Let $|X|=n$. How to find all number of subsets $X$ consisting of an even number of elements?
Let $A$ be a non-empty set and $n$ be the number of elements in $A$, i.e. $n:=|A|$.  I know that the number of elements of the power set of $A$ is $2^n$, i.e.  $|\mathcal{P}(A)|=2^n$.  I came across the fact that exactly half of the elements of $\mathcal{P}(A)$ contain an odd number of elements, and half of them an even number of elements.  Can someone prove this? Or hint at a proof?
Resolve build errors due to circular dependency amongst classes Let $X$ be a topological space. The diagonal of $X \times X$ is the subset $$D = \{(x,x)\in X\times X\mid x \in X\}.$$   Show that $X$ is Hausdorff if and only if $D$ is closed in $X \times X$.   First, I tried to show that $X \times X \setminus D$ is open using the fact that $X \times X$ is Hausdorff (because $X$ is Hausdorff), but I couldn't find an open set that contains a point outside $D$ and is disjoint to it... Why does unequal amounts of RAM in each slot decrease performance? describes the RAM specifications for many of the MacBook laptops. In the section for my laptop (13" Pro Late 2011), it says:     Additional Notes: For best performance, fill both memory slots, installing an equal memory module in each slot.   This isn't the first time I've heard of this phenomenon, and it's definitely not specific to Apple.   Why is this the case? Other than the obvious decrease in available memory, why would it be worse to run with 6GB (1x4GB + 1x2GB) than with 8GB (2x4GB)? How to deal with "java.lang.OutOfMemoryError: Java heap space" error? I am writing a client-side Swing application (graphical font designer) on Java 5. Recently, I am running into java.lang.OutOfMemoryError: Java heap space error because I am not being conservative on memory usage. The user can open unlimited number of files, and the program keeps the opened objects in the memory. After a quick research I found  and others saying on Windows machine the JVM defaults max heap size as 64MB.  Given this situation, how should I deal with this constraint?  I could increase the max heap size using command line option to java, but that would require figuring out available RAM and writing some launching program or script. Besides, increasing to some finite max does not ultimately get rid of the issue.  I could rewrite some of my code to persist objects to file system frequently (using database is the same thing) to free up the memory. It could work, but it's probably a lot work too.  If you could point me to details of above ideas or some alternatives like automatic virtual memory, extending heap size dynamically, that will be great. Maximum number of edges in a simple graph? I found that the maximum number of edges in a simple graph is equal to $$\sum\limits_{i=1}^{n-1} i$$  Where $n =$ number of vertices.  For example in a simple graph with $6$ vertices, there can be at most $15$ edges.  If there were any more edges then $2$ edges would connect the same pair of vertices and thus would not be a simple graph.  Is this a known theorem? C# 4.0 optional out/ref arguments I'm looking for an algorithm that sorts strings similar to the way files (and folders) are sorted in Windows Explorer. It seems that numeric values in strings are taken into account when sorted which results in something like  name 1, name 2, name 10   instead of  name 1, name 10, name 2   which you get with a regular string comparison.  I was about to start writing this myself but wanted to check if anyone had done this before and was willing to share some code or insights. The way I would approach this would be to add leading zeros to the numeric values in the name before comparing them. This would result in something like  name 00001, name 00010, name 00002   which when sorted with a regular string sort would give me the correct result.  Any ideas? The portion of the tangent to any conic intercepted between the point of contact and the directrix subtends a right angle at the focus of the conic? How can I uninstall software? Seeing there are various ways to install on Ubuntu. What are the various ways to uninstall items?  Which ways always work? How can I be sure something was uninstalled? I created a custom counter 'rtaskno' and a command '\rtask' as following:  \newcounter{rtaskno} \newcommand{\rtask}{%    \stepcounter{rtaskno}%    \thertaskno}   I like to use them as following:  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}  \section{Task \rtask. Blah Blah..