Let us define:  $\bigcap \emptyset  = \{ x|\forall A(A \in \emptyset  \Rightarrow x \in A)\} $  I understood that this cannot be defined. Somehow it enables Russell's paradox to exist. Why is that?
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...)
Remove Meta Privilege for MSE (only) What I am asking here is to please remove the privilege to participate in Meta for MSE.  This privilege is literally useless. You can't really have a meta site for a meta site. There's no Meta.Meta.SE here or any other SE site out there: Meta.SE covers for every site in SE, including itself.     When users find out that you don't really get anything new, it's kinda sad for those people. There is no use for it except just to be there and fill up the space between the first and third privilege. Privileges should be shown only if you earn something new, and this privilege to participate in the meta isn't new because it's where you report problems in all of SE, including Meta.SE.    So please remove the privilege to participate in meta just for Meta.SE.    This is not a dupe as this is a feature-request to remove it, not support on why it is there. How to insert text into a root-owned file using sudo? Check if specific id element exist MISSISSIPPI problem How many arrangement of the letters in MISSISSIPPI have at least 2 adjacent S's?  I was thinking that I can glue two of the S's together, so there will be 9 letters plus the special letter SS, and the number of arrangements will be $10!/(2!2!4!)$, this is not the correct answer, but what is wrong with my reasoning.  (the correct answer is $11!/(4!4!2!)-7!/(4!2!)\binom{8}{4}$, I understand the solution) Why does Ubuntu create a file ending in ~ for every file I create with a text editor like GNU Emacs? How to prove the divisibility rule for $3\, $ [casting out threes] The divisibility rule for $3$ is well-known: if you add up the digits of $n$ and the sum is divisible by $3$, then $n$ is divisible by three.  This is quite helpful for determining if really large numbers are multiples of three, because we can recursively apply this rule:  $$1212582439 \rightarrow 37 \rightarrow 10\rightarrow 1 \implies 3\not\mid 1212582439$$ $$124524 \rightarrow 18 \rightarrow 9 \implies 3\mid 124524$$  This works for as many numbers as I've tried.  However, I'm not sure how this may be proven.  Thus, my question is:     Given a positive integer $n$ and that $3\mid\text{(the sum of the digits of $n$)}$, how may we prove that $3\mid n$? Evaluate: $\lim_{n\to\infty}\left({2\sqrt n}-\sum_{k=1}^n\frac1{\sqrt k}\right)$ How do I change the screen resolution using Ubuntu command line? I installed Ubuntu and the resolution of the desktop is so big that it only shows about 50% of the screen, but is shows up normal while using command line.   What should I type in the command line to change the desktop screen resolution? I have two desktops (one of which I'm using as a server), my main machine has a weaker i5 cpu, my server has a stronger i7 cpu. On my main machine I am experiencing stutters because the CPU often runs at full load under regular use (strangely, I also use upwards of 10GB of RAM, no joke... The heaviest program I'm running is google chrome).  I was wondering if there was any possible way to share a cpu from one machine over network to another machine as an emulated processor, which could be used by the client machine in a similar sort of fashion as virtual machines use or "borrow" their host's cpu.  This way I could offload work onto the server's cpu when the main cpu is under full load to prevent annoyances like stutter, and possibly combine both of them for more intense tasks (like decryption, compilation, archiving or other multithreaded stuff)  Does this sort of software exist, or has anyone at least tried to make it? If not, is it feasible?  (This sort of program would also ideally be able to offload data onto the server's RAM for faster processing, since the now common gigabit connections get to have nearly the same speeds as a standard HDD this doesn't sound far fetched either)  To reiterate, what I am primarily looking for is the ability to offload work from a system's main cpu when it is under too much load and forwarding it to another computer's cpu over network. In comics does Batman have a rule against killing? Inspired by  question, I have to admit I don't know a lot about the Comic Book Batman, but I know a fair amount about his other incarnations.   I know for example, that the 1989 movie had him kill the Joker. He vehemently wanted the Joker dead and had no compunctions about taking a life. I also know that he had a rule against killing in The Animated Series that he never broke, and that he very nearly broke it in Batman Beyond.  But as far as the comic books go, when did Batman develop a rule against killing his foes? Does he ever kill them rather than deliver them to the proper authorities? are used by the Empire in a manner that is tactically identical to how a modern day army would use a tank, in that they are used as shock armour with powerful main guns which will punch through entrenched positions. (No combined arms tactics were used in the Battle of Hoth, but that's a different question altogether)  However, AT-AT walkers have a number of (rather lethal) drawbacks and no advantages that I can see compared to tanks.    