Here is my question:     Give a formal set theoretic argument to prove that, for all sets $X, Y$ and $Z$, $$X\setminus(Y\setminus Z)=(X\setminus Y)\cup(X\cap Z)\;.$$   I know this is true, using the identity found , so I'm unsure how to answer the question as it appears to be an identity in itself?!  How do you think I should answer it?  EDIT: This is my final answer (thank you for your help). Is it correct?  //My interpretation of the half of your answer $$\begin{align*} &amp;\text{Suppose }X\setminus(Y\setminus Z)\\ &amp;\implies a\in X\setminus(Y\setminus Z)\\ &amp;\implies a\in X\land a\notin (Y\setminus Z) &amp;\quad\text{[Definition of complement]}\\ &amp;\implies (a\in X\land a\notin Y)\lor (a\in X\land (a\in Y\land a\in Z)) &amp;\quad\text{[distributive law]}\\ &amp;\implies X\setminus Y\cup (X\cap Y\cap Z) &amp;\quad\text{[associative law]}\\ &amp;\implies X\setminus Y\cup (X\cap Y\cap Z)\subseteq X\setminus Y\cup (Y\cap Z) &amp;\quad\text{[}A\cap B\subseteq A\text{]} \end{align*}$$  //My half of the answer  $$\begin{align*} &amp;\text{Suppose }X\setminus Y\cup (X\cap Z)\\ &amp;\implies a\in (X\setminus Y\cup (X\cap Z)) \\ &amp;\implies a\in (X\setminus Y)\lor a\in(X\cap Z) &amp;\text{[definition of union]}\\ &amp;\implies (a\in X\land a\notin Y)\lor (a\in X\land a\in Z)&amp;\text{[def of complement and intersection]}\\ &amp;\implies a\in X\land (a\notin Y\lor a\in Z) &amp;\text{[Distributive Law]}\\ &amp;\implies a\in X\land (T\land (A\notin Y\lor a\in Z))&amp;\text{[Identity]}\\ &amp;\implies a\in X\land ((a\notin Y\lor a\in Y)\land (a\notin Y\lor a\in Z)) &amp;\text{[Excluded middle]}\\ &amp;\implies a\in X\land (a\notin Y\lor(a\in Y\land a\in Z)) &amp;\text{[Distributive law]}\\ &amp;\implies a\in X\land a\notin(Y\setminus Z) &amp;\text{[Definition of complement]}\\ &amp;\implies a\in X\setminus(Y\setminus Z)&amp;\text{[Definition of complement]}\\ &amp;\implies (X\setminus Y)\cup (X\cap Z) \subseteq X\setminus (Y\setminus Z) \end{align*}$$  Since A ⊆ B, and B ⊆ A, sets A and B are equal.
Show that  $A \setminus ( B \setminus C ) \equiv ( A \setminus B) \cup ( A \cap C )$  This is an exercise I was trying to do ( not homework ) and I got stuck as follows:  Working from $A \setminus ( B \setminus C )$:  $x \in A \wedge x \notin ( B \setminus C )$ $x \in A \wedge ((x \in B \wedge x \in C) \vee (x \notin B \wedge x \in C) \vee (x \notin B \wedge x \notin C) )$  At this point I'm not sure how to proceed.  Working from the other side $ ( A \setminus B) \cup ( A \cap C ) $:  $ x \in ( A \setminus B) \cup ( A \cap C ) $  $ x \in ( A \setminus B) \vee x \in ( A \cap C ) $  $ ( x \in A \wedge x \notin B ) \vee ( x \in A \wedge x \notin C ) $  $ x \in A \wedge (x \notin B \vee x \notin C ) $  $ x \in A \wedge x \notin ( B \cap C ) $  $ A \setminus ( B \cap C ) $  I could also use some formating tips like how display logical not and how to get the lines to display properly without blank lines inbetween. :-)
How can I remove all duplicates so that NONE are left in a data frame? In Windows there’re perhaps only a couple of important folders (by important I mean important in my logical picture of the Windows file system) in the installation drive (in my case C:\). Namely Program Files and Windows. I simply stay away from Windows folder and the “add remove program files” is good enough to handle the program files folder of Windows. Of course there’s a folder named Users where the users (who are not admins) can access only their folders.  Thus there’s a clear picture at some level in my mind of the Windows file system. In Ubuntu, when I reach the location /, there’s a huge list of folders, most of which I have no clue as to what they contain. The /bin folder seems to be the equivalent of the Windows folder in windows. The /usr folder seems like it’s the equivalent of the Users folder in Windows. But even the /home folder looks like it can fit the bill.  Please understand that I do understand, that Ubuntu (Linux) has a different character than that of Windows, i.e., there need not be exact equivalent of Windows functions, in Ubuntu. All I am looking for is a bit more clearer picture of the Ubuntu file system.   This question is a part of a bigger question which I am splitting up to make it more answerable. The original question can be found here: Pretty formattings for warning and info blocks I'm looking for pretty formatting for warnings, and others for some info or tips.  For the moment, I use the following formatting but maybe someone has one better idea.    To obtain this, I use the following code.  \documentclass[10pt,a4paper]{article} \usepackage[utf8]{inputenc} \usepackage[x11names,svgnames]{xcolor} \usepackage{graphicx} \usepackage{fourier-orns} \usepackage[framemethod=tikz]{mdframed} \usetikzlibrary{calc}  \makeatletter % Default settings for warnings     \newcommand{\warningSymbol}{\raisebox{0.8\depth}{\danger}}      \definecolor{warningColorText}{named}{Red3}     \definecolor{warningColorLine}{named}{Red3}     \definecolor{warningColorBack}{named}{LemonChiffon1}     \definecolor{warningColorBackSymbol}{named}{white}  % Inline mode     \newcommand{\warning}[1]{%         \textcolor{warningColorText}{\warningSymbol{}\,#1}%     }  % Block mode     \tikzset{         warningsymbol/.style={             rectangle,             draw  = warningColorText,             fill  = warningColorBackSymbol,             scale = 1,             overlay         }     }      \mdfdefinestyle{warning}{%         hidealllines      = true,         leftline          = true,         skipabove         = 12,         skipbelow         = 12pt,         innertopmargin    = 0.4em,%         innerbottommargin = 0.4em,%         innerrightmargin  = 0.7em,%         rightmargin       = 0.85em,%         innerleftmargin   = 1.1em,%         leftmargin        = 0.85em,%         middlelinewidth   = .2em,%         linecolor         = warningColorLine,%         backgroundcolor   = warningColorBack,%         fontcolor         = warningColorText,%         firstextra        = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)             node[warningsymbol] {\warningSymbol};         },%         secondextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%         middleextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%         singleextra       = {             \path let \p1=(P), \p2=(O) in ($(\x2,\y1-4)$)              node[warningsymbol] {\warningSymbol};         },%     }      \newmdenv[style=warning]{@Warning}     \newenvironment{Warning}{\let\warning\relax\begin{@Warning}}{\end{@Warning}} \makeatother  \usepackage{lipsum}  \begin{document}  \warning{IMPORTANT !} \lipsum[1]  \begin{Warning}      \lipsum[1] \end{Warning}  \end{document} crisscross, dillydally, riffraff, etc The diagonalizable matrices are not dense in the square real matrices I am stuck with the identity  $$ F_{2n} = \sum_{k=1}^n \binom{n}{k} F_k, $$  which happens to be formula . I am using induction, but so far without too much result.  $$ \sum_{k=1}^{n+1} \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n+1}{k} F_k = \\ F_{n+1} + \sum_{k=1}^n \binom{n}{k} F_k + \sum_{k=1}^n \binom{n}{k-1}F_k =\\ F_{n+1} + F_{2n} + \sum_{k=1}^n \binom{n}{k-1}F_k, $$   I am stuck with the sum $ \sum_{k=1}^n \binom{n}{k-1}F_k$, because, on the other hand   $$ F_{n+1}+ \sum_{k=1}^n \binom{n}{k-1}F_k = F_{2n+1} $$  Which would conclude the proof. What am I missing? Drupal unexpected "Fatal error: Class name must be a valid object or a string [...] common.inc" on every page "Fatal error: Class name must be a valid object or a string in C:\xampp\htdocs\uc\includes\common.inc on line 7506"  The error above showed up unexpectedly. I was adding a "Long text" field to a content-type, but when I expected to be brought to a saved field configuration page, I instead got a Fatal error on every page.  I had a look at includes\common.inc:  /**  * Get the entity controller class for an entity type.  */ function entity_get_controller($entity_type) {   $controllers = &amp;drupal_static(__FUNCTION__, array());   if (!isset($controllers[$entity_type])) {     $type_info = entity_get_info($entity_type);     $class = $type_info['controller class'];     $controllers[$entity_type] = new $class($entity_type); // Line 7506   }   return $controllers[$entity_type]; }   However, I just don't understand Drupal at a low enough level to understand the relationships between these variables and functions.  I also searched Drupal for people with a similar problem:  (1)  who claims that I have to install the submodule "Relation Endpoints Field", but searching Drupal, .  (2) :  DELETE FROM cache; DELETE FROM cache_filter; DELETE FROM cache_menu; DELETE FROM cache_page; DELETE FROM watchdog;   (via , misleading name, I know.)  Clearing those tables from the cache did not help.  (3) And another user . Because I can't find a module specifically pertaining to old search facets, I went into MySQL system table, and marked the Search module's line with a 0 status, indicating to stop loading the module altogether.  On that same link as (3), one user suggested enabling the , so I dropped all the tables, imported the latest stable .sql, and installed the module, but after recreating the problem (trying to add a field to a content-type), that as well concluded fruitless.  