Manifold

I've got an exercise I would like to discuss with you...I came to this idea because of the thread "even and odd numbers" which has a lot to do with it... Task (Source: V.A. Zorich, Mathematical Analysis 1, Springer-Verlag): a) Prove the equipollence of the closed interval [math]\{x\in\mathbb{R}~|~0\le{x}\le{1}\}[/math] and the open interval [math]\{x\in\mathbb{R}~|~0<x<1\}[/math] of the real line [math]\mathbb{R}[/math] both using the Schröder-Bernstein theorem and by direct exhibition of a suitable bijection. b) Analyze the following proof of the Schröder-Bernstein theorem: [math](card~X\le{card~Y})\wedge(card~Y\le{card~X}) \Rightarrow (card~X=card~Y)[/math]. Proof: It suffices to prove that if the sets [math]X,Y,Z[/math] are such that [math]X\supset{Y}\supset{Z}[/math] and [math]card~X=card~Z[/math], then [math]card~X=card~Y[/math]. Let [math]f:X\rightarrow{Z}[/math] be a bijection. A bijection [math]g:X\rightarrow{Y}[/math]can be defined, for example, as follows: [math]g(x)=\left\{{f(x),~if~x\in{f^n(X)\setminus{f^n(Y)}~for~some~n\in\mathbb{N},}\atop~{x,~otherwise.}\[/math] Here [math]f^n=f\circ{...}\circ{f}[/math] is the nth iteration of the mapping [math]f[/math] and [math]\mathbb{N}[/math] is the set of natural numbers. (Remark: N={1,2,3,...} in this terminology)

Well...[math](a<b)\wedge(b<a) \Rightarrow (a=b)[/math] is actually wrong, since you can't find a number that is greater and fewer than the given one at the same time...it would then hold: [math](1<2)\wedge(2<1) \Rightarrow (1=2)[/math] ...which is a double nonsense... In fact, for any x and y in R precisely one of the following holds: a) x<y, b) x=y, c) x>y Proof: Consider the axioms for the set of real numbers: 1. [math]\forall{x}\in\mathbb{R} (x\le{x})[/math], ; 2. [math](x\le{y})\wedge(y\le{x}) \Rightarrow (x=y)[/math], from this follows b)...this axiom is justified by the first one, and it is what you seem to have meant; 3. [math]\forall{x}\in\mathbb{R} \forall{y}\in\mathbb{R} (x\le{y})\vee(y\le{x})[/math], from this follows a) and c); ...the first axiom is quite tricky...in fact, [math]1\le{1}[/math] is a bit confusing at first...but it shows that what you assumed is impossible. As for the Schröder-Bernstein theorem it's not less meaningful than any other theorem...sets are objects, which are more complex than numbers...infinite sets in particular. It's more difficult to compare them with one another, than to do the same with numbers...I mean comparing the number of elements of two, their cardinality...and not showing that they are the same. For example, X is a proper subset of Y doesn't imply that the number of elements in X is less than that in Y...sometimes the converse is true...it's much more difficult to show that something like axiom 2. also holds for sets, both finite and infinite...so I think it's still meaningful to prove a theorem which backs up both cases, as in the example.

For jordan: ...well...better to say, that was the question which implies what I had thought previously: Does it mean that we could see "ourselves", our own galaxy when it was young...or follow its formation back in time? I think what I said about Andromeda above would have been true if the answer to my question had been "yes"... For Thales: Yes...I thought about that yesterday...I think it follows in particular that observing "ourselves" at any moment in time is principally impossible, because we must then see our own image, but the light emitted by our galaxy would reach us again just when it has all ended...and it would be as old as the universe itself...am I right?

If we look at the Andromeda galaxy through a telescope, we see it as it was 2 million years ago...but if we look at it with a naked eye, then, according to my proposition, we must see it as it is at the moment...which is complete nonsense...it's clear, we zoom in the image of the object, in the information brought by light...though from the past...but we don't zoom in the object itself...I think I got it.

you're right... well...I told about the axes because they are defined...but this doesn't mean by far that they exist...it all just loses sense here...that's why it's an empty set.

This should go: 1. Take the set N(2k+1)={1,3,5,...}...We construct the bijective mapping from N (N={1,2,3,...} is the set of natural numbers, =the set of positive integers, for those, who are accustomed to another terminology) into N(2k+1). By doing this we assign to each element from N an element from N(2k+1)...thus we introduced an equivalence relation between these sets. From the condition that both sets are infinite and equipollent, it follows that they possess the same number of elements, i.e. they have the same cardinality: card N=card N(2k+1); 2. Now take the set N(2k)={2,4,6,...}. We procede in the same manner...constructing the mapping and showing that card N=card N(2k); 3. We see that from card N=card N(2k+1) and card N=card N(2k) follows the fact that card N(2k+1)=card N(2k). Thus we can claim that there is an equal number of odd and even numbers. This is a more or less intuitive approach to this task...because the latter assertion in 3. should be proved strictly...and there is in fact a theorem extending the property from 3. on any set. This is the so called Schröder-Bernstein theorem: [math](cardX\le{cardY})\wedge(cardY\le{cardX}) \Rightarrow (cardX=cardY)[/math]

This seems to be a complex elliptical cylinder with respect to y- and z-axes...or simply an empty set.

Oh...I think I didn't get the initial question...I thought there is something special about the cartridge case that makes attraction to both possible...but I didn't think about a "sequence" of events - trying and getting the cartridge case nearer to the glass stick and then to the ebonite rod...I saw two independent experiments...and that was a silly mistake...That's why, I think the question is meant to be answered exactly the way you did it.

Alright...as I said my proof is a bit formal...The thing is that I'm studying set theory on my own at the moment...and that problem has been part of my work, so to speak...though I found it on another forum...I decided to take it as another exersise to test methods of set theory. The core of my solution is work with infinite sets and relations between them...I just need a check for correctness... (to be continued)

Hi! It is said that as we look deeper into space we look in the past...we see younger galaxies and objects, which formed at an early stage of development of our universe...and here is my question... Does it mean that we could see "ourselves", our own galaxy when it was young...or follow its formation back in time?

Hello! I've worked on a problem concerning even and odd numbers recently...sounds quite simple actually...though I'm not sure whether my (more or less formal) solution is right. It is asked to find out whether there are more even or odd numbers...The result I came to is that there is an equal number of them...How would you solve this problem?

Hello! Just wondering...didn't think about that at all in the past... Why is actually a light cartridge case made from silver paper attracted both to a positively charged glass stick and to a negatively charged ebonite one?