need help with the following question:  Let $S \subseteq P(\mathbb N)$ be defined inductively so that:   $\mathbb N \in S$  for all$ ~ a \in \mathbb N \Longrightarrow \mathbb N\setminus\{a\}\in S$ for all $A,B \in S \Longrightarrow A\cap B \in S$   Prove that if $A\subseteq \mathbb N~ $ and $~ \mathbb N $ \ $A$ is finite then $A \in S$  thanks!
$A \subseteq \mathcal{R}(N)$ and given that (by inductive definition):   $N ∈ S$. If $a \in R$, then $R \setminus \{a\} \in A$.    I need to prove that for each $A \in S$, $N\setminus A$ is finite. How can I prove this formally?
Unable to install LanguageTool 2.5 extension for LibreOffice 4.2 on Ubuntu 14.04 I am not able to install LanguageTool extension for LibreOffice. Getting exception when I try to install it.   p.s. Somebody says that the solution is installing openoffice.org-java-common package but Ubuntu 14.04 does not have this package in its repository.  System details:  OS: Ubuntu 14.04 LTS  LibreOffice: 4.2 (4.2.3.3)  LanguageTool: 2.5  Java: Oracle JDK 1.8.0_05     Exception message: Why doesn't saturation current in the photoelectric effect depend on the frequency of light absorbed ny the metal emitter? Where is this picture from? How do I use Reverse Image Search to find the source of an anime/manga image? So I see an image that I believe is from either an anime or a manga. What are some ways that I can go about finding what series it's from? Try these options:   (browser built-in support and plugins)        Note: Please note that as of May 9, 2016, ALL  questions are now . Therefore, you can't ask an identification request question on this site. However, such questions are still welcome  for users with at least  (globally on StackExchange).    Protip: If you are using Google Image Search in a non-English language,  or just use  for better, more accurate results. Error: invalid environment block. Press any key to continue...   The error message appears immediately after the boot screen on purple background. Ubuntu 12.04.1 LTS. If I use  two different router in the same network while both are running in the default router mode then we can access the setup page or setup interface from browser just by IP address of corresponding router. But when I use one of the router as repeater then I can only enter the setup page of the router which is running in router mode. Why router's setup page cannot be accessed when using the router as repeater? Now how can I change the settings of the router in repeater mode? The only way is to reset router with the hardware key then setup again. A Vista virtual machine I use only has a 10 GB virtual hard disk -- and I'm worried about it running out of space over time. I used  to check for the culprits..  And the primary culprit is that /winsxs or &quot;Windows Side-By-Side&quot; folder.  What is the Windows Side By Side folder? Well, that's .  All of the components in the operating system are found in the WinSxS folder – in fact we call this location the component store.  Each component has a unique name that includes the version, language, and processor architecture that it was built for.  The WinSxS folder is the only location that the component is found on the system, all other instances of the files that you see on the system are “projected” by hard linking from the component store.  Let me repeat that last point – there is only one instance (or full data copy) of each version of each file in the OS, and that instance is located in the WinSxS folder.   So looked at from that perspective, the WinSxS folder is really the entirety of the whole OS, referred to as a &quot;flat&quot; in down-level operating systems.  This also accounts for why you will no longer be prompted for media when running operations such as System File Checker (SFC), or when installing additional features and roles. That explains why the folder starts off big, but not why it gets larger over time – the answer to that question is servicing.   In previous versions of Windows the atomic unit of servicing was the file, in Windows Vista it’s the component.  When we update a particular binary we release a new version of the whole component, and that new version is stored alongside the original one in the component store.  The higher version of the component is projected onto the system, but the older version in the store isn’t touched.  The reason for that is the third part of why the component store gets so large.  Lots of other people :  A coworker recently received his every two year work laptop refresh and was in the middle of loading his new laptop with Vista. For some reason, his hard drive C: was partitioned to only 25GB and the rest of the space reserved for a much larger drive D:. Anyway, after loading many Windows Updates, he was left with hardly any drive space left on C:, and for some reason, the WinSxS directory had ballooned to over 8GB. While I knew what the WinSxS (Windows Side-by-Side) directory was for, I wasn't quite sure why that directory would explode in size so fast with normal operating system updates.  I don't really care about having all these duplicate files for safety, but I do care about the VM exploding due to an out of disk space error. So.. can /winsxs be safely pared down in any way? ..  