I have checked some post in this web but in this particular situation I am finding trouble with the negation. I have tried supossing by hypothesis that $\forall x. \phi$ but the contradictions of this does not mean $\forall x . \neg \phi$. I am also not sure at all of some notation. Once I have supposed something by hypothesis and got a contradiction, how should I justify what I have proved by contradiction in next line?  Thank you for your attention.
I am reading Velleman's "How to Prove it", and in one example he proves  $\neg \exists x P(x)  \implies \forall x \neg P(x)$    in the following way:    Suppose $\neg \exists x P(x)$. Let x be arbitrary. Suppose $P(x)$. Since we have a specific $x$ for which P(x) is true, it follows that $\exists x P(x)$, which is a contradiction. Therefore $\neg P(x)$. Since $x$ was arbitrary, we can conclude that $\forall x \neg P(x)$, so $\neg \exists x P(x)  \implies \forall x \neg P(x)$    What makes me confused is the line in bold text, that is "Suppose $P(x)$". The reason is that he writes that he is doing proof by contradiction, but $P(x) \not\iff \neg\forall x \neg P(x)$. He is not treating all possible cases where $\forall x \neg P(x)$ is false by assuming $P(x)$, does he? So how can this be a valid proof by contradiction?    Edit: I deleted my first example since it was clearly not representative for the problem.  Edit 2: So the thing that I am opposing I guess is that $\neg \forall x \neg P(x)$ is not $P(x)$. And from what I have undestood, a proof by contradiction should use the negation of the consequent which I do not think that Velleman does.
Remove all but numbers from NSString I am trying to create a 3d game in unity. I have a cube in unity and I want it to move left or right on swipe . It should move to a distance till where user swipe or take their finger off of a screen. I want it to move fast or the cube should move fast on swipe and cube should flow smooth. Like in the gif there is a circle and the user use their finger to move the circle till wherever the user swipe it and it should be instant and I want my cube to move like that at a distance till where user swipe their finger and it should be instant and fast.  I don't know how to do it. Any help is appreciated! Out of Bag Error makes CV unnecessary in Random Forests? Photoshop resize circularly Is there a way to do this circular resize of a pattern in photoshop (flat image) ? (I only want to resize the green disk, without changing the size of the interior white disk and without losing details of the green disk, just flatting them) Im trying to run the following code  \documentclass[border=10pt]{standalone} \usepackage{pgfplots} \pgfplotsset{width = 7cm, compat = 1.8} \begin{document} \begin{tikzpicture} \begin{axis}[domain = -2:2, domain y = 0:2*pi] \addplot3[contour gnuplot = {output point meta = rawz, number=10, labels=false}, samples = 41, z filter/.code=\def\pgfmathresult{-1.6}] {exp(-x^2) * sin(deg(y))}; \addplot3[surf, samples = 25] {exp(-x^2) * sin(deg(y))}; \end{axis} \end{tikzpicture} \end{document}   and I'm getting the error  Package pgfplots Error: sorry, plot file{Utitlled_contourmp0.table} could not be opened   through a quick search I found out that contour maps require pgfplots and gnuplot to talk to each other. In theory the way to make them interact is adding  !TEX option = --enable-write18   or   %!TEX option = --enable-escape   at the top of the file. Unfortunately this has not worked for me. Any suggestions? I am trying to update Xcode to 8.0. My OS is 10.11.6 El Capitan. However I am getting:     We could not complete your update   There is not enough disk space available to install the product.   I have got 14.22 GB free space. How much do I need?  