P | Q | P->Q  T | T | T (That makes sense)  T | F | F (That makes sense)  F | T | T (Why the anwser is T?)  F | F | T (Why the anwser is T?)
Provided we have this truth table where "$p\implies q$" means "if $p$ then $q$":  $$\begin{array}{|c|c|c|} \hline p&amp;q&amp;p\implies q\\ \hline T&amp;T&amp;T\\ T&amp;F&amp;F\\ F&amp;T&amp;T\\ F&amp;F&amp;T\\\hline \end{array}$$  My understanding is that "$p\implies q$" means "when there is $p$, there is q". The second row in the truth table where $p$ is true and $q$ is false would then contradict "$p\implies q$" because there is no $q$ when $p$ is present.  Why then, does the third row of the truth table not contradict "$p\implies q$"? If $q$ is true when $p$ is false, then $p$ is not a condition of $q$.  I have not taken any logic class so please explain it in layman's terms.       Administrative note. You may experience being directed here even though your question was actually about line 4 of the truth table instead. In that case, see the companion question  And even if your original worry was about line 3, it might be useful to skim the other question anyway; many of the answers to either question attempt to explain both lines.
Let us say we have the following Makefile  include config.mk  # Generate summary table. results.txt : $(ZIPF_SRC) isles.dat abyss.dat last.dat     $(ZIPF_EXE) *.dat &gt; $@  # Count words. .PHONY : dats dats : isles.dat abyss.dat last.dat  %.dat : books/%.txt $(COUNT_SRC)     $(COUNT_EXE) $&lt; $*.dat  .PHONY : clean clean :     rm -f *.dat     rm -f results.txt   where all the variables are defined in config.mk  LANGUAGE=python COUNT_SRC=countwords.py COUNT_EXE=$(LANGUAGE) $(COUNT_SRC) ZIPF_SRC=testzipf.py ZIPF_EXE=$(LANGUAGE) $(ZIPF_SRC)   Is there any tool available that can expand all the include lines within the makefile so as to produce a bigMakefile? If such a tool does not currently exist, can a python/perl/awk/bash script be cooked up to achieve this goal? Jack probs with heaphones/earbuds So I already asked this question, but I couldn't respond to the answers. I can't hear out of the left ear of any headphones/earbuds because I have to push the jack towards the Earth to hear out of it. It seems you guys think the jack is the problem when I'm pretty sure it's the computer (my Acer laptop). I also forgot to mention the headphones (PX22 Turtle Beaches) work with both ears, but I have to plug in the usb extension cord in order for that to work (it's also the same cord that enables the mic on my headphones), plus I lost the cord. Thanks 4 hewps. Prove the following using epsilon-delta definition:  Let $a \in \mathbb{R}$ and $f,g$ be defined on $\mathbb{R}$.  If  $$\lim_{x\to a} f(x) = 647 \tag 1$$  and $$\lim_{x\to a} g(x) = \infty \tag 2$$ then   $$\lim_{x\to a} f(x) + g(x) = \infty.$$  $$(1) \implies\forall \epsilon &gt; 0, \exists \delta &gt; 0, s.t. |x-a|&lt;\delta \implies |f(x)-647| &lt; \epsilon$$  $$(2) \implies \forall M &gt; 0, \exists \delta_2 &gt; 0, s.t. |x-a|&lt;\delta_2 \implies |g(x)| &gt; M$$  Not sure about this... I've set up the definitions. I believe we will have to take a minimum between the deltas for first two limits then use it in the third.  Anyone have any clues? I'm a little confused as to whether I should install Ubuntu on its own partition on my hard drive, use VirtualBox or another virtualization package to install it, or use Wubi to install it directly on top of my current OS (Win 7). I definitely want to learn and use Ubuntu, so this is not just for playing around with it.   Also, if I choose to partition, should I partition the hard drive myself or should I let the Ubuntu installation menu do it for me? I understand that I am going to need a main partition, for Ubuntu's core components, and also a swap partition. Then there is the option to add a partition for "home"- I don't understand what combination of these partitioning options I should choose, or whether it is better to partition in Windows before I install Ubuntu or just partition my hard drive when I install Ubuntu itself How to prove that a set is invariant? warning message at command line `sudo` Everytime I try to sudo something at the terminal, I get a warning message:  alexandre@XPS-15Z $ sudo mount file.iso /mnt/ -o loop [sudo] password for alexandre:  no talloc stackframe at ../source3/param/loadparm.c:4864, leaking memory   It's just annoying to get that all the time. How do I solve that? I'm on Ubuntu 14.04 (upgraded from 13.10) What is the difference between "What are you?" and "Who are you?"? I am often asked "Who are you?" and "What are you?", but I don't know the difference between these two questions.  Please explain it to me. What is the best way to filter a Java Collection? I want to filter a java.util.Collection based on a predicate. Proofs involving limsup and liminf I've been working with proofs involving $\limsup$ and $\liminf$, and I'm a bit confused regarding their general methodology. More specifically, I'm unsure about whether my approach to the following problem makes sense.  Problem. Let $(s_n)$ and $(t_n$) be sequences and suppose that there exists $N_0$ such that $s_n \leq t_n$ for all $n &gt; N_0$. Show that $\liminf s_n \leq \liminf t_n$ and that $\limsup s_n \leq \limsup t_n$.   