Math Is Fun Forum

What exactly can't you see? I see the formulas for surface area and volume. As far as I can tell, there aren't any other formulas there. Can you not see the formulas for surface area and volume?

Try it this way. First prove that R is uncountable (you can do this by proving the open interval (0, 1) is uncountable, if you need a clue how to do this then ask). Then prove that Q (the set of rational numbers) is countable (using Cantor's diagonal argument). Then prove that the union of two countable sets is countable. From here you can argue that if I (the set of irrational numbers) is countable, then

(how do you type union without LaTeX? I can only make ∩ ) is countable, contrary to R's uncountability. Thus I must be uncountable.

This is probably a slightly sloppy way to prove it, as I threw it together pretty quickly. But it works.

Just remember this. If a set is countable (as in countably infinite), then it may be written S = {s[sub]1[/sub], s[sub]2[/sub], ...}, that is, you can index the elements of S with the naturals, or in other words there is a one-to-one correspondence between S and N. So if you want to prove a set is countable, you should show that there exists an indexing of the set using the naturals. For example, to prove T = {1, 3/2, 2, 5/2, 3, 7/2, ...} is countably infinite, you just need to give such an indexing. Here is one: t[sub]n[/sub] = (n + 1)/2. Now to prove that a set is uncountable, you need to show that no such indexing will give you every element of the set. These are the standard methods you should use to prove countability and uncountability.

It really depends on what exact theory you want to learn and how deep you want to get into it. Some of the math in particular areas of theoretical physics can be extremely nasty. A good background in linear algebra, vector and tensor analysis, abstract algebra, and differential geometry (probably topology and functional analysis too) is probably what you'll want to have some freedom in roaming around the world of modern theoretical physics. Are there any particular theories you are interested in?

I have a question, but first I must know this: does anyone here use Miktex and WinShell?

Also, isn't this kind of a weird forum for this thread to be in? I always have to think for a second to remember where to find this topic.

(M, d) denotes the metric space given by the set M equipped with the metric d. It's a shorthand method for specifying the set and the metric used in a metric space, and is pretty useful when you are considering two metric spaces with different metrics, such as (S, d[sub]1[/sub]) and (T, d[sub]2[/sub]). It is also easier than saying "let M be a set and d: M × M -> R be a metric in M."

Topology is certainly abstract, and it might be the most abstract mathematics you have come across yet. What is the title of your book and the name of the author(s)? I taught myself topology as well, and I must say everything is a lot easier when you don't do it as abstractly at first. A good way to learn basic topology is to do a lighter version which is applied to real analysis. Chapters 3 and 4 of Tom Apostol's "Mathematical Analysis" (2nd Edition) do this pretty well. If you could get access to this book you could get a better grip on the basic concepts, since Tom doesn't throw anything too complicated at you. The problems are very doable as well, but some of them can still provide a challenge. If you want an entire book containing plenty of introductory point set and algebraic topology that is easy enough of a read for self-teaching, consider "Topology" (2nd Edition) James Munkres. It might be the best general introduction to topology out there. Michael Henle's "A Combinatorial Introduction to Topology" is another good elementary book (it was my first ). It's a Dover book, so it'll only cost you a little more than $10 (my copy says $12.95) and you should be able to find it at a bookstore like Borders. "Topology" by Hocking and Young is another good Dover book (priced at $13.95 on my copy), but I must say that it is rougher than any of the others that I have mentioned, and it probably won't help you out nearly as much if you are reading it by yourself.

If you can't find any of these books at a local library and want to purchase them, I recommend using www.bookfinder.com. This website searches through most online bookstores and lists results by price in ascending order. In particular, I saved $90 on my copy of the Munkres text thanks to this website, so as you can see it is a very nice tool in helping you save money.

We can get a little insight if we look into the definition:

It's not to hard to see from here that an open set is a neighborhood of its points. So really in most cases when you are talking about open (sub)sets containing a point, such as has been done a few times in here, you are talking about a neighborhood of that point. A simple example of a neighborhood is an open ball. If you would like to see some proof explicitly using the term "neighborhood", just ask.

