From a Point on a Line, to a Point in the Plane, to ... the Axiom of ChoiceDate: 01/26/2011 at 17:26:31 From: Alex Subject: One-to-One Correspondence Between Real Line and Real Plane I'm a high-school freshman enrolled in an Algebra 2 class exploring concepts in set theory and topology. I was wondering if it was possible to establish a one-to-one correspondence from a point on the real number line to a point on the two-dimensional real plane. That is, can you create a function that will transform any number (or point) on the real line into an ordered pair (or point) on the real plane and vice versa? I can't think of a tiling or correspondence that would transform the line into a plane. I know that the real plane is a two-dimensional manifold while the real line is one-dimensional, and infinitely thin, so I can't imagine how you can create a homeomorphism between them. I've considered - concentric circles formed from the real line, with a fractal-like tiling of them, thinking that it might produce the plane at infinity - an infinitely dense spiral from the origin - creating an ordered pair from the numerator and denominator of rational numbers -- but this creates duplicate points, so it's not one-to-one (e.g., 1/2 maps to (1,2); but 2/4 which is equal to 1/2 maps to (2,4)) I have a hunch this may be connected to Cantor sets and cardinality (from what I've read), but I haven't found a clear answer to my question. Date: 01/26/2011 at 23:09:31 From: Doctor Vogler Subject: Re: One-to-One Correspondence Between Real Line and Real Plane Hi Alex, Thanks for writing to Dr. Math. Yes, this can be done. For example, one way to do this is to start with a correspondence between the integers Z and two copies of the integers {0,1}xZ, such as even n -> (0, n/2) odd n -> (1, (n - 1)/2) Then every real number r can be written in decimal as a sequence of digits, so r = the sum of d_i * 10^i where each d_i is a digit (integer between 0 and 9) and the sum runs over all integers i. So you write the number as (d_k)...(d_2)(d_1)(d_0).(d_-1)(d_-2).... Now you just map your digits to the digits of a pair of real numbers by taking the d_n digit and mapping it to either the d_(n/2) digit of the first number or the d_((n - 1)/2) digit of the second number, according to whether n is odd or even. For example, it would map ... 123456789 -> (13579, 2468) 1234.567891234 -> (24.6813, 13.57924) pi (3.14159...) -> (3.4525599286638..., 0.11963873342433...) ... and so on. Of course, this is not the only way to do it. If you have any questions about this or need more help, please write back and show me what you have been able to do, and I will try to offer further suggestions. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ Date: 01/26/2011 at 23:38:09 From: Alex Subject: Thank you (One-to-One Correspondence Between Real Line and Real Plane) Thanks so much for your help! I'm assuming that this can be generalized to higher n dimensions using mod-n arithmetic. It's very counterintuitive that this sort of bijection exists and you did an excellent job of helping me understand it. Thank you. Date: 01/27/2011 at 23:01:55 From: Doctor Vogler Subject: Re: Thank you (One-to-One Correspondence Between Real Line and Real Plane) Hi Alex, Doctor Jacques points out that there is a catch to my answer; it isn't exactly a bijection because there are some real numbers that can be written in decimal in two ways. For example, my map would say to do the following: 1 -> (1, 0) 0.999999999... -> (0.9999999..., 0.99999999...) But this mapping is not quite well-defined: 0.999999999... = 1 http://mathforum.org/dr.math/faq/faq.0.9999.html Furthermore, you can occasionally get two numbers that go to the same point, as in 1 -> (1, 0) 1/11 = 0.090909090909... -> (0.9999999..., 0) There are ways to fix up this kind of problem -- by making special cases for those kinds of numbers. For example, with numbers that can be written in decimal in two ways (which are only rational numbers the denominators of which are divisible by no primes other than 2 or 5; that is, numbers of the form a/10^b for integers a and b), if you insist that these have to use the form that ends in repeating zeros rather than repeating nines, then you solve the problem of one number going to two possible pairs of numbers. You also avoid ever going to a pair (x, y) where both x and y end in repeating nines. But you still have to make an exception for the possibility of one of them ending in repeating nines and the other in repeating zeros, like 1/11 shown above. In fact, the problem is precisely the rational numbers of the form ... (11*a + 1)/(11*10^b) ... for integers a and b. You could fix up this problem by choosing to change the images of those numbers so that the images of both sets of numbers are interleaved among the images that used to be for the second set. Similarly with the images of another infinite subset of real numbers, such as ones of the form (11*a + 1)/(11*10^b) + ln(2) Or you could use a strategy similar to the one used for the Bernstein-Schroeder Theorem: http://en.wikipedia.org/wiki/Cantor%E2%80%93Bernstein %E2%80%93Schroeder_theorem I don't know if there is a cleaner way to make a one-to-one correspondence between R and R*R without having to fix up some subset of the points. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ Date: 01/27/2011 at 23:45:06 From: Alex Subject: Thank you (One-to-One Correspondence Between Real Line and Real Plane) Dear Dr. Vogler, Thanks! That's a lot to think about, but it's very interesting. By the way, could you please explain the Axiom of Choice to me and why it's so important in set theory? I saw it mentioned while I was reading about the Banach-Tarski Paradox. Thanks so much! Date: 01/28/2011 at 22:58:50 From: Doctor Vogler Subject: Re: Thank you (One-to-One Correspondence Between Real Line and Real Plane) Hi Alex, Oh, where to begin? If you've ever done any proofs, you'll find that they generally start with some assumptions or a hypothesis, and they frequently use other theorems proved previously, and they end with the conclusion of the theorem. Euclid's Elements was an early (perhaps the first) book that did this. But if you use smaller theorems to prove bigger ones, then where do you start? How can you prove the first theorem? The answer is that you have to start with some kind of initial assumptions or hypotheses. The name for those initial assumptions is "axioms." Euclid started with five axioms about geometry, and then he went on to prove lots of theorems about geometry. Well, most concepts in mathematics can be described in terms of set theory, so to build up all of mathematics in this way, the hard part is starting with the basic building blocks in logic and set theory. So there have been several attempts to do this, and it always turns out to be much more complicated than it would at first seem: someone comes up with some simple set of axioms, and then someone else proves something ridiculous from them; or someone shows that certain basic concepts can't be proven from your axioms. Well, the most widely-accepted set of axioms for set theory are the Zarmelo-Fraenkel (ZF) axioms, with or without the Axiom of Choice (with it, you call them ZFC). See also http://en.wikipedia.org/wiki/Zermelo_Fraenkel_set_theory Most of the axioms seem relatively basic and straightforward, but the Axiom of Choice seems obvious in some situations and almost ridiculous in others. So while it is usually accepted as an axiom and freely used in proofs where needed, sometimes mathematicians will point out when the Axiom of Choice is necessary for a proof or theorem to hold. - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ Date: 01/29/2011 at 08:24:39 From: Alex Subject: One-to-One Correspondence Between Real Line and Real Plane Thanks, but what does the Axiom of Choice actually mean? In other words, what does it state and how can it lead to ridiculous situations if it is so simple? Thanks again! Date: 01/29/2011 at 13:52:31 From: Doctor Vogler Subject: Re: One-to-One Correspondence Between Real Line and Real Plane Hi Alex, I'm sorry. I thought you were reading about the Banach-Tarski paradox. You don't consider that to be ridiculous? Well, then maybe the Axiom of Choice isn't so silly after all. On the other hand, if the Axiom of Choice is false, then an infinite product of infinite sets might be empty, which I find ridiculous. As for the precise statement of the Axiom, see: http://en.wikipedia.org/wiki/Axiom_of_choice - Doctor Vogler, The Math Forum http://mathforum.org/dr.math/ |
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