}   Later in the text, if I want to cross-reference 'Task 3', how can I do it? Where to put label? Can prematurely powering down a computer cause bad sectors in a hard drive? So if I'm correct, when you chose your starter gear and wealth you can either choose to roll some dice depending on your class and buy everything from that or you can choose the starter gear from your class and add the background items (including some gold). Here are my 2 questions:   If I choose the gear from my class, do I have to take all the equipment from only one of the (a) or (b) choices, or can I freely mix-and-match choices from the two sets? Can you spend the extra gold you get from your background on extra equipment (e.g. some soap and some fishing equipment) or do you have to start with that gold in its pouch?   The reason I ask is that I have recently created a level 1 wood elf druid and I mixed-and-matched the gear from the class, so that I chose a shortbow as a "(b) simple weapon" for the first point and a "(a) Scimitar" at the second point. My DM wasn't so happy about this, as he can't understand why I'm mixing (a) and (b).  He also wasn't so happy with me spending my 10gp from my outlander background. I decided that my druid wouldn't lay so much worth in 10gp compared to a bunch of stuff I got for that (e.g. some soap and some fishing equipment and some other useful things) and he seems to think that I'm supposed to just start with the gold instead. Mask and inherit the same systemd unit Suppose a package provides whatever.service file so certain events (including whatever.socket) cause a daemon to start with a certain configuration.  I want to disable this completely. In particular, I want the old socket and any other unit files that might refer to this service by name to be unable to start the daemon.  Instead, I want to create my own unit files for this daemon with a different .socket unit and different environment variables. I also want my override to automatically take account of possible upstream updates to the original unit file (which my current solution of copying whatever.service as whatever-modified.service and editing it unfortunately does not provide).  If I create the following symlinks and files: /etc/systemd/system/whatever.service → /dev/null /etc/systemd/system/whatever-modified.service → /usr/lib/systemd/system/whatever.service /etc/systemd/system/whatever-modified.service.d/fix.conf  systemd considers the new service to be masked as well and refuses to start it. How to avoid this? Consider the sequence $(\phi_n)_{n\in \mathbb{N}}$ of test-functions $\phi_n\in \mathcal{D}(\mathbb{R})$ defined by  $$\phi_n(x) = \dfrac{n}{\sqrt{\pi}}e^{-n^2x^2}.$$  I want to show that this is a delta sequence, in the sense that:  $$\lim_{n\to\infty}(\phi_n,\psi)=(\delta,\psi)=\psi(0), \quad \forall \psi\in \mathcal{D}(\mathbb{R}).$$  In that case we know that we have  $$(\phi_n,\psi)=\dfrac{n}{\sqrt{\pi}}\int_{-\infty}^{\infty}e^{-n^2x^2}\psi(x)dx.$$  The problem here is to compute this integral. The function $x\mapsto e^{-n^2x^2}\psi(x)$ has no singularities, hence contour integration wouldn't help. Integration by parts or substitution also doesn't seem to be of great help.  The major problem is that there is a function $\psi$ inside the integral about which we don't know really anything apart from the fact that it is $C^\infty$ and with compact support.  How can we compute this integral and in the end show that the sequence $(\phi_n)$ is a delta sequence?  Just to make clear, I'm here doing all of this in the context of the Riemann integral. Is a time series the same as a stochastic process? A stochastic process is a process that evolves over time, so is it really a fancier way of saying "time series"? How to prevent updating of a specific package? Should moderators be allowed to decline flags on their own comments? There are many examples of this, and I'd like to give a few:   A person who puts a lot of effort to help the community and earns reputation points. But that reputation is the motivation behind helping the community, not because they really want to. A student approaches a professor, and helps them in their teaching and doing research in order to have a good LOR (letter of recommendation). To a SEO, the content of the website must be good (well, this one is a little tricky, but you get the spirit.)   I'm not saying that the reputation/LOR/high-ranked is not deserved. It is. I just want to focus more on the selfish reason(s) behind that.  Any kind of answer can be accepted: single word, phrase, proverb or idiom. I want to see you.   I look forward to seeing you.      How can one say "to" in the first sentence is an infinitive marker and in the second sentence a preposition when we are given just the following two sentences and are asked to fill in the blanks?        I want to ____(see/seeing) you.   I look forward to __(see/seeing) you. In what cases I should skip 's in possessive case?