I know that the symmetric difference looks like that when having three sets A,B,C:    I want to make a prove with Venn diagrams to prove the associativity of the symmetric difference. What are some good ways to create this proof in a clear way?  I really appreciate your answer!
By $A\oplus B$ we denote the symmectric difference of two sets. The definition is $A\oplus B =(A\setminus B) \cup (B\setminus A)$. Now I hope to show that $A\oplus (B\oplus C) = (A\oplus B)\oplus C$. I remember that there's an elegant proof, but I forget its detail. (The first step is to show $A\oplus (A\oplus B) = B$, and maybe applying this formula, we can obtain associativity.)
How can I check the integrity of a downloaded Ubuntu CD? I've just downloaded a CD image of Ubuntu. I know the installer has a check CD option, but how can I check that the image is good before burning it to CD? What is the most portable method of launching a terminal emulator? Where do the Python unit tests go? If you're writing a library, or an app, where do the unit test files go?    It's nice to separate the test files from the main app code, but it's awkward to put them into a "tests" subdirectory inside of the app root directory, because it makes it harder to import the modules that you'll be testing.    Is there a best practice here? Is it possible to run a  job which needs the  command?  Like:   sudo rm somefile Can you grammatically end a sentence with "with"? Do you want to come with? Can I come with?  I seem to hear this construction more often in recent years, but it still grates on my ear. I know it's often said that one shouldn't , but I class that rule along with  (i.e. — pedantic tosh). Is it always informal/casual speech to end a sentence with &quot;with&quot;? Was it ever thus? Can my friend and I play minecraft together with different mods? Compute: $\lim_{n \rightarrow \infty}\sqrt{n}(A_{n+1} − A_n)$ where $A_n = \frac{1}{n}(a_1 + a_2 + \cdots + a_n)$ Access .mat file containing matlab classes in python Etymology of "history" and why the "hi-" prefix? According to , history comes from the same root as story. If they are from the same word, where does hi- come from? Is it just because of the English habit of taking names from other languages verbatim or is there something more to it? Diagonal straight lines in residuals vs fitted values plot for multiple regression I am observing strange patterns in residuals for my data:   [EDIT] Here are the partial regression plots for the two variables:      [EDIT2] Added the PP Plot   The distribution  seems to be doing fine (see below) but I have no clue where this straight line might be coming from. Any ideas?   [UPDATE 31.07]  It turns out you were absolutely right, I had cases where the retweet count was indeed 0 and these ~ 15 cases resulted in those strange residual patterns.   The residuals look much better now:   I've also included the partial regressions with a loess line. I need to save/compile a few times and a bunch of weird files (like foo.tex.Ak6664) appear in my Dropbox folder each time I need to compile between two and four times before I see changes I make in the editor (TeXstudio) and each time I compile, a new .tex file is created with random letters and numbers at the end.   For example, If I create the file hello.tex and compile two to four times I'll get the file,   hello.tex.QU6664      then I make some small changes in the editor and compile two to four times I'll get the additional file,   hello.tex.Ak6664   After continuing to do this (each time compiling between two to four times before I see the changes I made), I now have many files that look like  hello.tex.QU6664, hello.tex.Ak6664, hello.tex.YX6664, hello.tex.sN6664, hello.tex.Ss6664, ...   A little bit of additional information (in case it's relevant): Everything was working fine up until a few days ago, i.e. only needed to compile once and only the file hello.tex was created. I still only get one .aux,  .synctex and .pdf files. I use TeXstudio to write and compile. I don't think it has anything to do with the bibliography (I tried removing bibliography altogether and still having the same problem).  How do I prevent these additional files from appearing and only compiling once like I used to? Has anyone had a similar problem? What EXACTLY happens when I ask to delete my account? I have meticulously read through similar questions, answers and their comments but I still do not understand how the deletion is done. From what I understand I email the customer service and then they do what exactly?     1) They disable my account so that I can no longer use it?      2) They reassign all of my questions and answers to some other   account? To which one?      3) They delete all my questions and answers so that they go away from   public view?   What is the flexibility (if it exists) of the deletion procedure? Is it possible to erase the traces of my former association with the site. Can I request any of the following be done?     4) All of my questions and answers be deleted from public view?      5) Some of my questions and answers be deleted? If yes what criteria?      