Wikipedia:Reference desk/Archives/Mathematics/2013 January 15

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January 15

Empty Sets and Neighbourhoods definition of a Topological Space

(See the page Topological Space)

After an hour straining my head around this definition I still cannot understand how any topology can be defined this way, because it does not account for the null set. (In my mind, that is. I realize that I must be wrong, but cannot see why, so my reasoning is below. Very grateful to anyone who can point out the flaw(s) in my logic!)

According to this definition, from axiom 1), the null set cannot be a neighbourhood of any x in X since, by definition, the null set does not contain any elements.

Then, the null set cannot be an element of any N(x) (since it would then be a neighbourhood)

Then the function N does not contain the null set within any of its output sets, so X with N - the topological space - does not contain the null set.

66.91.94.61 (talk) 07:23, 15 January 2013 (UTC)

Open sets are not defined as a result of N - set $A$ is open if and only if for every $x\in A$, it happens that $N(x)\subseteq A$. Now, if $A=\emptyset$, there are no $x\in A$, so, vacuously, for every $x\in A$, it happens that $N(x)\subseteq A$. Therefore $\emptyset$ is open. --Fadiha (talk) 07:43, 15 January 2013 (UTC)

Just a small correction: I believe you mean "for every $x\in A$, $A\in N(x)$", since $N(x)$ is the set of neighborhoods of $x$.--80.109.106.49 (talk) 08:37, 15 January 2013 (UTC)

Okay, so then in $X$ = {1,2,3,4}, $T(X)$ = {{}, {1,2,3,4}}, what is the value of x such that $\emptyset \in N(x)$ ? 66.91.94.61 (talk) 09:57, 15 January 2013 (UTC)

Another question: is the domain of N only {$x:x\in X$} ? 66.91.94.61 (talk) 10:10, 15 January 2013 (UTC)

The empty set is open because the sentence "For all x, $x\in \emptyset$ implies that $\emptyset$ is a neighborhood of x" is true. See vacuous truth and the principle of explosion. Sławomir Biały (talk) 12:15, 15 January 2013 (UTC)

Also see http://www.youtube.com/watch?v=SyD4p8_y8Kw, in which somebody struggles to understand this very point. Staecker (talk) 12:41, 15 January 2013 (UTC)

Hypergeometric function

How can this integral be evaluated by the Hypergeometric function?--Almuhammedi (talk) 09:27, 15 January 2013 (UTC)

$\int \sin ^{n}x\ dx$

You don't need anything special for that for n an integer - the integral is a polynomial of sin and cos, see List of integrals of trigonometric functions. However if you really want a hypergeometric form then using Wolfram Mathematica online integrator [1] gives

$-\cos(x)\sin ^{n+1}(x){(\sin ^{2}(x))}^{{\frac {1}{2}}(-1-n)}{_{2}F_{1}}\left({\frac {1}{2}},{\frac {1-n}{2}};{\frac {3}{2}};\cos ^{2}(x)\right)$

which I suppose isn't too bad but I kinda wonder what on earth it is doing with those sines at the beginning as they look like they should cancel. Dmcq (talk) 11:43, 15 January 2013 (UTC)

Thanks for reply, but I was curious to know how it eventually reached that formula. I also know the other methods like reduction and complex polynomial expansion and separation.--Almuhammedi (talk) 15:45, 15 January 2013 (UTC)

It's a repeated integration by parts. Note that the computation is simpler for the definite integrals $J(n):=\int _{0}^{\pi }\sin ^{n}x\ dx$, because there's no boundary term, and one finds a simple recursive relation:

$J(n+2):=\int _{0}^{\pi }\sin ^{n+2}x\ dx=\int _{0}^{\pi }(-\cos x)'\sin ^{n+1}x\ dx=$

$\int _{0}^{\pi }\cos x(\sin ^{n+1}x)'\ dx+[-\cos x\sin ^{n+1}x]_{x=0}^{x=\pi }=(n+1)\int _{0}^{\pi }\cos ^{2}x\sin ^{n}x\ dx=$

$(n+1)\int _{0}^{\pi }(1-\sin ^{2}x)\sin ^{n}x\ dx=(n+1)J(n)-(n+1)J(n+2),$

whence $J(n+2)={\frac {n+1}{n+2}}J(n)$ .

BTW, it is worth noting that as a consequence $J(n)$ is a rational number for all odd 'n' , and a rational multiple of $\pi$ for all even 'n'. From the inequality

$J(2n+1)\leq J(2n)\leq J(2n-1)$

one then finds an inequality $p_{2n}<\pi <p_{2n+1}$: It's the Wallis product. pma 20:14, 22 January 2013 (UTC)

Calculating the multiplicative order

I try to wrap my head around some number theory stuff, especially the multiplicative order. Say I want to calculate ord₄₉(2). I would start the following way (similar to the example given at multiplicative order#Example):

2² = 4 ≡ 53 (mod 49)

2³ = 8 ≡ 57 (mod 49)

However that seems to bring me away from what I want to get. Does that mean I have to first reduce 53 mod 49 or something? -- Toshio Yamaguchi 15:22, 15 January 2013 (UTC)

The equations you write are true, but a little beside the point. You might as well just say that 4 is 4 and 8 is 8; choosing a different number that's congruent to 4 or 8 mod 49 is legitimate, doesn't hurt (except it makes the calculations harder), but doesn't help either.

