I've been wondering recently the following:  Let $\sim $ be a symmetric and transitive relation defined on $S $. Let $a \sim b $, which implies $b \sim a $ by symmetry, and by transitivity, $a \sim a $. Hence, $\sim $ is reflexive.  I can't think of any counterexamples, although any are welcome. Also, if there is some counterexample, where could be the flaw in my reasoning be?
An equivalence relation is defined by three properties: reflexivity, symmetry and transitivity.  Doesn't symmetry and transitivity implies reflexivity? Consider the following argument.  For any $a$ and $b$, $a R b$ implies $b R a$ by symmetry. Using transitivity, we have $a R a$.  Source: Exercise 8.46, P195 of Mathematical Proofs, 2nd (not 3rd) ed. by Chartrand et al
Test for existence of nested JavaScript object key Is it possible to export a single object from an entire scene? siunitx tabular numbers with separated uncertainty I would like to format the space reserved for a number shown in a table with separated uncertainty. It seems that when I use the table-format option the brackets around the mantissa disappear. This is illustrated in the example below:  \documentclass{article}     \usepackage{siunitx}  \begin{document}     \begin{table}   \begin{tabular}{                   S[separate-uncertainty = true]                   S[table-format=1.1(1)e1,                   separate-uncertainty = true]                   }     {Header} &amp; {Header}\\     1.0(2)e2 &amp; 1.0(2)e2\\   \end{tabular} \end{table}     \end{document}     I have also tried to use the bracket-numbers and multi-part-units = brackets without solving this issue. Is there any workaround to center the number and keep the brackets like in the first column of the example? What is the 'rev' dropdown supposed to do? I only see this on sites where I don't have edit privileges, and only where there have been multiple revisions:        What exactly is this supposed to do?  Intuitively, I would expect it to change the text in the box to what it was at that revision, but it doesn't. And, if that's it, why only where I need to suggest edits? Tweak showlabels/showkeys - wrap the label form in a .tpl.php file This is the code in my template.php file.  function studiga_theme_theme($existing, $type, $theme, $path) {         $items['add_property_form'] = array(         'render element' =&gt; 'form',         'template' =&gt; 'node--14',         'path' =&gt; drupal_get_path('theme', 'studiga_theme') . '/templates/node',     );      return $items; }    function custom_add_property_form($form, &amp;$form_state) {     $form['first_name'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('First name')),     );     $form['last_name'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Last name')),     );     $form['address'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Address')),     );     $form['city'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('City')),     );     $form['state'] = array(         '#type' =&gt; 'select',         '#options' =&gt; array(             'default' =&gt; 'State',             '...' =&gt; '...',         ),     );     $form['zip'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Zip')),     );     $form['email'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Email')),     );     $form['phone'] = array(         '#type' =&gt; 'textfield',         '#attributes' =&gt; array('placeholder' =&gt; t('Phone')),     );     $form['submit'] = array(         '#type' =&gt; 'submit',         '#value' =&gt; 'Submit',     );      return $form; }   This is the code in node--14.tpl.php.                                          Contact Information                                                                                                                                                                                                                                                                                                                                                                                                                                                                                  &lt;div class="row"&gt;     &lt;div class="small-12 medium-12 large-8 large-offset-2 columns"&gt;         &lt;?php print render($form['submit']); ?&gt;     &lt;/div&gt; &lt;/div&gt;     But no fields are appearing - what am I doing wrong? You can't have a straight forward  in a tpl.php file can you? We are using following code in .php script for generate .pdf file.  $result1 = exec("R CMD Sweave /var/www/html/r/r_latex_sweave.Rnw");  $result2 = exec("pdflatex /var/www/html/r/r_latex_sweave.tex");   but, we are getting this errors when we are executing .Rnw and .tex files.  ! I can't write on file `r_latex_sweave.log'. Please type another transcript file name:  ! Emergency stop !  ==> Fatal error occurred, no output PDF file produced!  Note: We have .tex and .Rnw folder in /var/www/html folder? contractive property A topological property $\mathcal{R}$ is contractive if $(X,\tau)$ has property $\mathcal{R}$ and if $\tau^{\prime} \subset \tau$, then (A, $\tau^{\prime}$) has property $\mathcal{R}$.  A topological space is $T_B$ if each compact subset  is closed.  A topological space is called a US-space provided that each convergent sequence has a unique limit.  A topological space is called a C- C-space provided that  the  compact subsets  are precisely the closed sets.     Is property C-C , US contractive,  hereditary?      Is property $T_B$  contractive? From this  by orlp:     If I strap a rocket booster to a rocket in space and fire it for one second, then the work provided is much higher when the rocket is flying fast compared to when the rocket was stationary. In both cases the rocket fires for the same duration but in the former case the rocket travels a much greater distance during this period. What gives? Is Ubuntu an open source software? How to run the Ubuntu ISO live image from Windows without USB or DVD?  This is a self answered question. Let $R$ be a commutative ring with 1 then why does $a\in N(R) \Rightarrow 1+a\in U(R)$? We have a restricted Windows 7 computer that hides and prevents non-admin users from accessing the C Drive using the following policies:       However, they are able to circumvent this by typing the following into Explorer: \\localhost\C$  How can I disable this path but allow other UNC paths. For example they are allowed to access a shared folder on a different computer. eg. \\192.168.2.1\SharedTransfer  Note: Simply Enabling the Group Policy: Remove Run menu from Start Menu will not work as this blocks all UNC paths.  Update  The user, called Site User, is not a member of the Administrator group directly or indirectly. It is only a member of the group Users.  Accessed the following from: Control Panel\All Control Panel Items\Administrative Tools\Computer Management (Local) &gt; System Tools &gt; Local Users and Groups   Local Users and Groups  - Groups       - Administators              - Administrator              - Service User (our admin user)       - Users              - NT AUTHORITY\Auntenticated Users (S-1-5-11)              - NT AUTHORITY\INTERACTIVE (S-1-5-4)              - Site User (user account in question) How do you secure wood framing to a concrete basement floor? Indian CGPA to US GPA system Reference: What is variable scope, which variables are accessible from where and what are "undefined variable" errors? Java equivalent of unsigned long long? In C++, I enjoyed having access to a 64 bit unsigned integer, via unsigned long long int, or via uint64_t. Now, in Java longs are 64 bits, I know. However, they are signed.  Is there an unsigned long (long) available as a Java primitive? How do I use it? Why does the x86-64 / AMD64 System V ABI mandate a 16 byte stack alignment? When I capture packets using the packet analyser wireshark while visiting  I can see packets that use the HTTP protocol (e.g.a GET request). My Get request uses 4 protocols: Ethernet II, IPv4, TCP and HTTP.  For the IPv4 I can see my own local IPv4 address of my pc as the source, for the destination I see the IPv4 address 91.198.174.225. When I go to this address in my browser I get to some wikipedia related website that says'wiki does not exist'. Why am I not seeing the Hydrogen article when I go to that ip address? I am looking for an English translation of Markov's 1958 paper, On insolubility of the homeorphy problem, which I remember coming across on a website for a computational topology course (taught by Afra Zomorodian).