My Math Forum How are integers a subset of R^2?

 Real Analysis Real Analysis Math Forum

December 26th, 2016, 03:40 PM   #12
Senior Member

Joined: Aug 2012

Posts: 1,854
Thanks: 513

Quote:
My most humble and sincere apology. I love this material out of Rudin and it would be ironic and sad if I chased someone away when I was trying to help.

I definitely agree that my first response was way off the mark. You asked a simple question and I gave a complicated answer at entirely the wrong level. That was an error on my part.

Joppy had already made the same point in a much more clear way, so I was acknowledging his priority in this regard. That remark wasn't actually intended for you but for everyone else.

Please don't leave here on my account. Most posters are much nicer than me. A few are worse.

Last edited by Maschke; December 26th, 2016 at 03:43 PM.

 December 26th, 2016, 04:34 PM #13 Member   Joined: Oct 2016 From: Arizona Posts: 60 Thanks: 15 Math Focus: Still deciding! It's okay. I was really hungry when I posted that last response, and I'm kind of like a T-Rex when I'm hungry. You do seem nice, and it's comforting to know that you are familiar with Rudin's book. For what it's worth, I do know the answer now. It IS what you said about $(\frac 1n, 0)$ for $n=1,2,3...$ It is hard when it's talking through a forum because no one knows each other. You don't know me, but if you did you would see how much I actually do know about the math I am asking. As I had said in my introduction (in New Users), I never ask actual "math problem" questions and some of the questions I ask may seem slightly obvious or different. I only ask clarification questions such as these. Sometimes it's the littler things that can throw me off. I am happy that I understand the proofs in Rudin's book all on my own. I am a self learner, so my background is a bit different. Now that I know this answer, I can study that chart I posted and can practice with perfect subsets of metric spaces, so thank you. It helps to know that I understand the context clearer. Sometimes, when communicating with other math people, I get upset because it seems like they don't realize that I actually know my stuff. Or they tell me things that I already know thinking I don't know them. but this is not a fault of anyone's it's just how it is because we don't really know each other. Anyways, all is well. Last edited by ProofOfALifetime; December 26th, 2016 at 04:39 PM.
 December 26th, 2016, 04:42 PM #14 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,579 Thanks: 541 Math Focus: Yet to find out. I should also apologise for my lack of conciseness and intervening in areas of math which i am not an expert (maschke is both wise and knowledgeable!). I was trying to stimulate some further conversation about having an intuitive sense of the real coordinate space, starting with (d) in your book. Namely, going from R^1 to R^2, and seeing that it is an extension from the 1d real number line, to the 2d coordinate system. And my use of notation was poor! ---------------------------------------------------------------------------------------------- Of course, there will always be a certain level of ambiguity in the tone of peoples responses when conversing online. I think we should all try to first view comments as if they were spoken with a positive or neutral and concise like tone. That way even if someone is trying to be nasty, you will be ignorant of it! . I will say though, that mathematicians or people talking about math have a certain way of expressing their arguments in such a way that can sometimes feel confronting at times. This is simply a result of the rigorous and definitive nature of the subject. Thanks from ProofOfALifetime
 December 27th, 2016, 11:55 AM #15 Senior Member   Joined: Sep 2016 From: USA Posts: 357 Thanks: 196 Math Focus: Dynamical systems, analytic function theory, numerics I would agree 100% with ProofofaLifetime. Certainly when starting to learn mathematics we were all taught to write things very precisely. In this case, what Rudin has said is completely wrong. By definition, $A$ is a subset of $B$ if whenever $a \in A$, then it follows that $a \in B$ also. In this case, every element of $B$ is an ordered pair and no integer is an ordered pair. In compute science this would be called a type error. What Rudin ${\bf means}$ however is that the integers are isomorphic with a subset of $\mathbb{R}^2$ with one possible isomorphism given by $n \mapsto (n,0)$. It is good that you noticed this subtlety as it shows you are understanding what you are reading. However, once you understand what he actually means you should allow for this slight ambiguity as it can't be avoided if math books are to remain under 1200 pages. Hope this helps. Thanks from greg1313
 December 27th, 2016, 12:19 PM #16 Global Moderator     Joined: Oct 2008 From: London, Ontario, Canada - The Forest City Posts: 7,788 Thanks: 1037 Math Focus: Elementary mathematics and beyond I would have thought that $\mathbb{R}^1$ is a subset of $\mathbb{R}^2$. Thanks from skeeter
December 27th, 2016, 12:45 PM   #17
Senior Member

