Let X be a set and let $P(X)$ denote the set of all subsets of X, that is $P(X)=   \{A|A\subseteq X\}$. given $A,B\in P(X),$ define $A \triangle B:=(A\setminus B)\cup(B\setminus A)$.      Prove that $(P(X), \triangle)$ is a group.    Using the definition of a group I assume I have to show that there exist an associativity, identity, and inverse.
$G$ is the set of all subsets of a set $A$, under the operation of $\triangle\;$: Symmetric Difference of sets.  $A$ has at least two different elements.  I need to check if this is a group, and if it does to show if the group is abelian and/or finite.  Associativity - easy from Symmetric Difference. Identity element - empty group. Inverse element - each element is inverse to itself. Am I right?  abelian? finite?
Is it possible to install an old version of Mac OS on a new Mac? Use of inequality $1 - \cos (x) \leq x^2 /2$ In  answer the value of $1 - \cos(x)$ has to be evaluated in order to find its upper limit, if it exists.  In particular, $x = 2 \pi / n$. The answer is related to the length of a side of a regular $n$-gon inscribed into a unit-radius circumference; because the perimeter of the $n$-gon is always less than $2 \pi$, the single side must always be less than $2 \pi / n$.  The inequality  $$1 - \cos (x) \leq \displaystyle \frac{x^2}{2}$$ (1)  is used and the proof is completed with   $$2(1 - \cos(x)) \leq (2 \pi / n)^2$$  $$\sqrt{2(1 - \cos(x))} \leq 2 \pi / n$$  But it is well known that the cosine is a function $f(x) \in [-1;1]$, so $1 - \cos (x) \in [0,2]$. By using this information, we would obtain  $$1 - \cos (x) \leq 2$$ (2)  The proof would provide  $$2(1 - \cos(x)) \leq 4$$  $$\sqrt{2(1 - \cos(x))} \leq 2$$  which is a completely different result.   Why in that case it is preferable to use (1) instead of (2)? How to choose when it is convenient to use (1) and when to use (2) in a proof? Shell Type:  &gt;echo $SHELL /bin/ksh   Cronjob :   15 * * * * /bin/ksh /wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh &gt; /wls_domains/resMGT/logs/bea/data/script.log  The script is as below:  ##log files and lookup file "wlr3queue.txt" are in the same path of the script ##/wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh  Script :  #!/bin/ksh  dt=`date +%Y-%m-%d` xy=`date|awk '{print $4}'|awk -F":" '{print $1}'` yz=`expr $xy - 1` if [ $yz -le 9 ] then yz=0$yz fi  if [ $xy -gt 0 ] then  for i in `cat wlr3queue.txt` do  sum=0  count=`gzgrep -i "$dt" admin_resMGT_access.log* | grep "$yz:[0-5][0-9]" | grep "$i" | wc -l`  if [ $count -gt 0 ] then name=`echo $i | cut -d'/' -f3`  gzgrep -i "$dt" admin_resMGT_access.log* | grep "$yz:[0-5][0-9]" | grep "$i" | awk '{print $8}' &gt; consumed_time.txt  for j in `cat consumed_time.txt` do j_ms=`echo "$j * 1000" | bc` echo $j_ms &gt;&gt; consumed_ms_time.txt done  for k in `cat consumed_ms_time.txt` do sum=`echo "$sum + $k" | bc` done  avg=`echo "scale=2; ($sum/$count) "| bc`  min=`cat consumed_ms_time.txt | sort -n | head -1`  max=`cat consumed_ms_time.txt | sort -n | tail -1`  else  avg=0  min=0  max=0  fi ( echo $yz","$count","$avg","$min","$max )&gt;&gt;/wls_domains/resMGT/logs/bea/data/$name$dt.csv   rm -f consumed_ms_time.txt  done  else  echo "script won't execute at this hour" &gt; temp.txt  fi   I have executed the following commands and the script ran successfully.  ./wlr3queuetransaction.sh sh -x wlr3queuetransaction.sh /bin/ksh /wls_domains/resMGT/logs/bea/wlr3queuetransaction.sh  /bin/ksh/ wlr3queuetransaction.sh.   How to debug? What to do? A very basic question concerning the $R^2$ of OLS regressions   run OLS regression y ~ x1, we have an $R^2$, say 0.3 run OLS regression y ~ x2, we have another $R^2$, say 0.4 now we run a regression y ~ x1 + x2, what value can this regression's R squared be?   