They have footpads in place of treads, which greatly increase their ground pressure, and which will cause them to easily sink as compared to treads They have a high center of gravity and can be easily tripped, which was exploited in the Battle of Hoth. (granted that this was canonically presented as an oversight, albeit a rather stupid one, since most martial arts strongly emphasise lowering your CG.) They are slower than a modern main battle tank despite their long legs, which one would normally expect to grant additional mobility (max speed 60km/h on flat terrain, compared to 72km/h on flat terrain for the ). This is exacerbated by the fact that AT-AT walkers are using fusion power cells as compared to diesel engines.   It would seem that it would make far more sense for the Empire to remove the legs and mount the main chassis on treads, which would make the vehicle far more effective. Why then, did they not do this? It clearly was not the case that imperial Engineers lacked imagination, as they did produce .   Possible answers which I find implausible as advantages:  1: AT-AT walkers are tall, and therefore have a surveying advantage over the battlefield / have a better vantage point to fire from  Orbital satellites can provide the observation, and in any case air/orbital support should always be called in, especially if fire support from above is required. How do you unpin and move the form in the Visual Studio designer? Finite order endomorphisms Why the finite order endomorphisms are diagonalizable over the complex numbers (or any algebraically closed field where the characteristic of the field does not divide the order of the endomorphism) with roots of unity on the diagonal? I have a shapefile and I want to change the order of the features. Is that possible? I want to sort the features in the shapefile because I'm embedding this shapefile in a mobile app. The app will display a list of features, and although I could do the sorting at load time, I'd prefer to have them sorted. Prove that the derivative of $x^w$ is $w x^{w-1}$ for real $w$ Can anyone give a proof of the derivative of this type of function? Specifically showing that  $\dfrac{d(x^w)}{dx} = wx^{w-1}$ for a real $w$?  I tried to use the Taylor series expansion for $(x+dx)^w$ and got the correct result. However, the proof of the Taylor series requires knowledge of the derivative of these functions. So this is essentially circular reasoning. I know that the same series is also given by the binomial expansion, but that's not entirely satisfactory either, because where's the proof that the binomial expansion works for all reals (isn't it only apparent for integers)? So far all of the arguments I've come across involve circular reasoning. I was thinking of showing that the binomial expansion is true for all reals using some form of proof by induction e.g. something like this.  I'm really not sure. Pure verbal nouns/deverbal nouns vs. gerunds This is a follow-up to a previous question which I am still trying to understand. I think I'm making progress in my understanding, but I would appreciate feedback to help me refine my thinking.  Here is the sentence:     Good writing requires hard work.   Originally, I thought that "writing" was acting as a gerund in this phrase.  After reading through the replies to that question along with back posts related to gerunds, I've since concluded that "writing" in this sentence is a , also termed a deverbal noun.  My reasoning?  —In the sentence above "writing" is modified by "good" and is thus acting more like a noun than a verb. —It has no direct object.  Is my thinking correct here?  If not, could someone politely help me to refine it?  However, what if I had written the following?     Writing well requires hard work.     or   To write well requires hard work.   In these examples, I'm thinking that "writing" would be a different kind of verbal noun—a non-finite verb acting as a gerund or infinitive, which is one piece of a larger complement noun clause.  The last bit is muddy for me, but I will leave it at that. Thank you in advance for your patience and insight. Create a tar archive split into blocks of a maximum size I need to backup a fairly large directory, but I am limited by the size of individual files. I'd like to essentially create a tar.(gz|bz2) archive which is split into 200MB maximum archives. Clonezilla does something similar to this by splitting image backups named like so:  sda1.backup.tar.gz.aa sda1.backup.tar.gz.ab sda1.backup.tar.gz.ac   Is there a way I can do this in one command? I understand how to use the split command, but I'd like to not have to create one giant archive, then split it into smaller archives, as this would double the disk space I'd need in order to initially create the archive. Are there cheap GPS or other navigator devices with downloadable maps for the whole world? I don't need a recommendation or a list but my knowledge of GPS and "navi"s and the differences between them is an utter mystery to me.  But often I'm lost in backstreets in random countries where access to something like a live Google Maps would be awesome. I absolutely cannot afford a smartphone with worldwide roaming though.  Can GPS's do what I want or am I barking up the wrong tree? If they can is it only high end ones than can? OS X Yosemite brought support for recording iPhone/iPad screen natively without having to using AirPlay applications like AirServer or Reflector.   How do I record the iPhone screen on macOS ? Update Garmin Nüvi 255 maps