Prove that if $A$, $B$, and $C$ are sets then $(A - B) \cup (A - C) = A - (B \cap C)$.   I have the proof for the first direction:  Let $x \in (A - B) \cup (A - C)$ be given. Hence, $x \in (A - B)$ or $x \in (A - C)$. Suppose $x \in (A - B)$. Hence, $x \in A$ and $x \notin B$. Since $x \notin B$ it is logically true that $x \notin (B \cap C)$. So, $x \in A$ and $x \notin (B \cap C)$. Hence $x \in A - (B \cap C)$ by definition. A similar argument works in the case where $x \in (A - C)$. So, $(A - B) \cup (A - C) = A - (B \cap C)$.  I'm confused as to how to go about proving the other direction. Would I assume that $x \in A - (B \cap C)$?
First part :  I want to prove the following De Morgan's law :  ref.(dfeuer)  $A\setminus (B \cap C) = (A\setminus B) \cup (A\setminus C) $    Second part:  Prove that $(A\setminus B) \cup (A\setminus C) = A\setminus (B \cap C) $    Proof:  Let $y\in (A\setminus B) \cup (A\setminus C)$     $(A\setminus B) \cup (A\setminus C) = (y \in A\;  \land y \not\in  B\;) \vee (y \in A\; \land y \not\in C\;)$      $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= y \in A\ \land (  y \not\in B\; \vee \; y \not\in C ) $      $\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;= y \in A \land  (\lnot ( y\in B) \lor \lnot( y\in C) )$   According to De-Morgan's theorem :  $\lnot( B \land C) \Longleftrightarrow (\lnot B \lor \lnot C)$ thus     $y \in A \land (\lnot ( y\in B) \lor \lnot( y\in C) ) = y \in A \land y \not\in (B \land C)$   We can conclude that  $(A\setminus B) \cup (A\setminus C) = A\setminus (B \cap C)$
We have a org wide email addressed which has been "Verified" into our sandbox. I understant org wide email addressed is unsupported metadata, so this was achieved as pre deployment step.   Into our repo, we have a workflow that send email alert to that org wide email address. "  Workflow   ... &lt;senderAddress&gt;noreply@mycompany.com.au&lt;/senderAddress&gt; &lt;senderType&gt;OrgWideEmailAddress&lt;/senderType&gt;    The problem we are facing is that we are still getting the following error even if this org wide email address is verified:      You must specify a verified Org Wide Email Address for this sender   type.   Is that a bug in the metadata api or expected behaviour ? Ideally we would like to not have to do that as a post deployment step.   PS: We are running DX under api version 43.0 Does anyone have a recommendation for a book to use for the self study of real analysis? Several years ago when I completed about half a semester of Real Analysis I, the instructor used "Introduction to Analysis" by Gaughan.   While it's a good book, I'm not sure it's suited for self study by itself.  I know it's a rigorous subject, but I'd like to try and find something that "dumbs down" the material a bit, then between the two books I might be able to make some headway. Use of 'const' for function parameters How far do you go with const?  Do you just make functions const when necessary or do you go the whole hog and use it everywhere?  For example, imagine a simple mutator that takes a single boolean parameter:  void SetValue(const bool b) { my_val_ = b; }   Is that const actually useful?  Personally I opt to use it extensively, including parameters, but in this case I wonder if it's worthwhile?  I was also surprised to learn that you can omit const from parameters in a function declaration but can include it in the function definition, e.g.:  .h file  void func(int n, long l);   .cpp file  void func(const int n, const long l)   Is there a reason for this? It seems a little unusual to me. I believe that umask is something that controls , but do not fully understand it.  After running umask 0644 in a , I cannot read the files I create with the command-line text editor nano. I noticed that the permissions of that file are set to 0022 instead of the default 0755.  How does umask work? I thought I could just remove the each digit in the umask from 0777, 7 - 6 = 1 and 7 - 4 = 3, so I expect the permissions to be 0133, but apparently, this is not the case.   