Prove that $f(\overline A) \subseteq \overline{f(A)}$
proof $f(\overline{A})\subseteq \overline{f(A)} \Leftrightarrow f$ continuous
The following happened playing Minesweeper:   How do I figure out which square to click? How do you return to the default view (the one you see when blender starts)? Proving $\,a\mid bc\!\iff\! \frac{a}{(a,b)}\!\mid c\ $ [general Euclid's Lemma] I tried to start by showing that $ \frac{a}{\gcd(a,b)} $ is always an integer, let's call it $d$, because $a$ is always a multiple of $(a,b)$ based on the definition of a g.c.d. I then tried to show that if $a | b$ then $a|bc$ from the theorem that states that $a|b$ implies that $a|bc$ for any integer $c$, but I cant seem to prove that $a | b$ or that $a|bc$ only if $d \mid c$ (where $d = \frac{a}{\gcd(a,b)}$ ). Any help would be much appreciated. Generate random numbers uniformly over an entire range I need to generate random numbers within a specified interval, [max;min].  Also, the random numbers should be uniformly distributed over the interval, not located to a particular point.  Currenly I am generating as:  for(int i=0; i&lt;6; i++) {     DWORD random = rand()%(max-min+1) + min; }   From my tests, random numbers are generated around one point only.  Example min = 3604607; max = 7654607;   Random numbers generated:  3631594 3609293 3630000 3628441 3636376 3621404   From answers below: OK, RAND_MAX is 32767. I am on C++ Windows platform. Is there any other method to generate random numbers with a uniform distribution? I would that the images would appeared for each overlay. Eg, in overlay 1 I have img1.pdf, in overlay 2 I have img2.pdf, and in overlay 3 I have img3.pdf. How I do this?  \begin{center}   \begin{tikzpicture}     \node (img1) {\includegraphics[width=.5\linewidth]{img1.pdf}};     \pause     \node (img2) {\includegraphics[width=.5\linewidth]{img2.pdf}};     \pause     \node (img3) {\includegraphics[width=.5\linewidth]{img3.pdf}};   \end{tikzpicture} \end{center} Is there a secular, non vulgar alternative to "for heaven's sake"? I know for heaven's sake, for Pete's sake, for God's sake and for Christ's sake. All of those, however are religious references. The only non-religious equivalent phrases I know are for fuck's sake and its variations. Are there any non-vulgar, non-religious alternatives? I am very used to the general construct of for X's sake and am looking for something along the same lines that I could use in cases like  C'mon, it's just a scratch for X's sake! That's not even true for X's sake!  It does not need to also be applicable as a general exclamation of frustration (Oh, for chrissake!) but that would be nice. Basically, I'm looking for a synonym for the &quot;Oh come on!&quot; sense of for fuck's sake rather than the Oh shit! one.  EDIT: Since this has gotten quite a bit of attention, I would like to urge those who answer to read the entire question, not just the title. I am not just looking for polite swearwords but for something that can be used in the example phrases above. Probability of 3 people in a room of 30 having the same birthday I have been looking at the birthday problem (http://en.wikipedia.org/wiki/Birthday_problem) and I am trying to figure out what the probability of 3 people sharing a birthday in a room of 30 people is. (Instead of 2).  I thought I understood the problem but I guess not since I have no idea how to do it with 3. Registry key / GPO to disable and block Windows 10 upgrade? Show all of my question/answers to me even if they are deleted Signature/date line with fixed width How does accepting an answer work? What does it mean to accept an answer? How do I accept an answer, and what are the rules? Can I accept my own self-answer? What are the reputation benefits? Which answer should I accept? Should I upvote the accepted answer? What should I do if none of the answers are suitable?      For more information, see "" in the . Temporarily suspend bash_history on a given shell? Is there a way to temporarily suspend history tracking in bash, so as to enter a sort of "incognito" mode? I'm entering stuff into my terminal that I don't want recorded, sensitive financial info. From the  article on Wikipedia, the following with regard to the Coriolis Effect on a rotating sphere:     By setting vn = 0, it can be seen immediately that (for positive φ and ω) a movement due east results in an acceleration due south. Similarly, setting ve = 0, it is seen that a movement due north results in an acceleration due east. In general, observed horizontally, looking along the direction of the movement causing the acceleration, the acceleration always is turned 90° to the right and of the same size regardless of the horizontal orientation.   