In this case are 2 past perfect ok
Canonical Post #2: What is the perfect, and how should I use it?
If everything in the universe doubled in size overnight, would it be noticeable? This is exercise 1.6.1 of   Suppose that 501 distinct integers are selected from 1 . . . 1000. Show that there are distinct selected integers $a$ and $b$ such that $a | b$. Show that this is not always true if 500 integers are selected.  This seems like it should be simple but it's has been stumping me for a few days.  Here's my idea for a proof:  Let $S$ be a set of 500 distinct integers between 1 and 1000 that don't divide each other. Let $S_{1-500}$ and $S_{501-1000}$ partition $S$ into sets containing the elements ≤ and > than 500, respectively. Then consider $2 S_{1-500}$ and $\frac{1}{2}S_{501-1000}$, the result of multiplying the elements of these sets by 2 and 1/2, respectively. Because elements of S are coprime, $S_{1-500}$, $S_{501-1000}$, $2 S_{1-500}$, and $\frac{1}{2}S_{501-1000}$, are disjoint. Also note that each of these sets is a subset of the integers from 1 to 1000. One can deduce that the cardinality of the union of these sets is 1000, and therefore they form a partition of the integers from 1 to 1000.  Now, I know this demonstrates that you can't add another number to $S$ that is coprime with all the elements in $S$, which means that a coprime subset of the integers from 1 to 1000 can have at most 500 elements. Can someone articulate why this is true for me? Also, this was in a section on the Pigeonhole principle, can someone rework the proof to utilize that, and possibly make it simpler? Thanks. So I've got the shout in my shouts list and I've unlocked all three parts of it, but I can't seem to be able to use it all, just the first part.  Am I missing something? I still have a quest for 'Learn the first word of Durnehviir's shout' but it's showing as me having already got it.  What gives? What is Ubuntu's status on the Meltdown and Spectre vulnerabilities? Why is using "for...in" for array iteration a bad idea? Nesting HTML- anchor tags To whom was Yoda referring when he said, "There is another?" In the original trilogy Yoda refers to another hope:     Obi-Wan: "That boy is our last hope."   Yoda: "No...there is another..."   The assumption being after seeing Return of the Jedi (Episode VI) that the other was Leia.  If, however, you consider the prequel trilogy canon, then this assumption depends greatly on your point of view*. Obi Wan knows about Leia (and for unknown reasons has discounted her as a hope), so Yoda must be referring to someone or something else.  Is there any canonical source as to what Yoda meant, or was uncle George setting us up for Episode VII?  *as a wise man once said      you will find that many of the truths we cling to depend greatly on   our own point of view.   Just to add a bit of clarity: I was thinking about who Yoda was referring to when he was talking to Obi-Wan. When talking to Luke, it is obvious that Leia is the other (as he talks about a 'Skywalker'). It is not so obvious (after watching the prequel trilogy) that he means it when talking to Obi-Wan. I asked this question for many people/professors without getting a sufficient answer, why in QM Lebesgue spaces of second degree are assumed to be the one that corresponds to the Hilbert vector space of state functions, from where this arises? and why 2-order space that assumes the following inner product:  $\langle\phi|\psi\rangle =\int\phi^{*}\psi\,dx$  While there is many ways to define the inner product.  In Physics books, this always assumed as given, never explains it, also I tried to read some abstract math books on this things, and found some concepts like "Metric weight" that will be minimized in such spaces, even so I don't really understand what is behind that, so why $L_2$? what special about them? Who and how physicists understood that those are the one we need to use? Kant introduces the notion of the noumenon in his critical philosophy; he also later demonstrates in his system that solipsism isn't possible.   Is the introduction of the noumenon essential to this demonstration, and thus justifies it's introduction? If $aRb \land bRc$ never occurs, is a relation considered transitive? How to find manually installed packages? I installed a plain Ubuntu 10.10 on my computer and installed some software via apt-get. Last week I managed to break everything and just started from scratch, and I need to reinstall my software. Is there some way to create a list with all the packages that I have installed manually?  So that it would give me a list like texlive, ... and not texlive, texlive-dep1, textlive-dep2, ... and all the standard packages removed?  If I could somehow figure out which programs out of the regular install I have removed, that would be awesome too!    Update 2015-05-23: I now use Ansible as configuration management on my systems. There I specify the packages to install. This serves the goal, installing all the programs on a fresh system, even better. So this question might be the entry into some light configuration management. The following you could have seen in  (sorry, I've commented there, so you won't see it anymore) in Chrome 35.0.1916.153 m (current version). Notice how the add comment link overlaps the bounty box:    The same you can observe with a comment box on other sites, e.g. at this time in :    Could you shift it a bit down ?  It seems that it happens on more than just Stack Overflow (maybe all sites ?). How to pick the title of a remote webpage Get variable from PHP to JavaScript I want to use a PHP variable in JavaScript. How is it possible? Short Story/Novella in Which People Living in What they Think is Small Town Fifties America Find Look Magazine and Don't Recognize Anything in It This is probably from the mid-to-late fifties.  It involved families living in what seemed to be normal, small town America of the time.  But two boys found in a garbage dump or in a nearby landfill a copy of "LOOK" magazine from their time period (1955? 1957?).  Their problem was that they and their families didn't recognize any of the people or events that were reported in the magazine.    The tone of the articles in the magazine matched their actual lives in that the magazine reported world events, trouble spots, celebrities, politics, and had advertisements for cigarettes and TVs, etc. but the facts were completely different. I think the magazine was a magazine from 'real' America and they were living in a 'fake' America, i.e., they had been brought up in a separate isolated environment with everything different but now had found something from the 'real' world. The tone of the story was that they were never supposed to know about any of this. Maybe, just maybe, part of it was a story about Marilyn Monroe but none of the people had ever heard of her. I can't remember much more than that but I have a nagging thought that they were part of some controlled environment experiment by the US Military. How should ambiguous tags be dealt with? If tag "foo" refers to two or more different technologies or concepts is it better to retag the less common usages or is it better to add additional tags to disambiguate?  My particular issue is with the  tag which mostly refers to the  but could also refer to the  programming language &amp; software platform. Is it better to retag the Curl language questions as  (any other suggestions?) or add some other tag such as "language"? What is a general rule for use of auxiliary verbs in sentence? Should we duplicate it or not? For instance,     It is available for every item and (is) used with . . . If $\mathrm{Tr}(A)=0$ then $T=R^{-1}AR$ has all entries on its main diagonal equal to $0$ Momentum is defined by the product of mass and velocity. Now a projectile out of a gun has to have high velocity to penetrate a human body, as its mass isn't significant. But to reach this velocity, due to inertia/the law of energy conservation, momentum on both sides is to be equal. As there is strong negative acceleration in the opposite direction of the bullet, it should result in a strong force in the opposite direction of the bullet ($F = m*a $).  Therefore I wonder why shooting a bullet with a handgun is not ripping your hand apart. The usage of singular and plural has always been confusing for me.   I often see sentences like these         People are using cell phones.   People are using a cell phone.      Does the first sentence mean everyone has a phone and they are all using their own?  Does the second sentence mean they are sharing one cell phone?   If I see a group of people holding a cell phone in their hand(s), &lt;- even this is confusing for me, should I use the first sentence then?   Another example: you see two men, and both of them are carrying a bag. Which sentence should I say/use?        They are carrying backpacks          They are carrying a backpack      Could you please make it clear for me?