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R set magnitudeby Lama
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#1
Jun704, 10:25 AM

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"real line" is used to mean real axis, i.e., a line with a fixed scale so that every real number corresponds to a unique point on the line. http://mathworld.wolfram.com/RealLine.html
If the realline scale is fixed, then how a part of the realline has the same magnitude of the realline? 


#2
Jun704, 10:43 AM

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Two sets have the same cardinality if you can find a onetoone correspondence that maps one set onto the other.



#3
Jun704, 10:47 AM

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Consider this an advance warning, Organic; if you try to use this thread for theory development, it will be deleted. You've used up all of your second chances.



#4
Jun704, 10:56 AM

P: 467

R set magnitude
Ok Hurkyl,
I got it, can you please answer to my question? 


#6
Jun704, 11:25 AM

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#7
Jun704, 11:35 AM

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I will assume by "part" you mean an interval whose endpoints are not the same. If by "magnitude" you mean simply the cardinality of the sets, then it's done by showing a 11 correspondence. For example, a 11 correspondence between the interval (0, 1) and the whole real line can be given by the function f(x) = (12x)/(x (1x)); this function maps from the interval (0, 1) onto the entire real line. If by "magnitude" you mean length, then they generally don't have the same magnitude; the real line has infinite length, while most intervals don't. 


#8
Jun704, 11:50 AM

P: 467

Hurkyl,
I am talking about a bijection between the realline and any arbitrary part (or a subset if you will) of it. So, If the realline scale is fixed, then how a part of the realline has the same magnitude of the realline? 


#9
Jun704, 11:56 AM

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What is a "fixed real line scale", and whatever has that to do with the existence of bijections



#10
Jun704, 11:57 AM

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Well, the choice of scale is irrelevant to cardinailty; the line consists of the exact same points no matter which ones I choose to call "0" and "1". So, the question reduces to:
I gave an example of how the interval (0, 1) has the same cardinality as the real line, by presenting the bijection f(x) = (1  2x) / (x (1  x)). Here is an example of how a part might not have the same cardinality as the real line: the set of points {0, 1, 2} has finite cardinality, but the real line does not have finite cardinality. 


#11
Jun704, 12:07 PM

P: 467

As you see, the words "fixed scale" are used. 


#12
Jun704, 01:03 PM

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#13
Jun804, 04:38 AM

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Organic, www, lama, dialog, whatever today, these things are all definitions. Why do you not accept that? Fixed scale is not a well defined term, and I would hesitate to say what it means, but presumably it is to do with distances being euclidean. That has nothing to do with cardinality. "size" is a vague word, and just as we would never say "amount" when referring to chemistry if we were being precise, you need to distinguish between the suggestive descriptions offered as an aid to understanding and the formal rigorous definitions that are used when we come to practise mathematics.



#14
Jun804, 04:23 PM

P: 27

I have found this thread http://www.physicsforums.com/showthr...438#post228438 .



#15
Jun904, 04:56 AM

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Please can someone lock this? Don't suppose he's on a static IP is he?



#16
Jun904, 05:11 AM

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Can you explain f(x) = (1  2x) / (x (1  x))?



#17
Jun904, 05:25 AM

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What do you mean "explain" it? It's a rational function defined for all real numbers except and 1. It has vertical asymptotes at x= 0 and x= 1 and a horizontal asymptote y= 0. What more do you want to know about it?



#18
Jun904, 05:47 AM

P: 27

This function maps from the interval (0, 1) onto the entire real line.
But it is not explain how it can be done if the real line has no fractal structure (self similarity over scales). So, where can I find a mathematical theory that explains why it can be done? 


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