Joined: Aug 2012

Posts: 1,854
Thanks: 513

Quote:
 Originally Posted by greg1313 I would have thought that $\mathbb{R}^1$ is a subset of $\mathbb{R}^2$.
Not according to the formal definition of a subset. It's only true by ignoring set theory and implicitly identifying the real line with a particular set of ordered pairs in the plane. That is, we have a sequence of maps

$\mathbb R \rightarrow \mathbb R \times \{0\} \rightarrow \mathbb R^2$

that sends the real number $x$ to the ordered pair $(x,0)$, which is an element of $\mathbb R^2$; and that we then identify the reals with that subset of ordered pairs.

This is so common we generally ignore it; but it's important to know in the back of our minds that whenever we say that "math is based on set theory," that's a lie. In fact we abandon set theory and adopt a structuralist perspective whenever it's convenient. In structuralism, objects are characterized by their relation to other objects without regard to their intrinsic properties. Thus we ignore the fact that a real number $x$ is not a set-theoretic element of $\mathbb R^2$; but that the points $(x,0)$ are isomorphic to the reals so we simply say that the reals are a subset of the plane.

This point need not be hammered on pedantically every day. But in the back of our minds we need to realize that sometimes we use the formalism of set theory; and other times we simply ignore it out of convenience.

This is the point I was attempting to make earlier. That we all agree that the integers are a subset of the plane even though set-theoretically they are not.

I agree with @SDK that Rudin's question was imprecise in a way that would be confusing to students.

Last edited by Maschke; December 27th, 2016 at 12:49 PM.

December 27th, 2016, 04:06 PM   #18
Senior Member

Joined: Feb 2016
From: Australia

Posts: 1,579
Thanks: 541

Math Focus: Yet to find out.
Quote:
 Originally Posted by Maschke This point need not be hammered on pedantically every day. But in the back of our minds we need to realize that sometimes we use the formalism of set theory; and other times we simply ignore it out of convenience. This is the point I was attempting to make earlier. That we all agree that the integers are a subset of the plane even though set-theoretically they are not.

What sort of consequences (if any) are there if we incorrectly view certain problems in a structuralist or formalist perspective (if this is correct usage of the terms)?
Is this in anyway similar to the platonic and formalist perspectives?

Last edited by skipjack; December 30th, 2016 at 07:39 PM.

 December 28th, 2016, 11:35 AM #19 Member   Joined: Oct 2016 From: Arizona Posts: 60 Thanks: 15 Math Focus: Still deciding! Thank you all for adding to the discussion. It has turned into a very interesting topic. I enjoy all of your responses.
December 28th, 2016, 11:43 AM   #20
Member

Joined: Oct 2016
From: Arizona

Posts: 60
Thanks: 15

Math Focus: Still deciding!
Quote:
 Originally Posted by SDK I would agree 100% with ProofofaLifetime. Certainly when starting to learn mathematics we were all taught to write things very precisely. In this case, what Rudin has said is completely wrong. By definition, $A$ is a subset of $B$ if whenever $a \in A$, then it follows that $a \in B$ also. In this case, every element of $B$ is an ordered pair and no integer is an ordered pair. In compute science this would be called a type error. Hope this helps.
Exactly, this is what was throwing me off. But, I learned that according to rudin he regards the set of complex numbers as $R^2$, so remembering that helped me as well. The reason it helped me is because we can write integers as complex numbers with imaginary part zero. (He regards them as ordered pairs so in other words, we make the 2nd coordinate $0$.)

 Tags integers, subset

 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post bschiavo Real Analysis 9 October 6th, 2015 11:17 AM redgirl43 Applied Math 1 April 21st, 2013 06:20 AM guru123 Algebra 3 January 15th, 2013 08:05 AM andreaugust Applied Math 16 June 22nd, 2012 11:42 AM sansar Linear Algebra 1 March 16th, 2009 07:42 AM

 Contact - Home - Forums - Cryptocurrency Forum - Top