Wikipedia:Reference desk/Archives/Mathematics/2011 February 1

= February 1 =

Can somone help with the following proof
Hi everybody/ I seem to have difficylties in understanding a specific thing in the followind proof of the following claime. Maybe somone can help? claim: There exist a countable non-weakly Frechet-Urysohn space. and here is the proof: Let x be an arbitrary point of the Stone-Cech reminder ω* of the discrete space ω. Then $$ X = \omega \cup \{x\} $$ is the desired example.Indeed, it is a countable space. Let us assume that X is Weakly Frechet-Urysohn. As $$ x \in \overline\omega $$ there must br a countable infinite disjoint familly F such that x(RZ)F. Let A and B be any two infinite subfamilies of F. Then both $$ x \in \overline {\bigcup \mathbf{A}} $$ and $$ x \in \overline {\bigcup \mathbf{B}} $$ must hold. But this is impossible if we choose A and B to be disjoint subfamilly of F as in this case $$ \bigcup \mathbf{A} $$ and $$ \bigcup \mathbf{B} $$ are disjoint subsets of ω and x is an ultrafilter of ω. The thing which I don't understand is this, Why does the fact that $$ \bigcup \mathbf {A}$$ and $$ \bigcup \mathbf {B}$$ are disjoint implies that it is not possible that both $$ x \in \overline \bigcup \mathbf{A} $$ and $$ x \in \overline \bigcup \mathbf{B} $$? Here are the required definitions: Definition: A point x is called weakly Frechet Urysohn point if whenever $$ x \in \overline{A} \setminus A $$ there exists a countable infinite disjoint family F of finite subsets of A such that for every neighborhood V of x the subfamily $$ \{ F \in \mathbf{F}: F \cap V = \emptyset \} $$ is finite. If every point of a space is a weakly Frechet-Urysohn point then this space is called a Weakly Frechet Urysohn space. Definition: A point x \in X and a countable infinite disjoint family F of X are said to be in the Reznichenko relation (Rz), written x(RZ)F, if the following holds: For every neighborhod V of x, the subfamily $$ \{ F \in \mathbf{F}: F \cap V = \emptyset \} $$ is finite. Thanks for any of you who will be able to help! Topologia clalit (talk) 17:27, 30 January 2011 (UTC)


 * For any set $$S\subseteq\omega$$ it holds that $$x \in \overline S$$ iff S is in the ultrafilter that represents x. This cannot be true for two disjoint sets. –Henning Makholm (talk) 14:58, 31 January 2011 (UTC)

Matching Socks in the Dark
''You are in a dark room with no light. You have 19 grey socks and 25 black socks. What are the chances you will get a matching pair?''

This is a question I came across on a list of wacky/difficult interview questions. Assuming it's not a trick (the question doesn't explicitly say you can only grab two socks, maybe I have a flashlight, etc.), I think the answer is 942/1122, but that seems too messy for a question like this. Anyone care to confirm/deny/correct?--68.51.73.79 (talk) 06:31, 1 February 2011 (UTC)
 * Doesn't seem messy. Calculate probability that both socks are grey.  Calculate probability that both socks are black.  Add together.  Done.  71.141.88.54 (talk) 07:00, 1 February 2011 (UTC)
 * 942/1122 is not too messy, but it's implausibly large. I got the same result at first -- it seems that we both thought that 19+25 was 34 . But it's really 44, so the chances are 942/1892. –Henning Makholm (talk) 09:38, 1 February 2011 (UTC)
 * [ec] I got $$\frac{19\cdot18+25\cdot24}{44\cdot43} = \frac{471}{946}$$. -- Meni Rosenfeld (talk) 09:39, 1 February 2011 (UTC)

Area of a quadrilateral
A Quadrilateral with sides taken in order in a plane ,Whose equations are given.What is the Formula for finding Area of Quadrilateral in terms of Coefficents of variables present in  equaion of lines(sides).In fact i have find out a simple formula for  area of triangle whose equation of sides are given .now I want to get a general formula.if no one has find out it ,then i will try for it for my interest. TRue Path Finder


 * First find the coordinates of the corners by solving each pair of equations for neighboring sides using Cramer's rule. Then insert into the shoelace formula. –Henning Makholm (talk) 19:49, 1 February 2011 (UTC)

I know this method ,but i want a general formula for quadrilateral.Also for Pentagon,hexagon ,... if equation of sides are given. — Preceding unsigned comment added by True path finder (talk • contribs) 02:32, 2 February 2011 (UTC)
 * Well, for a set number n of sides, substitute letters for the coefficients in your formulae, and follow Henning Malcolm's method. The result will be a formula for the area in terms of the equations' coefficients. (That doesn't work for an arbitrary number of sides. (Well, if you do it for a few values of n and see a pattern (but I doubt you will), then you can try to prove a more general rule, e.g. by induction on n. (How does the area change when we add a side?))) 99.40.234.78 (talk) 08:43, 2 February 2011 (UTC)