Union of disjoint countable sets is countable
Prove that the union of two disjoint countable sets is countable
Changing sides in tennis This  says:      A player has to win six games, and at least two more than the foe, to claim a tennis match. Players switch sides after every odd game, as in the first, third and fifth games, and so on. Players also switch sides at the end of a set if they're playing an odd number of games.   So, you start, 1-0, and change sides. If the game ends 6-0, 6-2 or 6-4 (or an even number of games), you play the first game of the new set, change sides, and you are back to the previous rule.  But what if a player wins 6-1 or 6-3? You win the set, then change sides. Now, you play a new set, do you change again at 1-0 (therefore after having played just one game), or wait until the 1-1 or 2-0, therefore now changing after an even number of games in the second set? A singularity, by the definition I know, is a point in space with infinite of a property such as density.  Density is Mass/Volume.  Since the volume of a singularity is 0, then the density will thus become infinite because Mass/0 = undefined  However, is it possible to have a singularity with a mass of 0? 0/0 is indeterminate, but would it be possible for a singularity to exist even if its mass and volume were zero? I`m using the Newton-Raphson method for obtaining MLE for parameters for maximizing my objective function. At each iteration, I want to check that is the Hessian matrix negative definite or not and I see the Hessian matrix is not negative definite at some iterations. So, I want to use the Fisher scoring algorithm ( of derivation) In this situation, the Fisher information matrix is indefinite!! Is it possible? What are the reasons for this? Are the second-order derivatives wrong? Or are values of expectations incorrect? How does the NSAutoreleasePool autorelease pool work? As I understand it, anything created with an alloc, new, or copy needs to be manually released. For example:  int main(void) {     NSString *string;     string = [[NSString alloc] init];     /* use the string */     [string release]; }   My question, though, is wouldn't this be just as valid?:  int main(void) {     NSAutoreleasePool *pool;     pool = [[NSAutoreleasePool alloc] init];     NSString *string;     string = [[[NSString alloc] init] autorelease];     /* use the string */     [pool drain]; } HUD is not working after upgrade My HUD is not working after I upgraded from 11.04 (and or 11.10).  It prompts for command and it tries loading for 4 seconds then gives up. decode a quoted printable message in php I have the string which that encoded in .  How do I decode this in php?  Lorem ipsum dolor sit amet, consectetuer adipiscing elit. Vestibulum felis = pede, vestibulum et, ullamcorper in, ultrices eget, metus. Suspendisse in n= ulla. Sed justo. Donec dapibus auctor neque. Pellentesque faucibus ante nec=  metus. Vivamus feugiat, velit rhoncus semper suscipit, libero augue venena= tis leo, nec ornare nulla elit eget diam. Nullam sollicitudin. Minimizing Application to system tray using WPF ( Not using NotifyIcon ) I have some questions about a suspended user's earning and activity.   Can suspended users get reputation back after the suspension period? If a user is suspended for 7 days and he/she earns reputation during this period. Then will it be calculated after completing suspension period? If a suspended user has awarded a bounty to other user and the awarded bounty marked as illegal. Then will that bounty rolled back to their account?  Can suspended users get any alert from comments or chat room using @username method? Can suspended users post comments on his/her question/answer? If another user replies or ask a question on suspended user's post. Then, can a suspended user give response back? I've read a lot of conflicting advice on whether it's advisable to store eggs in the refrigerator. The case against seems twofold: (1) that eggshells are porous, and eggs can take on unwelcome flavours, or even spoil faster in the fridge, and (2) that cooking with cold eggs can be problematic. But lots of people refrigerate eggs through perceived necessity.  Do eggs last longer in the fridge, and is the shelf life gain worth it? Or should I just stick them in the cupboard? As , the retag privilege serves a lot more confusion and a lot less of its benefit due to the suggested edit system. It must be done away with. It will be done away with. ♪  In light of this decided course of action, we're going to need a new snack for the road between "Create Tags" at 300 reputation and "Established User" at 1000 reputation. We'd like your help in thinking about some ideas on what would be a nice privilege to sit at this level. Since we're... a bit dry on our end.  The primary guideline in deciding what kind of privilege to put here is acknowledging its position in the overall hierarchy of privileges. 