The way I approached it was as follows:   Let $N &gt; N_0$. Then $\limsup_{N \rightarrow \infty} \{ s_n : n &gt; N \} \leq t_n$ as $s_n \leq t_n$, and $\limsup s_n $ is the largest possible limit of a subsequence of $s_n$. As $t_n : n &gt; N$ is (by definition) less than $\limsup_{N \rightarrow \infty} \{ t_n : n &gt; N \}$, the proof is complete.  I'm pretty sure this is incorrect, however, and I'm generally unclear about the method behind such a proof. Any help is appreciated! Iteration over std::vector: unsigned vs signed index variable What is the correct way of iterating over a vector in C++?  Consider these two code fragments, this one works fine:  for (unsigned i=0; i &lt; polygon.size(); i++) {     sum += polygon[i]; }   and this one:  for (int i=0; i &lt; polygon.size(); i++) {     sum += polygon[i]; }   which generates warning: comparison between signed and unsigned integer expressions.  I'm new in the world of C++, so the unsigned variable looks a bit frightening to me and I know unsigned variables can be dangerous if not used correctly, so - is this correct? Can multiple different HTML elements have the same ID if they're different elements? Change pivot or local origin of an object How can I change the local origin of an object, without changing the object location in the world space.  This is the object located at position (0,0,0) in the world space    This is the same object located at position (0,1,0) in the world space, but with its local origin offset so that the object is in the same place    Is there a way to do this in a single step? Using a naked black hole as a mirror? The YouTube video is a good example how the gravity of this merging binary black holes bend light around themselves. If $a^{n}-1$ is prime then $a=2$ and $n$ is prime? I was doing some basic Number Theory problems and came across this problem :     Show that if $a$ and $n$ are positive integers with $n\gt 1$ and $a^{n} - 1$ is prime, then $a = 2$ and $n$ is prime   My Solution : (Sloppy)        $a^{n}-1$ = $(a-1)$ . $(a^{n-1} + a^{n-2} + ... + a + 1)$   This means that $(a-1)$ | $(a^{n}-1)$    But $(a^{n}-1)$ is prime   So , $(a-1)$ = 1 $\Rightarrow$ $a = 2$   Now , let $n$ be composite   $n = kl$ , where $1 \lt k \lt n$ and $1 \lt l \lt n$   Now , $a^{kl}-1$ = $(a^{k}-1)$ . $(a^{k.{(l-1)}} + a^{k.{(l-2)}} + ... + a^k + 1)$   This means that $2^{n} -1$ is composite    Hence , we have achieved a contradiction      My Question : Am I correct ? Installing nvidia drivers on 18.04 I have an inkling that the following word usage is grammatically-correct; however, it is immensely difficult to search for confirmation, as all I receive for queries like &quot;Proper as an adjective&quot; are reams of results on &quot;Proper adjectives&quot;. The word itself I am using as a postpositive adjective, similar in form to the phrase &quot;time immemorial&quot;.  The shelter remains in name only; the shelter proper is in another town. Work in theory remained, although operations proper had mostly been outsourced.  Can anyone advise? I am trying to follow Blender Guru's beginner tutorial. He went to the Compositor tab and was using it to Denoise an image but when I got there, no nodes show up. Can you guys help? Badge Notification Repeats Not sure if someone else has already raised this - I've got eleven notifications that I've received a Caucus badge on English Language Learners. Perhaps one notification for each community that I'm a member of?  I am a member of the community, and did visit the election. Also, the badge hasn't been awarded more that once (so I think it's just the notification that's at fault).  Note that this seems to happen for other badges, it has been reported for: Caucus, Nice Answer, Enthusiast. I know how to detect when resistors are arranged in parallel or series arrangement and I can also find their equivalent resistance in simple circuits or when resistances are connected in form of triangle but what happens when the arrangement is complex like this :   Which resistors are parallel and which are in series ? How can I find the equivalent resistance in such cases ? Is there rule or method for figuring this out ? Read Post Data submitted to ASP.Net Form I have a working login form in an asp.net application.  Standard stuff with a username and password text box and a button to process the login.  Works fine.  I have a new requirement to allow the user to input the username and password from a separate plain html page that is not a part of my asp.net application.  I plan on achieving this using standard html - form, input, submit button etc.  The form action will be the URL of my asp.net login page and its method will be POST.    What I want to do in the C# code behind page of the asp.net login form, presumably in the Page_Load event, is to check if the request for the page contains a username and password value being passed in.  If it does then I need to read those values and process the login as if someone had clicked the login button on the asp.net page.  If not then I will display the login form as usual.    How do I check for the existence of, and read, the username and password values in the request for my page.