But eventually you get to

2⁶ = 64 ≡ 5 (mod 49)

Now you know that

2⁷ ≡ 10 (mod 49)

And you don't even have to compute 2⁷, you just multiply 5 by 2. And so on. Eventually you will discover that 2³⁶ ≡ 1 (mod 49). --Trovatore (talk) 20:00, 15 January 2013 (UTC)

Ooops — a couple silly blunders in the above. It first went wrong when I subtracted 49 from 64 and got 5. Of course, it's 15. You can figure out 2⁷ on your own. Also, the totient of 49 is 42, not 36, so I should have said 2⁴² ≡ 1 (mod 49). (That doesn't, by itself, tell you that the order is 42 — you have to rule out the possible smaller values 3, 6, 7, 14, 21.)

Trouble Reading an Article

Hello, I was wondering if you could help me figure out what the periods ('.'s) inside the expressions $(\forall xyz.xFy\wedge xFz\rightarrow y=z)$, $(\forall xyz.(x,y)\in F\wedge (x,z)\in F\rightarrow y=z)$, and $\{y\mid (\exists x.x\in A\wedge y=F(x))\}$ in the section on functions in the article explaining how historical decisions have decided how to implement mathematics in set theory mean. Can you guys help me?

Thanks,

     BCG999 (talk) 19:46, 15 January 2013 (UTC)

It looks like they're just to separate the quantifiers from the rest of the formula. I have seen this usage before but I don't think it's particularly widespread; most people use parentheses. --Trovatore (talk) 19:54, 15 January 2013 (UTC)

Not just quantifiers. Generally, a dot like this is equivalent to a left bracket whose matching right bracket is assumed to be at the furthest possible place which would make it a well-formed expression.—Emil J. 20:23, 15 January 2013 (UTC)

It's much more commonly used with the lambda calculus, to separate lists of bound variables from expressions, like this: $\lambda abc.\,a(bc)$. « Aaron Rotenberg « Talk « 21:02, 15 January 2013 (UTC)

Also, Haskell uses the dot if you write out type variables explicitly: forall a b. [a] -> [b] -> [(a, b)]. « Aaron Rotenberg « Talk « 22:52, 15 January 2013 (UTC)

I know I've seen that used in some book on finite model theory (or some such) before to add an extra condition on the quantifier; I've, actually, seen it in a few such books. Unfortunately, I'm away from my library, so not much help; just wanted to point out that it might mean more than just a divider. Though, every book I've seen it in mentioned what they were using it as short hand for; I also recall that the dot meant something diff for uni vs exi quants; don't know if that has anything to do with the above, though. Sorry if this reads poor, my laptop is on its last few minutes and am far from plugs!Phoenixia1177 (talk) 07:24, 16 January 2013 (UTC)

In $(\forall xyz.xFy\wedge xFz\rightarrow y=z)$ the dot means it is true that. In $\{y\mid (\exists x.x\in A\wedge y=F(x))\}$ the dot means such that. Bo Jacoby (talk) 09:43, 16 January 2013 (UTC).

So, shouldn't someone edit the article in which these formulas reside to make the points at which this punctuation mark sees usage more clear by replacing it with some other notation that's used more often and is easier to look up? Should we maybe take this discussion to the article's talk page?

BCG999 (talk) 18:31, 17 January 2013 (UTC)

P.S.: On another note, I didn't find your explanation of this notation where it should have been in the article on periods in the first place; should we fix this, too?

I converted the article to a more conventional notation. What I can’t figure out is the meaning of $\in \lceil TC(A)$ (twice) in Implementation of mathematics in set theory#The cumulative hierarchy; it looks like a typo, but I’m not sure what is the correct version (“\lceil” is quite a long string to be generated by an accidental slip of a finger).—Emil J. 19:01, 17 January 2013 (UTC)

     Huh; I didn't notice that because I'm not to that point in the article yet and am kind of going through things as I read on down the page. I'll take your word for it, but I also noticed that we might be able to shorten $\forall x,y,z\,\left(\left(x,y\right)\in F\wedge \left(x,z\right)\in F\to y=z\right)$ to $\forall x,y,z\,\left(\left(x,y,z\right)\in F\to y=z\right)$. Am I right?

     BCG999 (talk) 19:13, 18 January 2013 (UTC)

No, not at all. The condition says that F cannot contain two pairs with the same left coordinate, and different right coordinates. Your condition says something about triples, but F does not contain any triples (well, unless the encoding of triples is such that some triples are also pairs by accident). PS: Could you please stop inserting extra indents in your posts on talk pages? It interferes in a quite confusing way with the usual talk page indentation conventions. See also WP:INDENT.—Emil J. 19:26, 18 January 2013 (UTC)

Sorry, but I just assumed that I should type things out in the way that I normally write them down; I usually denote paragraphs with indentation instead of blank lines between paragraphs. Anyway, I should have known that the condition about which we were originally speaking dealt not with individual objects grouped together, but with ordered pairs, and should therefore include two of them inside itself.

BCG999 (talk) 19:40, 18 January 2013 (UTC)