In an exercise, it asks to prove $\bigcap \varnothing$ is equal to the set of all sets.  I understand that there are many proofs online but I have a specific question (mostly regarding mathematical logic) regarding my proof.  From my understanding, given a set $S$ (where all of its elements are also sets), $x \in \bigcap S \iff \forall \, Y\in S, x \in Y$.   Now I will try and show the set of all sets is contained in $\bigcap \varnothing$.  Let $x$ be an arbitrary set - now $x \in \bigcap \varnothing \iff \forall \, Y\in \varnothing, x \in Y$.  My question is why the statement $\forall \, Y\in \varnothing, x \in Y$ vacuously true?  From my understanding, a statement is vacuously true in the case of an implication (ie. $p \implies q$ where $p$ is false) but I fail to see how $\forall \, Y\in \varnothing, x \in Y$ can be written in an implication (I tried writing it in the form $\forall Y: Y \in \varnothing \implies x \in Y$ but this is not the same statement)
Let $\mathbb S = \varnothing$.      Then from the definition: $ \bigcap \mathbb S = \left\{{x: \forall X \in \mathbb S: x \in X}\right\}$      Consider any $x \in \mathbb U$.      Then as $\mathbb S = \varnothing$, it follows that: $\forall X \in \mathbb S: x \in X$ from the definition of vacuous truth.      It follows directly that: $\bigcap \mathbb S = \left\{{x: x \in \mathbb U}\right\}$      That is: $\bigcap \mathbb S = \mathbb U$. ()   Proofwiki uses the above "proof" to "prove" that intersection of the empty set is the whole universe.   My question is, is the use of vacuous truth really allowed in axiomatic set theory, like ZFC? I don't see how the use of vacuous truth is justified.   The next problem I can think of is that we cannot really "define" the elements of empty set (to my knowledge, there is no element in empty set) so how can we then prove as the above proof did? This seems to contradict the use of vacuous truth.  And of course, there is issue of using the whole universe as a set, and I don't think this is allowed.... (Maybe proof above is using a different axiomatic set theory, as I am using ZF-minded thoughts...)
i.i.d. random variables uniform distribution Two rotating discs with same angular momentum when brought in contact completely stop. Why is the angular momentum not conserved in this case? Two discs mounted on different thin, lightweight rods oriented through their centres are made to rotate about their axes seperately such that the angular momentum of the two about their respective axes is same in both magnitude and direction. When both are brought in contact, they stop due to force of friction. Why is the angular momentum not conserved about their axes in this case?(Was positive before contact, but zero after the process completes, no external force present either) Setting background-image using jQuery CSS property What tense should I use after the conjunction 'lest'? Must it always be an infinitive? What Thunderbolt to USB 3.0 cables/adapters exist for OS X and Mac hardware? I have the Late 2013 Macbook Pro, which has two USB 3.0 ports and two Thunderbolt 2 ports. I currently use both USB 3.0 ports and one TB2 port. I want to connect a third USB 3.0 device to the machine, but I don't want to use a USB splitter on one USB port as I'm concerned the devices would draw too much power from the bus. I have no current use for the second TB2 port and I would love to be able to use it for this purpose.   Browsing extensively online, I've found numerous docks, such as , that connect via a Thunderbolt cable. This would work, but it's an expensive, 'bulkier' solution (and not the one I'm looking for).   I'm aware that a USB 3.0 device wouldn't be able to take advantage of Thunderbolt's I/O speeds, but it's certainly compatible, which is all I'd need. After all, Apple even sells a . I also know that Thunderbolt is not only used for digital video purposes (along with the TB > Firewire option, there exists ).   There are almost no Thunderbolt to USB 3.0 cables/adapters on the market. I'd think this hardware would be more prevalent given the prominence of USB devices out there. What adapters are available for OS X? SQL Switch/Case in 'where' clause Java: Difference between the setPreferredSize() and setSize() methods in components How do you prevent web pages from appearing in the Google search index? I have some web pages that Google has indexed. I have now added a robots.txt file to exclude them, but should I also add a noindex tag? If so, then how do I do so? Installing Ubuntu on a SSD Going to install Ubuntu 10.10 on new intel x25M 80GB SSD. It will be fresh install. I have been googling for past few days and getting overwhelming articles/blogs/Q&amp;As. One particularly very useful being:     But with so many suggestions and differences of opinions (on different links) this simple OS install process seems to be daunting task to me and I really want to stick with Ubuntu (although have used for very short period of time).  Can someone help me by answering few questions (yes, they are repeated because I couldn't comprehend the answers elsewhere)   Which file system (ext2/3/4 or something else)? (consider SSD life) Can it be changed after installation? Should I partition the disk? (as we do in traditional HDD) for now, no plan of dual booting. Only Ubuntu will live on scarce space of 80GB SSD. I have 2 GB RAM, should i still allocate swap space (if I don't allocate swap space, can I still hibernate the machine)? will swap space impact SSD life? Should I consider putting additional 1GB RAM to avoid swap space? What is partition aligning? Is it needed to be done before installing the Ubuntu OS or can be done later?   