Administrators should not, for any reason, take it upon themselves to clean out the /winsxs directory - doing so may prevent Windows Update and MSI from functioning properly afterwards.  Preventing accidental deletion from the directory is accomplished by putting a strong security descriptor on the directory that inherits to its children.  If this /winsxs folder gets any larger, I'm going to have to re-image this virtual machine.. which I really don't want to do! I work in D8, I create a custom module first_module. By modules create a url and display all nodes by *.html.twig file.  Write code in controller.php  public function contentthree() {    $connection = Database::getConnection();    $query = $connection-&gt;select('node', 'n');   $query-&gt;fields('n', ['nid', 'type']);    $groups = $query-&gt;execute()-&gt;fetchAll();    foreach ($groups as $group) {     $events[] = [$group-&gt;nid, $group-&gt;type];   }    return [     '#theme'  =&gt; 'first_module_display',     '#events' =&gt; $events,   ]; }   In *.module  function first_module_theme($existing, $type, $theme, $path) {    return [     'first_module_display' =&gt; [       'variables' =&gt; ['events' =&gt; NULL],       'template'  =&gt; 'first-module-list',     ],   ]; }  function _get_node_details() {    $output = 'Welcome D8';    $output .= 'Node Details';    return $output; }   and create templates first-module-list.html.twig and display data  {%for key, data in events %}  {%for key1, data1 in data %}     {{ data1 }}  {% endfor %}  {% endfor %}   work fine and showing data correct. But, my question is how to call function _get_node_details() and how to render node details in first-module-list.html.twig files? I am writing a document for which I need the margin to be exactly as follows: left, right, top and bottom text margin all equal to 1in, distance from bottom of the paper to page number equal to 0.75in (I don't have any footnote or header to set).  I tried to use the fullpage package in LaTeX but it does not give me the 0.75in to the page number. Is there a way to force that? I tried to set manually the margins for a 8.5x11 paper but I am not succeeding.    I tried but after writing:  \documentclass[PhD]{class_file} \usepackage[margin=1in,footskip=0.25in]{geometry}   I get the error message \paperwidth (0.0pt) too short.  I don't see the \paperwidth is set to 0.0pt anywhere though..  Also it seems to me that depending on which machine I use to compile my tex file, I get different margins, even when I set them up manually (without using geometry). Is it possible?    Using the geometry package things seem to work a bit better now (I changed machine and the problem with the paper width previously mentioned disappeared).   \usepackage[margin=1in,footskip=0.25in]{geometry}   gives me without problem upper, right and left margin of 1in. But the lower margin is 1.2in and I can't change it to be 1in even by using bottom=1in or bottom=0.8in. I really don't see how this can happen. Maybe a problem with the spacing between lines? The class files I am using can be fund at   I erased the margin settings of uclath12.clo as the margin I need are different from the ones that this file gives. Thank you. I have installed Ubuntu alongside Windows 8.1. But the wifi is not working in Ubuntu (perhaps the drivers are not present), which worked fine in windows. Also I cant connect with Ethernet since my laptop does not have an Ethernet port. So is there any way for me to connect to the internet in ubuntu? Edit: Output of the terminal command  lspci -knn | grep Net -A2:  vikram@vikram-Inspiron-11-3148:~$ lspci -knn | grep Net -A2 01:00.0 Network controller [0280]: Broadcom Corporation BCM43142 802.11b/g/n [14e4:4365] (rev 01)     Subsystem: Dell Device [1028:0018]     Kernel driver in use: bcma-pci-bridge vikram@vikram-Inspiron-11-3148:~$    I did what Pilot6 suggested. Here is the output:    vikram@vikram-Inspiron-11-3148:~$ sudo dpkg -i *.deb [sudo] password for vikram:  Selecting previously unselected package bcmwl-kernel-source. (Reading database ... 163804 files and directories currently installed.) Preparing to unpack bcmwl-kernel-source_6.30.223.141+bdcom-0ubuntu2_amd64.deb ... Unpacking bcmwl-kernel-source (6.30.223.141+bdcom-0ubuntu2) ... Selecting previously unselected package dkms. Preparing to unpack dkms_2.2.0.3-1.1ubuntu5.14.04_all.deb ... Unpacking dkms (2.2.0.3-1.1ubuntu5.14.04) ... Setting up dkms (2.2.0.3-1.1ubuntu5.14.04) ... Setting up bcmwl-kernel-source (6.30.223.141+bdcom-0ubuntu2) ... Loading new bcmwl-6.30.223.141+bdcom DKMS files... First Installation: checking all kernels... Building only for 3.16.0-30-generic Building for architecture x86_64 Building initial module for 3.16.0-30-generic ERROR (dkms apport): kernel package linux-headers-3.16.0-30-generic is not supported Error! Bad return status for module build on kernel: 3.16.0-30-generic (x86_64) Consult /var/lib/dkms/bcmwl/6.30.223.141+bdcom/build/make.log for more information. modprobe: FATAL: Module wl not found. update-initramfs: deferring update (trigger activated) Processing triggers for man-db (2.6.7.1-1ubuntu1) ... Processing triggers for initramfs-tools (0.103ubuntu4.2) ... update-initramfs: Generating /boot/initrd.img-3.16.0-30-generic vikram@vikram-Inspiron-11-3148:~$ Non-uniform distribution of p-values when simulating binomial tests under the null hypothesis I heard that under the null hypothesis the p-value distribution should be uniform. However, simulations of binomial test in MATLAB return very different-from-uniform distributions with mean larger than 0.5 (0.518 in this case):   coin = [0 1]; success_vec = nan(20000,1);  for i = 1:20000     success = 0;     for j = 1:200         success = success + coin(randperm(2,1));     end     success_vec(i) = success; end      p_vec = binocdf(success_vec,200,0.5);     hist(p_vec);   Trying to change the way in which I generate random numbers didn't help.  