I think it's clear the $R^2$ for the multiple regression should be no less than 0.4, but is it possible for it to be more than 0.7? I want to create a table like this using Latex        ---------------------------------       |               sets            |       ---------------------------------       | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | --------------------------------------- | I1  | * |   | * | * |   | * |   | * | ---------------------------------------   Here is what I wrote  \begin{table} \caption{Multiprogram sets} \label{multiprogram}  \begin{tabular}{| c | c | c | c | c | c | c | c | c |}   \cline{3-9}   \multicolumn{8}{}{}    &amp;   Sets  \\   \hline   \multicolumn{1}{}{}   &amp; 1 &amp; 2 &amp;  3 &amp; 4 &amp; 5 &amp;  6 &amp; 7  &amp; 8 \\   \hline    astar         &amp;  &amp; * &amp;  &amp; * &amp;  &amp;  &amp; * &amp;   \\   \hline  \end{tabular} \end{table}   But it doesn't work! The output look like            -----------------------------                                  sets | ---------------------------------------         1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | --------------------------------------- | I1  | * |   | * | * |   | * |   | * | ---------------------------------------   Any way to fix that? Copying protected files between servers in one line? I'd like to copy squid.conf from one server to another.   The servers don't talk to each other. I'd like to go through my workstation. Both servers have the file, so it will be overwritten on the target. The files have 600 permission and are owned by root. root login via ssh is disabled (PermitRootLogin no). I'd like to do it in one line, if possible, since it will be a part of a setup guide.   I know to do  ssh source 'tar czpf - -C /etc/squid/ squid.conf' | \     ssh target 'tar xzpf - -C /etc/squid/'   to copy files between servers and preserve permissions. However, in this case I will get "Permission denied".  I also know I can do this:  ssh -t source 'sudo cat /etc/squid/squid.conf'   This way the -t allows sudo to ask for the admin password before outputing the content of the file.  The problem is, I don't know how to combine those techniques into something that will ask for the sudo password on each server, and transfer the file to its destination. Is this possible?  UPDATE: Here's the best I could come up with:  ssh -t source 'sudo tar czf /tmp/squid.tgz -C /etc/squid squid.conf' &amp;&amp; \ ssh source 'cat /tmp/squid.tgz' | \     ssh target 'cat &gt;/tmp/squid.tgz' &amp;&amp; \ ssh -t source 'sudo rm /tmp/squid.tgz' &amp;&amp; \ ssh -t target \     'sudo tar xzf /tmp/squid.tgz -C /etc/squid &amp;&amp; sudo rm /tmp/squid.tgz'   Calling this a one-liner seems like a stretch. I think I'll just break it down to separate steps in the setup guide. I have a table that I want to insert on a page, but at least one (perhaps both) of the following conditions are met:   The table is too wide to fit within the text block or page. That is, I'm exceeding some horizontal restriction. The table is too tall to fit within the text block or page. That is, I'm exceeding some vertical restriction.   What are my options to make this table fit? If it doesn't fit, regardless of my attempts, what other options exist? What is the difference between g++ and gcc? I have windows 7 in Arabic. Now I dowloaded QGIS 2.14.3-2 in French.  But when I installed it shows Arabic menu/tools...  How can I change the menu language to French? Why do corticosteroids harm COVID-19 patients? states that steroids could do more harm than good in treating coronavirus (COVID-19), referring to .      Understanding the evidence for harm or benefit from corticosteroids in   2019-nCoV is of immediate clinical importance.   