I need help proving "For each $x,y,z$ in $\mathbb{N}$, if $x&lt;y$, then $xz&lt;yz$."  This must be done using Peano's axioms and the definitions of addition, multiplication, and ordering.   I have tried induction on z, and the base case works out but I run into an issue when it comes to the inductive part seeing as I am no longer dealing with equality.   Maybe induction is not the correct approach to this proof, and maybe a direct $P\implies Q$ proof would be more logical, however once I assume $x&lt;y$, I don't have much to work with to deduce $xz&lt;yz$, besides the definition of order.
Let $a,b,c\in \mathbb{Z}$. If $ac&gt;bc$ and $c&gt;0$, then $a&gt;b$   I think I would use contrapositive to prove this. But I am not sure proper contrapositive statement.   Is it If $a&lt;b$, then $ac&lt;bc$, and $c&lt;0$.? If so, contrapositive is not a good choice, isn't it?
How do you clear a stringstream variable? I've tried several things already,  std::stringstream m; m.empty(); m.clear();   both of which don't work. How do I counter-knife an enemy? When you press down the right stick to counter the attack, do you just press it, repeatedly press it, or press and hold? I've been knifed a few times from the front and each time I try to counter I fail. Also, if you believe you are about to be knifed and hold down the right stick until you are knifed, will that counter an attack, or do you have to press it down just at the right time? Which process scheduler is my linux system using? I'm not quite sure if stackoverflow or serverfault is the forum to use for this question, but ill try here.  Is there some command I can use or a file I can check in order to find out which process scheduler I'm using on my Linux system? I am not looking for the I/O scheduler, but the process scheduler, and please, do not refer to guides or howtos for what a scheduler is, what I'm simply asking is if there is a command or file I can run/check to see this. I could check the doc for my kernel but again I'm just wondering if there is a command or file i can check.  I might have been a little redundant above but that's because the posts I've seen people seem to confuse process scheduler with the i/o scheduler, and they don't seem to answer the actual question, just provide links to what a scheduler is which is not what I'm after here, so apologies if the above comments sounded rude. fetching list of friends using facebook 2.0 What is an example of "this" assignment in C#? Does anybody have useful example of this assignment inside a C# method? I have been asked for it once during job interview, and I am still interested in answer myself. Differences in development between .NET and Mono How can two dates be compared in a shell?  Here is an example of how I would like to use this, though it does not work as-is:  todate=2013-07-18 cond=2013-07-15  if [ $todate -ge $cond ]; then     break fi                              How can I achieve the desired result? A symmetry of graph $X$ is a permutation of the vertices that also happens to be a permutation of the induced edges. In particular, the distances between vertices are preserved by a symmetry. Show that the set of symmetries of $X$ is a permutation group of $\operatorname{V}(X)$. Compute the cycle index of the group for the Petersen graph.   Alright I am super confused. I have no idea how to even start. Can someone point me in the right direction. How do I show that something is a permutation group? How do I compute the cycle index of a graph? When adding NOT NULL columns with a DEFAULT value - does PostgreSQL optimize this operation?  In case the table has n rows, a not-optimized alter-table-add-column would yield n writes of the default value - which could be very painful, obviously. With the optimization the DB would instantaneously create the new column, store just one copy of the default value which would be returned when no non-default value is found for that column in a suitable index data structure.  For example  such an . What causes the 'Connection Refused' message? This is a  about Connection Refused    We see a lot of questions to the effect      When I try to connect to a system I get a message      Connection refused      Why is this ? Diagonalizable matrix with parameters $$     \begin{bmatrix}     -2 &amp; -2 &amp; 0 &amp; 0 \\     -b &amp; -b &amp; 0 &amp; 0 \\      0 &amp; 0 &amp; -4 &amp; 4 \\      0 &amp; 0 &amp; b &amp; -b \\     \end{bmatrix} $$  I have this matrix and I need to find the values for $b$ that this matrix won't  be diagonalizable.  For what I got this matrix is diagonalizable for every $b$ value.  Please write me also the final answer, thank you very much. I'm describing a situation that happened in the past. To explain it, I want to use a description that is both true now and true when the situation happened.  Specifically, I want something like:     She touched me where my neck met my collarbone.   