UPDATE: I do not know the exact figure, though I did manage to update with 17.21 GB free space later on. Hi I want to try Ubuntu 20.04 without installing it due to some driver issues in my laptop. But I already have 2 OS ( Windows 10 and Ubuntu 20.04 [Dualboot] ), so will trying Ubuntu mess with the already installed Operating systems? And I know how to make live usb and booting from it but don't know how to exit it and then bring my laptop back to normal. Tried finding procedure on internet but reliable results are not found. Let $x,y,z&gt;0$: Prove that:   $\frac{1}{(x+2y)^2}+\frac{1}{(y+2z)^2}+\frac{1}{(z+2x)^2} \geq\frac{1}{xy+yz+zx}$  I tried to apply Cauchy - Schwarz's inequality but I couldn't prove this inequality! What books are recommended for learning calculus on my own? I recently graduated with a degree in bachelor of science with a focus interactive and multimedia design. I had to opportunity to take 1 C++ course and 1 HTML course. I was also only required to take one math class, introduction to calculus (and it was hardly calculus, it was more of a math class for designers)  Since graduating I have pursued a career in web development. I'm considering going back for a MS in computer science or MS in web technologies.   I feel my math has suffered dramatically as I have always been behind most others on my math studies (middle school through college). I want to learn calculus as I feel it's an integral part of any CS degree.   Does anyone have any recommendations on any great books that I can read through and teach my self a bit of calculus or other advanced math topics?  Thanks for any advice I want to make a Launcher that will do privileged (requires root) commands, but I don't want to see the terminal or have to type my password into the terminal. So I found the command, gksudo that allows me to enter my password so that I can escalate my privilege level. Now I want to couple this, so that I can change the partition priority using this command cgpt add -i 6 -P 0 -S 0 /dev/sda (this command is privileged) and have it automatically reboot from this one launcher.  I setup a Luncher like this:  Name: ChromeOS  Description:  Command: gksudo gnome-terminal -e "cgpt add -i 6 -P 0 -S 0 /dev/sda;reboot"  Comment:  But it does not work. Any tips? Where to ask "subjective and argumentative" questions on programming? True for every formula. For a formula $ \gamma = r \iff (p_1 \vee p_2) $ there is true that: $\rho \models \gamma \iff \rho(r) = \max(\rho(p_1), \rho(p_2))$  Is it a true for every formula r? Is there something similar to sprintf() in C#?  I would for instance like to convert an integer to a 2-byte byte-array.  Something like:  int number = 17; byte[] s = sprintf("%2c", number); What is the purpose of the word 'self'? Make Vim show ALL white spaces as a character As shown in various episodes ("Mirror, Mirror," "Phantasms," etc.), other universes exist, oftentimes containing the same characters, albeit in different situations than in the prime universe. I'm curious, though, whether individuals that apparently exist outside of the multiverse, such as members of the Q, would not be duplicated elsewhere. It seems reasonable to me, but I'm not sure. I am using a linux server which has 128GB of memory and 24 cores. I use top to see how much it is used. Its output is pasted at the end of the post. Here are two questions:  (1) I see that each of the running processes occupies a very small percentage of memory (%MEM no more than 0.2%, and most just 0.0%), but how the total memory is almost used as in the fourth line of output ("Mem:  130766620k total, 130161072k used,   605548k free,   919300k buffers")? The sum of used percentage of memory over all processes seems unlikely to achieve almost 100%, doesn't it?  (2) how to understand the load average on the first line ("load average: 14.04, 14.02, 14.00")?   Thanks and regards!  Edit:  Thanks!  I also really like to hear some rough numbers based on used percentage of memory to determine if a server is heavily loaded, since I once became the one who cramed the server without understanding the current load.  