It's unclear to me how exactly the use of corticosteroids (which is a common treatments for Asthma patients) can do harm other than the vague explanation of "steroids also impair the immune system's ability to fight viruses".   How exactly does the use of steroids harm the body in case of fighting a virus such as SARS-CoV-2 (2019-nCoV) that causes COVID-19? How to get Scilab 6.0.1 working on Ubuntu 18.04 LTS? I have installed clean Ubuntu MATE 18.04 LTS and installed Scilab from repository here with    sudo apt-get update sudo apt-get install scilab   After installation I tried to launch it from GUI (Applications → Other → Scilab) - it does not start. Then I tried to launch it from terminal with scilab command. Terminal output is as follows:  $ scilab /usr/bin/scilab-bin: error while loading shared libraries: libjava.so: cannot open shared object file: No such file or directory   so it can't find Java. I reported  about this problem.  I checked existence of libjava.so:  $ dpkg -S libjava.so   openjdk-11-jre-headless:amd64: /usr/lib/jvm/java-11-openjdk-amd64/lib/libjava.so   so it is in place.  What should I do to get Scilab 6.0.1 working on Ubuntu 18.04 LTS? I'm developing a site using Drupal 8 beta-14. I've created a view block of different terms and now I want to display it using code. How can I display it programatically?  I used to do it in Drupal 7 using this code but I'm confused about Drupal 8.  $block = module_invoke('block', 'block_view', '4'); $text_block = render($block['content']); Intuition behind logistic regression Recently I began studying machine learning, however I failed to grasp the intuition behind .   The following are the facts about logistic regression that I understand.   As the basis for hypothesis we use . I do understand why it's a correct choice, however why it's the only choice I don't understand. Hypothesis represents the probability that the appropriate output is $1$, therefore the domain of our function should be $[0,1]$, this is the only property of sigmoid function I found useful and appropriate here, however many functions satisfy this property. In addition, sigmoid function has a derivative in this form $f(x)(1-f(x))$, but I don't see the utility of this special form in logistic regression.   Question: what so special about sigmoid function, and why we cannot use any other function with domain $[0,1]$? The cost function consists of two parameters ${\rm Cost}(h_{\theta}(x),y)=-\log(h_{\theta}(x))$ if $y=1, {\rm Cost}(h_{\theta}(x),y)=-\log(1-h_{\theta}(x))$ if $y=0$. In the same was as above, I do understand why it's correct, however why is it the only form? For example, why couldn't $|h_{\theta(x)}-y|$ be a good choice for the cost function?  Question: what is so special about the above form of cost function; why cannot we use another form?   I would appreciate if you could share your understanding of logistic regression. Trig integral $\int ( \cos{x} + \sin{x}\cos{x}) \, dx $ Assume we have:  $$ \int (\cos{x} + \sin{x}\cos{x}) \, dx$$  Two ways to do it:  Use $$\sin{x}\cos{x} = \frac{ \sin{2x} }{2} $$ Then $$ \int \left(\cos{x} + \frac{\sin{2x}}{2} \right) \, dx = \sin{x} - \frac{ \cos{2x} }{ 4 } + C. $$ The other way, just see that $ u = \sin(x), du = \cos(x)dx $, $$ \int ( \cos{x} + \sin{x}\cos{x}) \, dx = \sin{x} + \frac{\sin^2{x}}{2} + C. $$  Now the part I don't see fully is, why aren't these results completely equal?  Taking the 2nd result, \begin{align} \sin{x} + \frac{\sin^2{x}}{2} &amp;= \sin{x} + \frac{1}{2} \, \left( \frac{1 - \cos{2x}}{2} \right) \\ &amp;= \sin{x} + \frac{1}{4} - \frac{\cos{2x}}{4}. \end{align}  So you have to absorb $\frac{1}{4}$ into $C$ for them to be equal.  