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Logic soundness and completeness I have to do a number of similar type questions, but I am having trouble grasping the general concepts around soundness and completeness. I have read up on the general definitions and think I understand them, but when asked to show something like the following I come up blank. Thanks for any ideas/help.   Show that the following propositional proof system, PCa, is sound but not complete.  PCa  Connectives (and constants) ∨, ∧, →, ↔, ¬, 0, 1  Axioms All tautologies of the form F→F  Rules of inference Hypothetical Syllogism timestamp, modification time, and created time of a file On finding polynomials that approximate a function and its derivative (extensions of Stone-Weierstrass?) this code always returns 0 in PHP 5.2.5 for microseconds:  &lt;?php $dt = new DateTime(); echo $dt-&gt;format("Y-m-d\TH:i:s.u") . "\n"; ?&gt;   Output:  [root@www1 ~]$ php date_test.php 2008-10-03T20:31:26.000000 [root@www1 ~]$ php date_test.php 2008-10-03T20:31:27.000000 [root@www1 ~]$ php date_test.php 2008-10-03T20:31:27.000000 [root@www1 ~]$ php date_test.php 2008-10-03T20:31:28.000000   Any ideas? Does Stack Exchange use caching and if so, how? Recently, I read this  about how they have some of their infrastructure laid out. I was curious to know, does Stack Exchange use caching (both at the application layer and/or database layer) and if so, how? How did Bill and Percy get 12 OWLs? In Prisoner of Azkaban, Hermione took 12 subjects, and needed a Time-Turner to do so. However, it is mentioned in the series that both Bill and Percy Weasley got 12 OWLs, so would have taken 12 subjects as well, so both would have needed Time-Turners for three years to do this. Doesn't this seem a little unusual, given how strict the Ministry is over the use of Time-Turners,  that they would give them to several Hogwarts students? Is there a Linux equivalent of Windows' "resource files"? Why is the Yang-Mills gauge group assumed compact and semi-simple? What is the motivation for including the compactness and semi-simplicity assumptions on the groups that one gauges to obtain Yang-Mills theories?  I'd think that these hypotheses lead to physically "nice" theories in some way, but I've never, even from a computational perspective. really given these assumptions much thought. What is the function __construct used for? I have been noticing __construct a lot with classes. I did a little reading and surfing the web, but I couldn't find an explanation I could understand. I am just beginning with OOP.  I was wondering if someone could give me a general idea of what it is, and then a simple example of how it is used with PHP? Where can you ask about Career Advice? The  specifically says not to ask  for career advice but people do it anyway.  Should I consider the Help Center outdated or only loosely enforced?    And what about a career advice site on the network?  There are 70+ sites, and most of them are for relatively narrow interests (which is not a criticism, I enjoy looking through the many sections).  A career section seems like it would be very popular.  Also, a careers section would help keep career questions off the other sites. The question asks to prove that -  $$\frac {\sin x} {\cos 3x} + \frac {\sin 3x} {\cos 9x} + \frac {\sin 9x} {\cos 27x}  = \frac 12 (\tan 27x - \tan x) $$  I tried combining the first two or the last two fractions on the L.H.S to allow me to use the double angle formula and get $\sin 6x$ or $\sin 18x$ but that did not help at all.   I'm pretty sure that if I express everything in terms of $x$, the answer will ultimately appear but I'm also certain that there must be another simpler way. I have a problem with a virus in windows 7, I don't know what kind of virus is infected my OS, I installed kasersky antivirus, and clear all detected viruses. After that I can't open any antivirus sites, can't ping any antivirus sites. example:  C:\Windows\system32&gt;ping kaspersky.com Pinging kaspersky.com [38.117.98.231] with 32 bytes of data: General failure. General failure. General failure. General failure.  Ping statistics for 38.117.98.231:     Packets: Sent = 4, Received = 0, Lost = 4 (100% loss),   When I ping to other sites, I get reply normally: example woth pinging to google.com  C:\Windows\system32&gt;ping google.com  Pinging google.com [31.7.160.224] with 32 bytes of data: Reply from 31.7.160.224: bytes=32 time=2ms TTL=61 Reply from 31.7.160.224: bytes=32 time=2ms TTL=61 Reply from 31.7.160.224: bytes=32 time=1ms TTL=61 Reply from 31.7.160.224: bytes=32 time=1ms TTL=61  Ping statistics for 31.7.160.224:     Packets: Sent = 4, Received = 4, Lost = 0 (0% loss), Approximate round trip times in milli-seconds:     Minimum = 1ms, Maximum = 2ms, Average = 1ms   I tryed any antispyware/antimalware softwares, but null results.  Anyone else encountered this problem? Is there any way to do n-level nested loops in Java? In other words, can I do something like  for() {     for {        for {        }     } }   Except N times? In other words, when the method creating the loops is called, it is given some parameter N, and the method would then create N of these loops nested one in another?  Of course, the idea is that there should be an "easy" or "the usual" way of doing it. I already have an idea for a very complicated one. Why shift-cmd-n fails as a custom shortcut for Firefox "New Private Window"? I created the shiftcommandn shortcut for Firefox in Syst Prefs to match other browsers' behavior.     It registers in the Firefox Menubar.     But pressing the shortcut opens a new non-private window, and it loads there the first item in my "Top Sites".   I imagine I could create another shortcut to override this behavior, but what is this behavior? I don't see anything highlighted in the Firefox Menubar when I hit the shortcut. Or might there be another solution (other than using a different shortcut to open new private windows)?   Problem exists on both Mojave and High Sierra. Default Firefox shortcut (shiftcommandp) still opens a new private window. Understanding shape and calculation of confidence bands in linear regression How to differentiate read and unread messages in android ListView? How to only substitute lines that match several patterns in sed? $ echo -e 'CH12\nCH23au' | sed '/^CH/s=^=&lt;b&gt;=' &lt;b&gt;CH12 &lt;b&gt;CH23au   I know I can match lines starting with CH by ^CH but how can I match multiple patterns?   Example:  Input:  CH12 CH23au   Output:  CH12 &lt;b&gt;CH23au   How to only put &lt;b&gt; where there is ^CH and au in the line? As I understand this, firewalls (assuming default settings) deny all incoming traffic that has no prior corresponding outgoing traffic.  Based on  and , reverse SSH tunneling can be used to get around pesky firewall restrictions.  I would like to execute shell commands on a remote machine. The remote machine has its own firewall and is behind an additional firewall (router). It has an IP address like 192.168.1.126 (or something similar). I am not behind a firewall and I know the remote machine's IP address as seen from the Internet (not the 192.168.1.126 address). Additionally, I can ask someone to execute ssh (something) as root on the remote machine first.  Could anyone explain me, step by step, how reverse SSH tunneling works to get around the firewalls (local and remote machines' firewalls and the additional firewall between them)?  What is the role of the switches (-R, -f, -L, -N)? Problem: Suppose $f$ is continuous for $x\ge 0$, differentiable for $x&gt;0$, $f(0)=0$, and $f'$ is monotonically increasing.  Define $g(x)=\frac{f(x)}{x}$ for $x&gt;0$. Prove that $g$ is monotonically increasing.  Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 6. It is stated as a problem in Spivak's Calculus and I can't wrap my head around it.