I think he might be "open relative to." In a metric space (M, d), the open ball of center a ∈ M and radius r is defined as B[sub]M[/sub](a, r) = {x ∈ M: d(x, a) < r}. Now if S is a subset of M, a point s ∈ S is called an interior point of S if some ball B[sub]M[/sub](s, r) lies entirely in S. As is the case in basic topology, S is called open if all of its points are interior points.

Now to answer Stanley's question, let X = R, Y = [-1, 1], and E = (0, 1], with each space having the standard Euclidean metric. 1 is an interior point of E relative to Y since B[sub]Y[/sub](1, 1) = (0, 1] lies entirely in E. Clearly all the other points of E are interior points relative to Y, so E is open relative to Y. But 1 is not an interior point of E relative to X, since B[sub]X[/sub](1, ε)
= (1 - ε, 1 + ε) will never be entirely contained in E with ε > 0. Then not all of the points of E are interior relative to X, and thus E is not open relative to X.

By hypothesis, we have a point x = x[sub]0[/sub] ∈ [a, b] such that

We want to show linear dependence, so consider the system

Since

it follows by a theorem (stated without proof here ) that there is an infinite number of pairs of nontrivial solutions of this system for c[sub]1[/sub] and c[sub]2[/sub]. Consider one such nontrivial pair and write the linear combination as

y[sub]3[/sub](x) is also a solution to our differential equation. We also have that

So y[sub]3[/sub](x) is a solution of

satisfying y[sub]3[/sub](x[sub]0[/sub]) = 0, y'[sub]3[/sub](x[sub]0[/sub]) = 0.

But y(x) = 0 is also a solution satisfying y(x[sub]0[/sub]) = 0 and y'(x[sub]0[/sub]) = 0. By the uniqueness of the solutions of our differential equation, y[sub]3[/sub](x) and y(x) = 0 are identical. Then

But by the construction of y[sub]3[/sub](x), c[sub]1[/sub] and c[sub]2[/sub] are not both zero. Then this equation has a nontrivial solution, and therefore y[sub]1[/sub](x) and y[sub]2[/sub](x) are linearly dependent.

darn, I didn't know darn was uncensored on this forum. Is this supposed to be the case? Also, I like the haiku mathsyperson posted. It was quite enjoyable.

I hate to reply to off-topic comments, but Jimi Hendrix is grossly overrated. Then again, I suppose it is hard to say who the greatest guitarist is, because music is so subjective. So in the end it is just my view that Hendrix is outclassed by many other guitarists, arguing about it will get us nowhere.

Anyway... something on topic... why did you go to the hospital if you had a giant footprint on your shirt? You should have gone to the laundromat instead.

Letting 0[sup]0[/sup] be 1 is a standard convention, which helps eliminate special cases where some theorems break down. There are some combinatorial and set-theoretic justifications for this convention. However, in analysis 0[sup]0[/sup] is treated as an indeterminate. Moreover, consider x[sup]x[/sup] = e[sup]x ln x[/sup]. There is no real number corresponding to ln 0, so x[sup]x[/sup] is not defined at x = 0; in other words 0[sup]0[/sup] is undefined. The calculator says it is equal to 1 because of the common convention mentioned above; but a real decent calculator would notify you something like "Warning: 0^0 replaced by 1" .

In "advanced" texts I notice that "log" is used instead of "ln". I'm not sure why they don't save themselves a letter and stick with "tradition", but I can see how they justify replacing the base 10 log notation with base e: base 10 logarithms just aren't that commonly used farther down the road. I think that ln is the clearest notation though. Is ln used in England?

I would bet even as much as 9/10 math majors wouldn't know who he is, unless they are particularly interested in mathematics competitions (that is how I first learned of him). But who knows, I've never asked anyone if they know of him before.