Since my neck is still attached to my collarbone (thankfully), I'm wondering if I shouldn't use the present tense here instead:     She touched me where my neck meets my collarbone.   Which is preferable, and why? iOS 7 Locked Bug via Ubuntu 13.10 I wanted to bring up an issue with the latest Ubuntu release that I hope can be solved. Although I will agree with the community that iOS 7 is not the most open platform release, it still is my desire to be able to grab Photos, Music, and Files off my Phone similar to what I would do to any other device.  Every time I connect my device to Ubuntu, I receive the following notification:  The device "Justin's iPhone" is locked. Enter the passcode on the device and click "Try again".   On my iPhone, I select the following   Unfortunately, this is where problems start to arise. It appears that Ubuntu either does not recognize the permissions from my iPhone or the iPhone is not allowing itself to truly be unlocked. As we can see the permissions are   Browsing askubuntu, I encountered the following   After running the code, I receive the following error:  justin@justin-CT15:~$ idevicepair unpair &amp;&amp; idevicepair pairERROR: Device d1fea2c56e51199f901eb34020a1581ea31a2a50 is not paired with this host   If I call the command again, i receive another error:  justin@justin-CT15:~$ idevicepair pair ERROR: Could not pair with the device because a passcode is set. Please enter the passcode on the device and retry.   Ultimately, I am stuck to experiment with solutions that end up not working. I am troubled by the lack of information and support for the iOS 7 platform on Ubuntu's latest release. If anyone has a workable solution or experiment for a workaround, I am happy to try  Thanks for reading and have a wonderful day! Prevent Word from Including Bullet Letter when Copying Text I'm copying quite a bit of text from Word into a text-based application, line by line.  The word text is structured like this:        Topic      a. Item A      b. Item B      When I highlight Item A, copy, and then paste into the text-based application, I get a.    Item A rather than just Item A as desired.  How can I prevent Word from including the bullet letter when copying? Twice today I noticed that my reputation increases with 15 on Stack Overflow, but when I click to see my account, the change is being reverted and none of my recent answers are marked as accepted. Then why does the reputation on the top of the page change ? When running sudo apt-get update I get these following errors:  Err archive.ubuntu.com quantal InRelease Err archive.ubuntu.com quantal-updates InRelease Err archive.ubuntu.com quantal-backports InRelease Err archive.ubuntu.com quantal-security InRelease Err archive.ubuntu.com quantal Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-updates Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-backports Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Err archive.ubuntu.com quantal-security Release.gpg   Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] Reading package lists... Done W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-updates/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-backports/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-security/InRelease   W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-updates/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-backports/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Failed to fetch archive.ubuntu.com/ubuntu/dists/quantal-security/Release.gpg  Unable to connect to archive.ubuntu.com:http: [IP: 91.189.92.177 80] W: Some index files failed to download. They have been ignored, or old ones used instead.   I am using Ubuntu 12.04. This is the command I used after installing Ubuntu from window installer. I am sure my laptop is connected to internet. What can I do now? WiFi repeatedly asks for password (Ubuntu 12.10 fresh) I'm sure the passphrase is correct 100% but Wifi keeps asking for authentication and never connects. What can I do? I've tried removing the networks, resetting the router, changing the password, rebooting Ubuntu, disabling IPv6.  Also, it worked until earlier today and then stopped. All my other devices log in fine. Getting "System program problem detected" pops up regularly after upgrade Efficient evaluation of functions defined by NIntegrate I have a conjecture that any two positive coprime integers $a$ and $b$ can be added to form any integer greater than or equal to $(a-1)(b-1)$.   This is true for $3$ and $5$, which can form any integer greater than or equal to $8$, where $8=3+5, 9=3+3+3, 10=5+5; k=3n+8$ or $3n+9$ or $3n+10$.  It is true for $4$ and $7$, which can form any integer greater than or equal to $18$, where $18=4+7+7, 19=4+4+4+7, 20=4+4+4+4+4, 21=7+7+7$, etc. How can it be proven that this will work for any two coprime numbers?