What is umask exactly? Explain it to me like I was a "Linux noob" How do I calculate with umask? What are use cases for umask? Finding the variable name passed to a function Floor function properties: $[2x] = [x] + [ x + \frac12 ]$ and $[nx] = \sum_{k = 0}^{n - 1} [ x + \frac{k}{n} ] $ I'm doing some exercises on Apostol's calculus, on the floor function. Now, he doesn't give an explicit definition of $[x]$, so I'm going with this one:     DEFINITION Given  $x\in \Bbb R$, the integer part of $x$ is the unique $z\in \Bbb Z$ such that $$z\leq x &lt; z+1$$ and we denote it by $[x]$.   Now he asks to prove some basic things about it, such as: if $n\in \Bbb Z$, then $[x+n]=[x]+n$  So I proved it like this: Let $z=[x+n]$ and $z'=[x]$. Then we have that  $$z\leq x+n&lt;z+1$$  $$z'\leq x&lt;z'+1$$  Then $$z'+n\leq x+n&lt;z'+n+1$$  But since $z'$ is an integer, so is $z'+n$. Since $z$ is unique, it must be that $z'+n=z$.  However, this doesn't seem to get me anywhere to prove that  $$\left[ {2x} \right] = \left[ x \right] + \left[ {x + \frac{1}{2}} \right]$$  in and in general that   $$\left[ {nx} \right] = \sum\limits_{k = 0}^{n - 1} {\left[ {x + \frac{k}{n}} \right]} $$  Obviously one could do an informal proof thinking about "the carries", but that's not the idea, let alone how tedious it would be. Maybe there is some easier or clearer characterization of $[x]$ in terms of $x$ to work this out.  Another property is $$[-x]=\begin{cases}-[x]\text{ ; if }x\in \Bbb Z \cr-[x]-1 \text{ ; otherwise}\end{cases}$$  I argue: if $x\in\Bbb Z$, it is clear $[x]=x$. Then $-[x]=-x$, and $-[x]\in \Bbb Z$ so $[-[x]]=-[x]=[-x]$. For the other, I guess one could say:  $$n \leqslant x &lt; n + 1 \Rightarrow  - n - 1 &lt; x \leqslant -n$$  and since $x$ is not an integer, this should be the same as $$ - n - 1 \leqslant -x &lt; -n$$  $$ - n - 1 \leqslant -x &lt; (-n-1)+1$$  So $[-x]=-[x]-1$ To find out how something is called is the one thing I don't know how to use Google for.  If I have a vector $v$ and a symmetric matrix $H$, then the operation $v^T H v$ gives me some sort of projection of the matrix $H$ on the vector $v$. For example if $H$ is the Hessian, the operation gives me the second derivative in the direction of $v$.  What is the name of this operation? Generating DLL assembly dynamically at run time Currently I have some code that is being generated dynamically.  In other words, a C# .cs file is created dynamically by the program, and the intention is to include this C# file in another project.  The challenge is that I would like to generate a .DLL file instead of generating a C# .cs file so that it could be referenced by any kind of .NET application (not only C#) and therefore be more useful. I'm trying to remove some characters from file(UTF-8). I'm using tr for this purpose:   tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat    File contains some foreign characters (like "Латвийская" or "àé"). tr doesn't seem to understand them: it treats them as non-alpha and removes too.  I've tried changing some of my locale settings:  LC_CTYPE=C LC_COLLATE=C tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat LC_CTYPE=ru_RU.UTF-8 LC_COLLATE=C tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat LC_CTYPE=ru_RU.UTF-8 LC_COLLATE=ru_RU.UTF-8 tr -cs '[[:alpha:][:space:]]' ' ' &lt;testdata.dat   Unfortunately, none of these worked.  How can I make tr understand Unicode? Frequent Write to Non-volatile Memory How do I insert a line at the top of a text file using the command line? 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A fair coin is tossed until a head comes up for the first time. The probability of this happening on an odd number toss is?  How do i approach this problem ? $\sum (-1)^n/n$ fails the p-series test, but passes the alternating series test? "OpenID 2.0 for Google Accounts is going away"; will this affect Stack Exchange? Combine rows / concatenate rows Particles rotation not working correct "Within" and "in" when referring to time