6) Can I have all of my contributions be assigned to some other   account which runs as sort of a "collective account" and already has   tons of content so that it doesn't happen like my contributions just   move to some other url but still stay grouped under that location and   therefore that location can be discovered as representing my account?      7) Can I have my contributions dispersed through several "collective   accounts?      8) Can my current public name be removed from any comments to my posts   so that my posts cannot be identified later as once belonging to me?   Please be polite with me. I wouldn't have asked if I have found the answer in the other questions. The other questions are also old and it is not clear what the situation is as per 05/2012.  UPDATE (for those interested):  I asked the Stackoverflow team about the procedure. Apparently none of the points 2-8 I asked above are possible, only 1 will take place. The link to my user page from my contributions is going to be removed and replaced with non-clickable text of an anonymized user name like "user1234". Whether this number is the same as my current user_id is not clear. That's all the information I got.  I'm going now to rename myself to "Anonymous" then request the account deletion. Those interested in the procedure will witness the exact result on this page.  Farewell. I can ask any of:     Do you know a breakfast place nearby?   Do you know a nearby breakfast place?   Do you know a good breakfast place?   but I really can't ask:     Do you know a breakfast place good?   Is there a general rule for determining whether an adjective must come before the noun or may come, Spanish-style, after it? Windows Power Options Clarification So I was looking through the Windows 7 power options via Control Panel today, and I realized I had a couple of unresolved questions as to the nature of various choices. I was wondering if someone could give me a clear and concise explanation of the difference between the following four selections:   Sleep Mode Hibernate Mode Standby Mode Turn off hard disks after...   For example, regarding option 4 - does this mean that after the elapsed time limit, the computer will go into standby, hibernate, sleep, or just turn off? Below is an image of the options (albeit on a Windows XP system, but I think the same concepts should apply): What is the meaning of 'though'? Okay, of course I've used different popular online dictionaries and the translation makes sense in the most cases.    Sometimes I find usages of 'though' where I don't get the meaning of the word.   An example I've found on twitter     That face, though. [Image of a dog looking weird]   The tweet came without any context or conversation.  When I translate this example to my native language - german - this makes absolutely no sense. At least for me.  So is the above usage some kind of slang? What is the meaning of though in this case?  For the completeness an example that makes absolutely sense for me     Though he tried very hard, he failed the course. I wanted to create a WP plugin. So I requested and my request was approved by WordPress.org But I don't want to develop that plugin anymore. How can I delete the repository forever or request its deletion. Windows 7 / 8 tool for amplifying sound / equivalent to pavucontrol? I'm watching live streamed videos, but the volume sometimes is barely loud enough.  With Windows, I can only turn the system volume and the flash volume up to maximum, and then bad luck if that's still not loud enough.  On Linux, on the same notebook, I have the same problem, but I can use pavucontrol (pulseaudio volume control) to amplify the sound volume beyond 100%.  Unfortunately, I can't always work on Linux, since Visual Studio and SQL-Server only run on Windows.  Is there any tool or way to amplify volume on Windows 7 / 8 , without having to resort to run Linux in a virtual machine? How do I use method overriding in this case? Do mankind and manmade activities/constructions have any effect on the rotation of the Earth? Is the Echo of a Echo Knight actually a creature? I've always been certain that the echo created by an Echo Knight fighter using their 3rd-level feature Manifest Echo is not defined in game terms as a creature, since the description of the echo never explicitly tells it is a creature. But the presence of people believing the echo to be a creature in lieu of the fact that it is immune to all conditions and might make saving throws (game characteristics typically associated with creatures only) brought this doubt to my mind.  from Jeremy Crawford implies it is just an image and not a creature, further confirming that the feature should tell you, but  implies that the echo is an object whilst the feature doesn't explicitly say so. (Included for reference and insight, since tweets are not eligible as official ruling) Is the echo a creature?