500 reputation is an early-intermediate user, one who has started to really get the hang of the site and its function. It can be something that provides convenience in how they currently can act on the site, or it could be some new functionality that fits as an early ability (especially considering there's no new abilities until 2000 reputation's "Edit Everywhat" ability).  I look forward to hearing what creative ideas y'all can provide. Area 51 is miscounting my proposal followers I know of one person who is following my proposal who doesn't show up on the proposal webpage.    My proposal is for quantitative finance: .  It shows 19 people following the proposal (on the side of the page), and the total listed at the top of the page is 19 followers.  Yet one user is not listed: .  This makes me wonder if other followers are also not being counted? Reputation and inbox notification shows up again after refresh Why don't I need to install any Drivers for Ubuntu Show that if $g \circ f$ is injective, then so is $f$. The Problem:  Let $X, Y, Z$ be sets and $f: X \to Y, g:Y \to Z$ be functions.  (a) Show that if $g \circ f$ is injective, then so is $f$.  (b) If $g \circ f$ is surjective, must $g$ be surjective?  Where I Am:  So, I really have trouble with these, for some reason. I can draw pictures and make sense of the problems, but writing down proofs is very difficult for me.   Basically, for (a), I ended up with some complicated statement involving an implication implying another implication and then tried to derive a contradiction. It just got so convoluted that I couldn't make sense of it anymore, and I know there's a quick, elegant way to show it.  For (b), I know that $g$ need not be surjective. Once again, though, proving it directly from definitions has given me a bit of a headache.  Any help here would be appreciated. Thanks in advance.  The Proofs!  Ok, I did it. Thanks for the help, everyone! Let me know if there's anything wrong with these proofs, or if they could bet any better.  (a) Suppose $f$ is not injective. Then $$ f(x_1)=f(x_2) \implies x_1 \ne x_2 \text{        }(*).$$ Let $f(x_1)=y_0=f(x_2)$ and let $g(y_0)=z_0$. Then $$ (g \circ f)(x_1) = (g \circ f)(x_2) = g(y_0) = z_0. $$ Since $g \circ f$ is injective, $$ (g \circ f)(x_1) = (g \circ f_2) \implies x_1 = x_2. $$ However, this contradicts $(*)$. Therefore, $f$ must be injective.  (b) Suppose $g$ is not surjective. Then  $$ \forall y \in Y, \exists z \in Z \text{ such that } g(y) \ne z \text{         }(**).$$  Since, $g \circ f$ is surjective,  $$ \forall z \in Z, \exists x \in X \text{ such that } g(f(x)) = z \text{       } (***). $$  Let $f(x) = y$. Then,  $$ g(f(x)) = g(y) = z. $$  Because of $(***)$, this is true for all $z \in Z$, which contradicts $(**)$. Therefore, $g$ must be surjective. I created one event receiver and deployed it, but only users who are site collection admins are able to view this web part. I have included the SPSecurity.RunWithElevatedPrivileges but the same thing. What am I missing?  Below is my code   public override void ItemAdding(SPItemEventProperties properties)     base.ItemAdding(properties);     SPSecurity.RunWithElevatedPrivileges(delegate()        {SPWeb web = properties.OpenWeb();         base.ItemAdding(properties);         string name= web.Lists[properties.ListId].Fields["first name"].InternalName;          var list = web.Lists["location names"];    });} Unparsed HTML shows on search result page When you search on any of the SE network sites, with a tag (so that a description is shown); it shows the tag info content box, with the highlight-search-term span elements.  For eg. I searched ; ; and the results are as shown:                  Notice the &lt;span class='search-highlight'&gt; boxes around searched tag terms. Can I visit other Schengen countries on a long-term Schengen visa or residency permit? How can I create launchers on my desktop? babel shorthand "| doesn't work in macros The package babel with the ngerman option (and most other languages as well, but not english) provides the shorthand "|, which prevents a ligature at the given position (see p. 81 of the , e.g. Auf"|lage will result in the word Auflage without an fl-ligature (as is desired because, in German, you don't want ligatures across morpheme borders).  However, when placed in a macro, "| doesn't work anymore. Why is this and how can I circumvent it?  \documentclass{article}  %\usepackage[T1]{fontenc}% doesn't help \usepackage[ngerman]{babel}  \newcommand{\foo}{"|}  \begin{document}  ff  f"|f  f\foo f  \end{document} Show that f(x) = inf{g(x1)+h(x2)} is convex subject to x1+x2=x and where g(.) and h(.) are convex functions.  Can I just go about this by using the regular definition of a convex function or which properties should I use? I know that the sum of convex functions is convex, but the infimum part confuses me. Thank you