Intended usage - heavy browsing, programming, regular video/music and some other non-CPU/RAM-intensive programs. Will keep big files in an external hard drive.  laptop config - 3yr old Vaio, Core2 Duo, 2GB RAM  More references:        (resizing/creating does not seem a simple process after look this) Expression for a choice which isn't really one Does $ \int_0^{\infty}\frac{\sin x}{x}dx $ have an improper Riemann integral or a Lebesgue integral? In  wikipedia article for improper integrals,  $$ \int_0^{\infty}\frac{\sin x}{x}dx $$ is given as an example for the integrals that have an improper Riemann integral but do not have a (proper) Lebesgue integral. Here are my questions:   Why does this one have an improper Riemann integral? (I don't see why $\int_0^a\frac{\sin x}{x}dx$ and $\int_a^{\infty}\frac{\sin x}{x}dx$ converge.) Why doesn't this integral have a Lebesgue integral? Is it because that $\frac{\sin x}{x}$ is unbounded on $(0,\infty)$ and Lebesgue integral doesn't deal with unbounded functions? I just came onto a project with a pretty huge code base.   I'm mostly dealing with C++ and a lot of the code they write uses double negation for their boolean logic.    if (!!variable &amp;&amp; (!!api.lookup("some-string"))) {        do_some_stuff();  }                                      I know these guys are intelligent programmers, it's obvious they aren't doing this by accident.   I'm no seasoned C++ expert, my only guess at why they are doing this is that they want to make absolutely positive that the value being evaluated is the actual boolean representation. So they negate it, then negate that again to get it back to its actual boolean value.   Is this correct, or am I missing something? Evaluating the supremum of a function as n tends to infinity Problem: Let $g:[0,1]\to\mathbb{R}$ be a continuos function such that $g(1)=0$. Show that $\sup\limits_{x\in[0,1]}|x^ng(x)|\to0$ as $n\to\infty$.  My approach: Let $f(x)=x^ng(x)$. Then, $f(x)$ is continuos on $[0,1]$. Using the extreme value theorem we can say, that $|f(x)|$ is bounded on $[0,1]$ and attains its bounds, i.e., $\exists x_0\in[0,1]$ such that $|f(x_0)|=\sup\limits_{x\in[0,1]}|f(x)|$.  Now, $|f(x_0)|=|x_0^ng(x_0)|$. If $x_0=0$, then we are done. So, let us assume $x_0\neq0$. By the Archimedean property of $\mathbb{R}$, $\exists$ a $y\in\mathbb{R}$ such that $y=x_0^n\leq1$, since $x_0\in[0,1]$.   Hence, $\lim\limits_{n\to\infty}\sup\limits_{x\in[0,1]}|x^ng(x)|=\lim\limits_{n\to\infty}\sup\limits_{x\in[0,1]}|f(x)|=\lim\limits_{n\to\infty}|f(x_0)|=\lim\limits_{n\to\infty}|yg(y^{\frac{1}{n}})|\leq\lim\limits_{n\to\infty}|g(y^{\frac{1}{n}})|=g(1)=0.$  I will be glad if anyone can tell me if I have done any mistake and if so please help me to correct it.  Thank you. Android design convention for saying a user needs to complete an action first? I've read the Android design conventions, but can't see a good solution.  I have a 3 tab mobile application. Tabs 2 and 3 need a selection made on screen 1 before they will work. The user enters at Screen 1, and the tab options are along the top. If they attempt to open screen 2 or 3, I would like to advise them to make a selection first. 'notifications' are the closest answer I can see from the Android design guidelines. Anybody have any other solutions?  Thanks  Andy Why does locking the rear tires on a vehicle cause it to spin? are often used in motor-sports and stunt driving to turn around tight corners.  Essentially, the driver applies the handbrake as he enters the turn, in order to "kick out" the back end and slide the vehicle around the corner.  However, applying the handbrake hard enough to lock the rear wheels while traveling straight will cause the rear of the vehicle to "kick out".  Given enough speed, the vehicle will spin 180 degrees.  Why do sliding wheels on a car want to "get in front" of rolling wheels? Comparing Arrays of Objects in JavaScript Finding files by only knowing extension? I'm somewhat disorganized in my file saving habits, so I wind up using the Search function in Windows Explorer more often than one should. I was wondering, is it possible to have Windows find all files of a certain extension ( like, say for example all .rar files) if it is, how? MiKTeX graphics version 1.3b bug on Windows 10 (\set@curr@file undefined) How can I make Devanagari fonts render properly in Ubuntu 16.04? Devanagari fonts appear broken like this:     It should appear this way    I have the same problem in browsers and office applications.  Is there any way to fix it? Density of random variables Let $X$ and $Y$ be independent random variables with respective densities: $f_X (x) = xe^{-\frac{x^2}{ 2}} 1_{]0,+\infty[} (x)$ and $ f_Y (y) =  \frac{1}{\pi \sqrt{1-y^2}} 1_{]-1,1[} (y)$  Let $U:=XY$ and $V:=X\sqrt{1-Y^2}$  I try to check whether U and V are independent and which law they follow.  What I did:  After some calculations, I found the density of $(U,V)$:  $ f_{(U,V)}(u,v)=\frac{1}{\pi } e^{-\frac{u^2}{ 2}}1_{\mathbb{R}}e^{-\frac{v^2}{ 2}}1_{\mathbb{R}_+^* }$ Which mean that $(U,V)$ follows a $ \mathcal{N}(0,1)$ law (because $u^2+v^2=x^2$) and that  U and V are independent because $f_{(U,V)}$ can be factored as the products of densities of $U$ and $V$  My question:  There is a factor $\frac{1}{\pi}$ and I don't know to which density it will belong: to $U$ density or to $V$ density?  Many thanks!