3,4,5,6..... Set B: ...10^-2 , 10^-1 , 10^0 , 10^1 , 10^2... A-->B f(1) = 10^0 f(2n) = 10^n for n >= 1 f(2n+1) = 10^-n for n >= 1 B--->A f(10^0) = 1 f(10^n) = 2n for n>=1 f(10^-n) = 2n+1 for n>=1 I tried to match all even numbers in set A with the positive whole number powers of 10 and the odd numbers with the negative whole number powers of 10. 1 from A and 10^0 from B were left over, so I matched those two together. --------------------------------------------------------------------------- ***** 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. A: [0,1] B: [3,8] A-->B f(x) = 5x+3 B-->A f(x) = (x-3)/5 ---------------------------------------------------------------------------------- ******* 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. A: 1/1, 1/2, 1/3... B: 0, 1/1, 1/2, 1/3... A-->B f(1) = 0 f(1/n) = 1/(n-1) for n>1 B-->A f(0) = 1 f(1/n) = 1/(1+n) ---------------------------------------------------------------------------------- *******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. A: [0,1] B: (0,1) C: {0, 1, 1/2, 1/3,...,1/n,...} A-->B f(0) = 1/2 f(1/n) = 1/(n+2) for n >=1 f(x) = x for x in [0,1] that do not belong to C B-->A f(1/2) = 0 f(1/n) = 1/(n-2) for n>2 f(x) = x for x in (0,1) that do not belong to C
Emeritus