I would really appreciate any explanation here. Is my C1/D visa valid for transit in the US, if I want to go on a vacation to Mexico? Prove the divergence of the sequence $\left\{ \sin(n) \right\}_{n=1}^{\infty}$. I am looking for nice ways of proving the divergence of the sequence $\left\{x_n\right\}_{n=1}^{\infty}$ defined by $$x_n=\sin{(n)}.$$ One (not so nice) way is to construct two subsequences: one where the indexes are picked such that they lie in the intervals  $$I_k=\left(\dfrac{\pi}{6}+2\pi(k-1),\dfrac{5\pi}{6}+2\pi(k-1)\right)$$ and one where they lie in $$J_k=\left(\dfrac{7\pi}{6}+2\pi(k-1),\dfrac{11\pi}{6}+2\pi(k-1)\right).$$ If ${x_n}$ converges, then all its subsequences must converge to the same limit, but here the first subsequence has all its values in the interval $\left[ \frac{1}{2}, 1\right]$ while the second has its values in $\left[-1,-\frac{1}{2}\right]$. Contradiction.  Filling the details of this proof is rather tedious and not very elegant. Anyone has a better idea? If I select a feature and run self.iface.actionIdentify().trigger() it will open the identify window but won't open the selected feature's data, how can I make it open the data of the selected feature? Dynamic Server Name in SQL Statements (from clause) K-means - comparing solutions with SSwithin elbow-method: minimum "too early", or non-monotonic curve Are There Any Mechanics for Removing Possession? So I'm thinking of a few adventure hooks for my players, and I thought about the possibility of them removing a demon that has possessed a noble. Are there currently any mechanics or rules surrounding possession and the removal of entities?  If not, does anyone have any play-tested homebrew rules? Controlling Volume Mixer Similarities and differences between correlation and regression If I want to investigate how two continuous variables are linked, what is the difference between calculating the correlation coefficient (Pearson's $r$) versus calculating the (simple linear) regression coefficient?  I see people who, if the regression coefficient is significantly different from zero, talk about the two variables as if they are correlated, which is confusing as it suggests that the two coefficients (correlation, regression) are the same thing.  Having said that, isn't $r$ a measure of the (regression line) slope anyway? I'm confused!! Testing Classification on Oversampled Imbalance Data I am working on severely imbalanced data. In literature, several methods are used to re-balance the data using re-sampling (over- or under-sampling). Two good approaches are:   SMOTE: Synthetic Minority Over-sampling TEchnique () ADASYN: Adaptive Synthetic Sampling Approach for Imbalanced Learning ()   I have implemented ADASYN because its adaptive nature and ease to extension to multi-class problems.  My question is how to test the oversampled data produced by ADASYN (or any other oversampling methods). It is not clear in the mentioned two paper how they performed their experiments. There are two scenarios:  1- Oversample the whole dataset, then split it to training and testing sets (or cross validation).  2- After splitting the original dataset, perform oversampling on the training set only and test on the original data test set (could be performed with cross validation).  In the first case the results are much better than without oversampling, but I am concerned if there is overfitting. While in the second case the results are slightly better than without oversampling and much worse than the first case. But the concern with the second case is if all minority class samples goes to the testing set, then no benefit will be achieved with oversampling.  I am not sure if there are any other settings to test such data. tag:* links incorrectly when using Save And Copy Profile To All Stack Exchange Accounts The  tag in my Stack Exchange profile links to different sites depending on where it's being viewed:  Profile on StackOverflow:  [tag:CSS] =&gt; https://stackoverflow.com/questions/tagged/css   Profile on StackExchange:  [tag:CSS] =&gt; https://stackexchange.com/questions/tagged/css   Profile on CodeReview:  [tag:CSS] =&gt; http://codereview.stackexchange.com/questions/tagged/css    I'd like to have these all link to https://stackoverflow.com/questions/tagged/css. You can ; why not add the ability to specify the site with tags? &lt;!-- tag: stackoverflow --&gt;[tag:css] &lt;!--tag: meta --&gt;[tag:feature-request]  The current behavior also cause a bug on my :    All tags send me to find a Panda aka 404 Page Not Found.