Shouldn't they be equal straight away? Why does a free-falling body experience no force despite accelerating? For $f$ in dual space, there exists $x$ with norm 1 and $f(x)=\|f\|$ if space is reflexive (and nontrivial) Let $X\ne\{0\}$ be a reflexive space and let $f\in X^*$, where $X^*$ is the dual of $X$. I want to know: in general, does there exist an $x\in X$ with $\|x\|=1$, and $f(x)=\|f\|$, where $\|f\|$ is defined as $\sup\{|f(x)|:x\in X,\|x\|=1\}$?  I know this is true for $\mathbb{R}^n$ with the norm from the standard inner product, but I'm wondering if it is true in general. I have tried as following:  1. Turn off: export DISPLAY=:0.0 &amp;&amp; xset dpms force off  2. Turn on: export DISPLAY=:0.0 &amp;&amp; xset -dpms   Which works but that is a temporary solution, which does not fix my problem yet. When i reboot system it start again and again the same auto turn off mode.   How can i now remotely using SSH turn it off forever?  IMPORTANT:  - KIOSK (ATM machine will use this option) - 24/7 advertising display/slides will use this   Optional Note: this guide i followed   , but permanent solution did not worked. How do I use large bitmaps in .NET? I'm trying to write a light-weight image viewing application. However, there are system memory limitations with .NET.  When trying to load large bitmaps (9000 x 9000 px or larger, 24-bit), I get a System.OutOfMemoryException. This is on a Windows 2000 PC with 2GB of RAM (of which 1.3GB is used up). It also takes a lot of time to attempt loading the files.  The following code generates this error:  Image image = new Bitmap(filename); using (Graphics gfx = this.CreateGraphics()) {     gfx.DrawImage(image, new Point(0, 0)); }   As does this code:  Stream stream = (Stream)File.OpenRead(filename); Image image = Image.FromStream(stream, false, false); using (Graphics gfx = this.CreateGraphics()) {     gfx.DrawImage(image, new Rectangle(0, 0, 100, 100), 4000, 4000, 100, 100, GraphicsUnit.Pixel); }   Also, it is enough to do just this:  Bitmap bitmap = new Bitmap(filename); IntPtr handle = bitmap.GetHbitmap();   The latter code was intended for use with GDI. While researching this, I found out that this is in fact a memory issue where .NET tries to allocate twice as much as is needed in a single contigous block of memory.    I know from other applications (Internet Explorer, MS Paint etc.) that it IS possible to open large images, and rather quickly. My question is, how do I use large bitmaps with .NET?  Is there anyway to stream them, or non-memory load them? Why does Maximum Likelihood estimation maximize probability density instead of probability I am trying to understand Maximum likelihood estimation but it looks like I am missing something rather elementary.          suppose we have an iid random sample $X_1, X_2,..., X_n$ for which the probability density function of each $X_i$ is $f(x_i; \theta)$ where $\theta$ is an unknown parameter. Then, the joint probability density function of $X_1, X_2,..., X_n$ is given by:          $f(X_1=x_1,X_2=x_2,\cdots,X_n=x_n)=\prod_{i=1}^{n} f(x_i, \theta)$      In Maximum Likelihood estimation, we try to maximize $f$ as a function of $\theta$.  Question: Why do we maximize probability density instead of probability? In what way does it make sense?  Edit: As explained in the answers, the probability of choosing a finite number of sample points from a continuous probability distribution is zero, so maximizing probability doesn't make sense, but how does maximizing probability density make sense? What is the relationship between lands and mana in Magic and what do each of them do?  A good answer should especially include how a new player should think about them.    Note:  This is an attempt to create a canonical question about the relationship between lands and mana in Magic.  Beginners are often confused about this given the same symbols are used to mean different things across basic lands, non-basic lands, and mana costs.  This can lead to confusion about , about , or even about .