My intuitive (but possibly incorrect) understanding is that if there are two points, Point A and Point B, at different latitudes in the Northern hemisphere, the Eastward velocities of these points are different because they are at different distances from the Earth's axis of rotation, and this causes the Coriolis effect for a rotating sphere.  If a projectile is fired due North from Point A near the Equator towards Point B near the North Pole, the projectile will start off with the higher Eastward velocity of Point A, and will land to the East of Point B, which is moving East at a slower velocity than the projectile.  Firstly, is that correct?  If that IS correct, that brings me to the quote off Wikipedia. The quote implies that, if a projectile is fired due East, it will experience a Coriolis force to the South. My intuitive explanation based on differences in velocities between origin and destination does not account for a Southwards movement at all, since the Point A and Point B velocities are identical if the points are on the same latitude.  What am I missing? Would a projectile fired due East or due West experience any North or South drift caused by the Coriolis effect (or anything else for that matter)?  Why? All odd primes except $5$ divide a number made up of all $1$s Okay, so I have worked on this problem and even though I can see it is true I just don't know how to show it.  Here it goes.  Question:  Show that every odd prime except $5$ divides some number of the form $111 \ldots 11$ ($k$ digits, all ones).  Thank you my friends! URL regex passed by regex buddy but failed by Dart Missing links to responses to comments on nomination posts In the responses tab in my profile, responses to comments on nomination posts include a working link in the page source, but because there's no title the links are empty and useless.    When listed in the inbox such posts do have a title, though in this case the link is to the post and not directly to the comment. (Actually, the links in responses end up being redirected and also go to the post and not to the comment, but they try to go to the comment.)  In Supercollider:    In Global Profile: Suppose John has 5 sweets. Is there any difference between the following two sentences?     Jack has 3 times as many sweets as John.      Jack has 3 times more sweets than John.   I prefer the first construction and would know unambiguously that Jack has 15 sweets in this case. However in the second construction I would be inclined to think that Jack has 20 sweets, since it seems to suggest 15 sweets in addition to the original 5. Using shell's read command with live editing functionality (readline like) Is there a standard (POSIX) way of asking the user some data from within a shell script, with  for example, while allowing live edition of the text being typed (what readline does)?   I know bash has read -e varname that allows for the person launching the script to use keyboard arrows for exemple, to edit or correct what has just been typed without deleting the last entered characters with backspace.   However, read -e is . And still, it is pretty cumbersome to delete all that has been wrote if you realise you made a mistake at the beginning of your long sentence... (Last Jedi spoilers ahead, if you haven't seen the movie, please don't read on.)  So, fridge logic moment for me:      If Vice Admiral Holdo had told Poe Dameron about the plan to just hold on long enough to make a covert run to Krait, then Poe wouldn't have felt the need to send Finn and Rose on their little side-trip, and the code-breaker wouldn't have overheard anything nor sold them out, and the First Order would've had no reason to notice the cloaked ships.   I do understand that there's a lesson in there about following orders, even if said orders come from a pretty face in a pretty dress, but that's kind of an out-of-universe reason. What was Holdo's reasoning? Surely she knew enough about Poe to realize that treating him like a mushroom1 would not result in anything good.  1 "kept in the dark and fed shit" I'm trying to get my csv table aligned to the left of my graph. The graph is appearing below (and indented) the table. Any ideas?  \begin{minipage}{0.5\textwidth} \pgfplotstabletypeset[col sep=comma, columns/distance/.style={column name={Col1}, column type={|c|}}, columns/loss/.style={column name = {Col2},column type={c|}}, every head row/.style={before row=\hline}, after row={\hline} ]{graph-data/data.csv} \end{minipage}  \begin{minipage}{0.5\textwidth} \begin{tikzpicture} \begin{axis}[ xlabel = Distance ($m$), ylabel = Loss ($dB$)] \addplot table [col sep=comma]{graph-data/data.csv}; \end{axis} \end{tikzpicture} \end{minipage}