Is There a One-to-One Correspondence Between (0,1) and the Real Line?

  • Thread starter Thread starter cragar
  • Start date Start date
  • Tags Tags
    Interval
Click For Summary

Homework Help Overview

The discussion revolves around the relationship between the interval (0,1) and the real line, specifically exploring whether there exists a one-to-one correspondence between them. The subject area includes concepts from set theory and real analysis, particularly focusing on countability and uncountability.

Discussion Character

  • Exploratory, Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants discuss various methods to establish a mapping from (0,1) to the real line, including the use of functions like tangent and transformations involving multiplication and shifting. Questions arise about the validity of these approaches and the nature of the mappings.

Discussion Status

There is an active exploration of potential functions that could establish the desired correspondence. Some participants provide suggestions and corrections regarding the proposed mappings, indicating a collaborative effort to refine the approach. The discussion is ongoing, with no explicit consensus reached yet.

Contextual Notes

Participants are working within the constraints of homework guidelines, which may limit the types of solutions or methods they can use. The discussion also reflects a need to clarify definitions and assumptions related to the mappings and the properties of the intervals involved.

cragar
Messages
2,546
Reaction score
3

Homework Statement


show that the interval (0,1) is uncountable iff [itex]\mathbb{R}[/itex]
is uncountable.

The Attempt at a Solution


Can I take the interval (0,1) and multiply it by a large number and then a large number and eventually extend it to the whole real line. So now (0,1) can be mapped to the whole real line. Then can I use cantors diagonal argument to show that the real line is uncountable?
 
Physics news on Phys.org
cragar said:

Homework Statement


show that the interval (0,1) is uncountable iff [itex]\mathbb{R}[/itex]
is uncountable.

The Attempt at a Solution


Can I take the interval (0,1) and multiply it by a large number and then a large number and eventually extend it to the whole real line. So now (0,1) can be mapped to the whole real line. Then can I use cantors diagonal argument to show that the real line is uncountable?

They are probably just looking for a 1-1 function between (0,1) and the real line. 'multiply it by a large number' isn't going to get you there. Can't you think of any functions that map the real line to an open interval?
 
tan(x), will that work
 
cragar said:
tan(x), will that work

tan(x) will map (-pi/2,pi/2) to R, right? Can you fix the function up so the interval is (0,1) instead of (-pi/2,pi/2)?
 
can i divide everything in the interval by pi and then shift it to the right by 1/2
 
cragar said:
can i divide everything in the interval by pi and then shift it to the right by 1/2

You CAN do anything you want if it works. Try it and see. What's your answer for a function mapping (0,1) to R?
 
okay so [itex]tan(\pi(x-\frac{\pi}{2}))[/itex] should do the trick for the mapping.
at this point can I show the reals are uncountable.
 
Last edited:
cragar said:
okay so [itex]tan(\pi(x-\frac{\pi}{2}))[/itex] should do the trick for the mapping.
at this point can I show the reals are uncountable.

Well, that's a 1-1 correspondence between (0,1) and R alright. Edit: Oh, wait. Don't you mean [itex]tan(\pi(x-\frac{1}{2}))[/itex]? Try the endpoints again.
 
Last edited:
ok you your right. so now that have a one-to-one correspondence between (0,1) and the real line.
If I show that the real line is uncountable using cantors diagonal arguement. will that complete the proof. Thanks for your help by the way.
 

Similar threads

Undergrad (0,1) is uncountable using binary expansions?
  • · Replies 15 ·
Replies
15
Views
4K
Proving Injectivity of Shroeder-Berstein Theorem for R to R^2 x R^2
  • · Replies 5 ·
Replies
5
Views
2K
If A is dense in [0,1] and f(x) = 0, x in A, prove ∫fdx = 0.
  • · Replies 2 ·
Replies
2
Views
2K
Ring of continuous real-valued functions
  • · Replies 14 ·
Replies
14
Views
3K
Undergrad Proof of a fact about real numbers - transcendental and algebraic
  • · Replies 7 ·
Replies
7
Views
3K
What is true for limit of f (x,y) as (x,y)→(0,1)?
  • · Replies 3 ·
Replies
3
Views
2K
Functions and Analysis with a fixed-point
  • · Replies 2 ·
Replies
2
Views
3K
Well-ordered subsets of real numbers
  • · Replies 12 ·
Replies
12
Views
7K
Proving that the two given functions are linearly independent
  • · Replies 21 ·
Replies
21
Views
3K
Mapping Function for (0,1) into Open Unit Square
  • · Replies 10 ·
Replies
10
Views
2K