Is swap regarded as almost the same as memory? For example, when memory and swap are almost of same size, if the memory is almost running out but the swap is still largely free, may I just view it as if the used percentage of memory + swap is still not high and run other new processes?  How would you consider together CPU or memory (or memory + swap) usage? Do you become worried if either of them reaches too high or both?  Output of top：  $ top    top - 12:45:33 up 19 days, 23:11, 18 users,  load average: 14.04, 14.02, 14.00 Tasks: 484 total,  12 running, 472 sleeping,   0 stopped,   0 zombie Cpu(s): 36.7%us, 19.7%sy,  0.0%ni, 43.6%id,  0.0%wa,  0.0%hi,  0.0%si,  0.0%st Mem:  130766620k total, 130161072k used,   605548k free,   919300k buffers Swap: 63111312k total,   500556k used, 62610756k free, 124437752k cached    PID USER      PR  NI  VIRT  RES  SHR S %CPU %MEM    TIME+  COMMAND  6529 sanchez   18  -2 1075m 219m  13m S  100  0.2  13760:23 MATLAB 13210 timothy   18  -2 48336  37m 1216 R  100  0.0   3:56.75 absurdity 13888 timothy   18  -2 48336  37m 1204 R  100  0.0   2:04.89 absurdity 14542 timothy   18  -2 48336  37m 1196 R  100  0.0   1:08.34 absurdity 14544 timothy   18  -2  2888 2076  400 R  100  0.0   1:06.14 gatherData  6183 sanchez   18  -2 1133m 195m  13m S  100  0.2  13676:04 MATLAB  6795 sanchez   18  -2 1079m 210m  13m S  100  0.2  13734:26 MATLAB 10178 timothy   18  -2 48336  37m 1204 R  100  0.0  11:33.93 absurdity  12438 timothy   18  -2 48336  37m 1216 R  100  0.0   5:38.17 absurdity 13661 timothy   18  -2 48336  37m 1216 R  100  0.0   2:44.13 absurdity 14098 timothy   18  -2 48336  37m 1204 R  100  0.0   1:58.31 absurdity 14335 timothy   18  -2 48336  37m 1196 R  100  0.0   1:08.93 absurdity 14765 timothy   18  -2 48336  37m 1196 R   99  0.0   0:32.57 absurdity 13445 timothy   18  -2 48336  37m 1216 R   99  0.0   3:01.37 absurdity 28990 root      20   0     0    0    0 S    2  0.0  65:50.21 pdflush 12141 tim       18  -2 19380 1660 1024 R    1  0.0   0:04.04 top  1240 root      15  -5     0    0    0 S    0  0.0  16:07.11 kjournald  9019 root      20   0  296m 4460 2616 S    0  0.0  82:19.51 kdm_greet     1 root      20   0  4028  728  592 S    0  0.0   0:03.11 init     2 root      15  -5     0    0    0 S    0  0.0   0:00.00 kthreadd     3 root      RT  -5     0    0    0 S    0  0.0   0:01.01 migration/0     4 root      15  -5     0    0    0 S    0  0.0   0:08.13 ksoftirqd/0     5 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/0     6 root      RT  -5     0    0    0 S    0  0.0  17:27.31 migration/1     7 root      15  -5     0    0    0 S    0  0.0   0:01.21 ksoftirqd/1     8 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/1     9 root      RT  -5     0    0    0 S    0  0.0  10:02.56 migration/2    10 root      15  -5     0    0    0 S    0  0.0   0:00.34 ksoftirqd/2    11 root      RT  -5     0    0    0 S    0  0.0   0:00.00 watchdog/2    12 root      RT  -5     0    0    0 S    0  0.0   4:29.53 migration/3    13 root      15  -5     0    0    0 S    0  0.0   0:00.34 ksoftirqd/3 I have an ec2 instance on AWS which already installed a few Wordpress-es in it.  I want to set things up to let more people handle their own projects, with me also being able to go into the server using terminal if I still want to.  I have been following tutorials such as  and a few others.  I figured all of them just teach users how to sftp into their own home folder, also the steps keep saying addUser but some users already existed and removing them might cause some issue.  How can I give existing users the permission to use SFTP for Wordpress projects?  Let's say all my Wordpress projects are already under /var/www/ which has already been set up with www-data:www-data.  Really new with setting things like this.  Thanks in advance for any help. I have been having extreme difficulties with this integral. I would appreciate any and all help. $$ \int \sqrt{\tan x} ~ \mathrm{d}{x}. $$ Making an elastic waistband on clothing?