My Math Forum  

Go Back   My Math Forum > College Math Forum > Real Analysis

Real Analysis Real Analysis Math Forum


Reply
 
LinkBack Thread Tools Display Modes
May 13th, 2011, 03:28 PM   #1
Newbie
 
Joined: May 2011

Posts: 1
Thanks: 0

Unit Interval vs. Unit Square Cardinality Question

Hello,

I'm currently studying sets and cardinality on my own, and I've come across some non-intuitive results I can't comfortably "fix".

I've come to understand that the unit interval [0, 1] and the unit square [0, 1] x [0, 1] have the same cardinality because a one-to-one correspondence can be created between them.

The only example I have seen of such a correspondence is as follows:
Any real number, A, between 0 and 1 inclusive can be represented as digits: 0.a1a2a3a4a5...
If (x, y) is a point in the unit square written as (0.x1x2x3x4... , 0.y1y2y3y4y5...) the corresponding point on the unit interval can be written as 0.x1y1x2y2x3y3..., basically "interlocking" the coordinates together.
In reverse, A is mapped to (0.a1a3a5..., 0.a2a4a6...) on the square.

My problem comes from the ability of some rational numbers to have two decimal expressions.
For example, suppose I want to find the point in the unit square where four-fifths is mapped to using the above method.
Four-fifths can be written as 0.8000.... or 0.7999....
Using the first expression, A = 0.8000... and x = 0.800... and y = 0.000..., so 0.8 maps to (0.8, 0)
Using the second expression, A = 0.7999... and x = 0.7999... and y = 0.999..., so 0.7999... maps to (0.7999..., 0.999...) or (0.8, 1)

Two different map results for the same input (just expressed in a different way)? How is this possible?

You can go on to see more surprising results in the reverse direction.
Mapping (0.8, 0.2) to the unit interval, the coordinate pair can take on four different expressions resulting in four unique mapping locations:
(0.8, 0.2) maps to (0.82)
(0.7999..., 0.2) maps to (0.72999...)
(0.8, 0.1999...) maps to (0.81099...)
(0.7999..., 0.1999...) maps to (0.71999...)

Does this show that the function is not one-to-one? Does this show that the two expressions one can use for the same number are actually different? How can this be justified?

Thank you for your time.
miss_direction is offline  
 
May 13th, 2011, 04:00 PM   #2
Senior Member
 
Joined: Jun 2010

Posts: 618
Thanks: 0

Re: Unit Interval vs. Unit Square Cardinality Question

miss_direction,

The function is not well-defined. To remedy this, you can simply settle on one form of the decimal expansion in the ambiguous cases. Then you will recover a well-defined function which is also a bijection.

-Ormkärr-
 is offline  
May 13th, 2011, 08:36 PM   #3
Member
 
Joined: Dec 2010
From: Miami, FL

Posts: 96
Thanks: 0

Re: Unit Interval vs. Unit Square Cardinality Question

If 4/5 gets mapped to two distinct points then the function is not injective.
guynamedluis is offline  
Reply

  My Math Forum > College Math Forum > Real Analysis

Tags
cardinality, interval, question, square, unit



Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes


Similar Threads
Thread Thread Starter Forum Replies Last Post
Solving Poisson equation in a unit square gasper7 Real Analysis 4 May 23rd, 2013 11:55 PM
Find the triangles to the nearest square unit sivela Algebra 1 February 19th, 2011 07:05 PM
Any conformal map from square to unit circle? lumiere137 Complex Analysis 0 October 19th, 2010 05:00 AM
equidecomposability of unit interval jason.spade Real Analysis 3 March 5th, 2010 03:34 PM
Solving Poisson equation in a unit square gasper7 Calculus 0 December 31st, 1969 04:00 PM





Copyright © 2018 My Math Forum. All rights reserved.