Still, the problem remains  The Fatal error shows up on every page, making the site completely non-functional. Any help or direction would be immensely appreciated. Thank you for looking. Legal issues for a "fangame" of a commercial franchise? I've played around in the past of the 2D Flash fan-made version of the popular Valve game Portal.  It has basically the exact same mechanics, but as a 2D side-view flash game.  Lately I've toyed around with creating a simplified flash version of some of my favorite franchises (such as Zelda, Mario, Splinter Cell, etc.).  Of course, nothing is "stopping" me from just using the same mechanics and just changing the name to something not copyrighted.  I have witnessed projects that have tried to directly port a 3D game into a 2D simplified version, with the exact same plot and puzzles (Ocarina of Time in 2D comes to mind), and I have seen them being shut down.  But I am not likely going to exactly duplicate any game...just borrow some of the source material and basic underlying mechanics.  For example, if I were to make a Zelda-influenced fan game, I would have items such as the Hookshot, Boomerang, and all other sorts of ones from the Gameboy Zeldas; just with a custom plot and dungeons.  With likely the same mechanics.  What are the legal issues involved in this?  I used to think it was a "Don't do this ever; it's a bad idea."  But then I've seen the Portal clone, and now I'm curious. Get full name twice in Bibliography I use biblatex with style=authoryear and when there are two references of the same author in the bibliography there is just a dash. How can I get the full name in the second entry too? $f$ has the form $f(x) =ax^2+bx+c$. Every differentiable function $f:R \rightarrow R$ with the property that $(2h)f′(x) =f(x+h)−f(x−h)$ for all $x \in R$ and all $h$ has the form $f(x) =ax^2+bx+c$.  I would like to get some help on this one. Taking partial derivatives was an idea, but is that legal? Video lectures on graduate level Classical Electrodynamics This is a rather broad question. Does anyone know of good video lectures for graduate level classical electrodynamics? Repeated error  I have been repeatedly encountering the following error when my laptop freezes. It may occur multiple times during the day, and should take several minutes to recover.  Source Windows  Summary Hardware error  Date ‎11/‎1/‎2018 10:11 AM  Status Not reported  Description A problem with your hardware caused Windows to stop working correctly.  Problem signature Problem Event Name: LiveKernelEvent Code:   144 Parameter 1:    3003 Parameter 2:    ffffad8c0b1a75c8 Parameter 3:    40010002 Parameter 4:    0 OS version: 10_0_17134 Service Pack:   0_0 Product:    256_1 OS Version: 10.0.17134.2.0.0.256.48 Locale ID:  1033   More details  Per my best reflection, the hardware error occurs when I am using the laptop on its docking station. Here is what I observe:   While using the laptop as normal, the "freeze" will occur and I can hardly operate the laptop: mouse cursor won't move continuously and Alt+Tab won't be able to switch across two windows. Shortly after (usually several minutes), things will go back to normal. Upon inspecting through the Performance tab in Task Manager, there had been no spike in anything, be it CPU, Memory or Disk. When I open the View reliability history tool, I will find another entry of Windows Hardware Error   Machine specification:   Thinkpad T480, i7-8650U (newly replaced module thanks to the manufacture)   With the same pair of docking stations, an older unit of T480 has had exactly the same problem.  Docking station: . Same issue has occurred on two identical docks with  slightly different monitors and accessories. OS specification: OS build 17134.376. Most recent collection of "reliability report":    Possible solutions   First, I would like to learn what exactly went wrong, hardware-wise: the manufacturer's support team did not bother too much with the precise error messages, and have ordered me to reinstall the Windows OS up to 5 times in total. [This is what they called "debugging".]   Please advise if you can decipher the recurrent "Hardware error".   Secondly, I will try to push the manufacturer to replace the docking stations, and see if the same "freezing problem' persists. 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Or am I affected with some malware or virus? AudioSessionSetActive and AudioSessionSetProperty is deprecated in iOS 7