Let $x \sim y$ be defined as meaning that the ordered tuple $(x, y)$ is in some set $S$. If this is the case, an equivalence relation on the set $S$ is defined as a subset of $S \times S$ with the following properties:   For all $x$ in $S$, $x \sim x$. If $x \sim y$, then $y \sim x$. If $x \sim y$ and $y \sim z$, then $x \sim z$.   I think this is an over-determined definition, because (1) is a consequence of (2) and (3), as long as it contains anything in addition to the empty set:  Assume that the equivalence relation contains at least point, $x \sim y$. By (2), it contains $y \sim x$. By (3), since $x \sim y$ and $y \sim x$, we also have $x \sim x$. Since we put no constraints on $x$ except that it be a part of the relation, this implies that for any point $x \sim y$, we have $x \sim x$. (We also have $y \sim y$, since $x \sim y$ implies $y \sim x$.) So why do we need to specify (1)?
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
Detecting pattern at the end of a line with grep Postfix Domain Whitelisting in Combination with Sender Authentication I'd like to whitelist several domains in my Postfix configuration in such a way that all e-mails from those domains are accepted as long as they're properly authenticated using SPF and/or DKIM, should it be used by that domain. This means that e-mails coming from those domains should not be subjected to the usual DNSBL and Greylisting process, only to a SPF and DKIM verification.  Right now I have managed to save e-mails coming from those domains from the Greylisting milter by configuring a sender_access file and using the postmap utility to convert it to a suitable hash-database.  main.cf:  ... parent_domain_matches_subdomains = yes smtpd_recipient_restrictions =   permit_mynetworks   permit_sasl_authenticated   reject_rbl_client zen.spamhaus.org   reject_unauth_destination   check_sender_access hash:/etc/postfix/sender_access   check_policy_service unix:private/tumgreyspf ...   sender_access:  domain1.com OK domain2.net OK   I've found out about  and also found a , but it appears like that is a DNSBL-only measure and is only slightly helpful in my case. I normally use the iPhone timer to sleep the device after I've dozed off to some music or podcast. Unfortunately this means that when I ask Siri to set a timer for some cooking, Siri waits the allotted time and turns off the phone rather than notify me that the time is up. Any solutions to this? Can I add new methods to the String class in Java? Command line option to check which filesystem I am using? Is there a command that will show which file system (ext3, ext4, FAT32, ...) the various partitions and disks are using?   Similar to how sudo fdisk -l lists information about disks and partitions? TAR 6 GB worth of files into separate 1.9 GB TAR files I have 28,000 files of size 6.7 GB in the directory aaa/input. I need create TAR files in the directory aaa/output, such that each TAR file cannot exceed 1.9 GB, so I have to tar the 28,000 files into 4 TAR files (6.7/1.9) as Log_(current date)_01.tar Log_(current date)_02.tar ... etc.  such that each tarball is less than 1.9 GB. Is there a way to do this using a Unix script? Can the rogue repeatedly hide in combat to sneak attack the same enemy? One of my players is a rogue and has been, on his turn, using Cunning Action to hide as a bonus action and immediately afterwards attacking. I have been ruling that this does work to grant him advantage as an unseen attacker as I believe that is the RAW, but it seems a little strange. Is he really, in the span of 6 seconds, ducking behind a corner and popping back out and this completely disorientates his target? I guess I just wanted to make sure I was ruling correctly. What's this I hear about First Edition Unix being restored? that the source code of First Edition Unix had been restored from tapes.  The response was amazement and a request for more information.  What have people done?  Who did it?  When?  And how? The question is pretty straight forward. I can get the exit code of the previous command by using $?. But what exactly is it?  I don't think it is a variable, because you can't have any special characters in the name of variable other than _.  foo?foo=10   will result in the error: foo?foo=10: command not found  So if it is not a variable, what is it? Are there others like it? HTML 5 Video stretch Software Name: Oracle SQL Developer Version 17.2.0.188  I have three tables: orders, order_details, and products.  I wanted to run a query to update units_in_stock in the products table based on quantity in order_details, where the associated orders row has been marked as shipped, and has a shipping date.  Here's what I tried:  UPDATE Products SET (Products.Unit_In_Stock = Products.Unit_In_Stock - Order_Details.Quantity) INNER JOIN (     Orders INNER JOIN Order_Details ON Orders.Order_ID = Order_Details.Order_ID     ) ON Products.Product_ID = Order_Details.Product_ID WHERE (         ((Orders.Shipped_Date) IS NOT NULL)         AND ((Orders.This_Order_Was_Shipped) = "YES")         );   When I run this, I get:  Error at Command Line : 1 Column : 45   Line 1, Column 45 is the equal sign in the very first statement that said  Products.Unit_In_Stock = Products.Unit_In_Stock - Order_Details.Quantity   Can you help me get this working? Linux: Case-INSENSITIVE Filesystem I am trying to create a simple page using Bootstrap within salesforce. I need this page to have the standard salesforce header and sidebar.  When I use the standard bootstrap library via a CDN and the page is affecting all the CSS styling for my standard salesforce headers and side bars. Any help on how to prevent this ? I am trying to create a schema page in LaTeX. It shall look like this.   How can I create schema code which look like this image? There should be also a click to go function. When the user clicks on the user_schema link he should be redirected to that page.  What classes I can use for this? Any suggestions would be appreciated.    \documentclass{article} \usepackage[utf8]{inputenc}  \begin{document} \maketitle  \section{User schema} \mySchema{Users} {     _id: ObjectId - Id do registro     name: String - Nome do usuário     date: Date - Data de Cadastro     ren_date: Date - Data de renovação do Cadastro     email: String - Lorem ipsum dolorsitamet, consecteturadipisicingelit, sed do eiusmodtemporincididunt ut labore et dolore magna aliqua. Ut enim ad minimveniam, exercitationullamcolaborisnisi ut aliquipexeacommodoconsequat. Duisautereprehenderit in voluptatevelit esse cillumdolore eu fugiatnullapariatur. Occaecatcupidatat non proident, sunt in culpa quiofficiadeseruntmollit.     access_log: Array [          {              date: String - Data da ocorrência.             user: ObjectId - Usuário que executou.             action: String - Ação Executada.         }         {              date: String - Data da ocorrência.             user: ObjectId - Usuário que executou.             action: String - Ação Executada.         }     ]     \link{group}{group_id}: ObjectId - Id do grupo. } \end{document}   Here is the image how should it appear: Fundamental Theorem of Calculus. Fundamental Theorem of Calculus says  Let $f$ be an integrable function on $[a,b].$ For $x$ in $[a,b],$  let $F(x)=\int_{a}^{x}f(t)dt.$ Then $F$ is continuous on $[a,b].$ If $f$ is continuous at $x_{0}$ in $(a,b),$ then $F$ is differentiable at $x_{0}$ and $F^{'}(x_{0})=f(x_{0}).$  Now my question is why we are saying that this is "Fundamental Theorem of whole Calculus".  It is clear that it connects intergal and differential calculus. But i am not getting the exact concept, why this is so important as its name is fundamental theorem of calculus. Why we says that it is fundamental theorem of calculus?  Please suggest me. Thanks in advance. When I tried to install Matlab using the Software Center, it prompted me for the Location of MATLAB installation(s). Since I am trying to install it, I don't know what to do. What is the plural form of "Software"? Does anybody know where to download the QGIS toolbar icons? What is the difference between # and . when declaring a set of styles for an element and what are the semantics that come into play when deciding which one to use? What causes a java.lang.ArrayIndexOutOfBoundsException and how do I prevent it?