I am reading An Introduction To Set Theory by Professor William A. R. Weiss, and during chapter 3, on the proof of $\forall x \forall y \exists z \ z = x\times y$ (the euclidean product exists), there's the following:  We obtain: $$\forall t \in y \ \exists q \ q=\left\{ v : \exists u \in x \ v=\langle u,t \rangle \right\}$$ By Extensionality, in fact, $\forall t \in y \ \exists ! q \ q=\left\{ v : \exists u \in x \ v=\langle u,t \rangle \right\}$.  This step seems intuitively clear, but it also doesn't seem like an special case, it seems like this logic could be applied to any set built using builder notation. So my question is: For any arbitrary set $x$ and any abritrary formula $\Omega$, does $\exists q \ q=\left\{ v:\Omega \right\}$ imply, by Extensionality, that $\exists ! q \ q=\left\{ v:\Omega \right\}$?
says:  Note that there is one axiom for every such predicate φ; thus, this is an axiom schema. To understand this axiom schema, note that the set B must be a subset of A. Thus, what the axiom schema is really saying is that, given a set A and a predicate P, we can find a subset B of A whose members are precisely the members of A that satisfy P. By the axiom of extensionality this set is unique  (Emphasis added) My question is why, by the axiom of extensionality, is this set unique? The Axiom of Extensionality is about set equality; what is the relation between the Axiom of Extensionality with uniqueness of this set?
What is purpose of /etc/shadow and shadow cache files in Linux operating system? What is the purpose of the /etc/shadow file in Linux Operating system?  Also, is it same for SUSE clients? There is one shadow cache file is maintained what is purpose of that? How do I set a custom resolution? I tried to use xrandr to set 1680x1050 as a new mode to VGA output, but it says:  sudo xrandr --addmode VGA-0 1680 X Error of failed request:  BadMatch (invalid parameter attributes) Major opcode of failed request:  140 (RANDR) Minor opcode of failed request:  18 (RRAddOutputMode) Serial number of failed request:  35 Current serial number in output stream:  36 Why does GPS depend on relativity? Logo Pack - What should I include? What files should i put in a logo pack to give to clients? Should I put small, medium and large version of my logo as well as the vector file. Can I use I2C over 2m cables? Retrieving HTTP status code from loaded iframe with Javascript MS Access to Oracle easy Conversion / Migration USB 2.0 device (scanner) does not work with xhci_hcd on USB 3.0 system Repeat the same reference in footnote on different pages Alternative definitions of wifebeater I was a li'l nonplussed to find out that the word  can mean other things besides "a man who beats up his wife". Its definition reads:        One who (usually as a repeated practice) beats one’s wife, or a husband prone to violence   A kind of sleeveless shirt, often but not exclusively worn as an undershirt.   (uncountable, UK, slang) Stella Artois, a brand of lager beer.       reckons that sense 2 is American and originates "apparently from the association of such a garment with men who commit domestic violence". It does not register the UK slang. I'm guessing that Stellas also have an unfortunate association with domestic violence.  Are these alternative senses popular in their respective countries or are they perhaps nonce words that have lingered on? Any idea how these disparaging associations came about? I expect that the brewers of Stella Artois are none too happy about it. Solve the integral $\int_0^\infty x/(x^3+1) dx$ I'm new here!  The problem: integrate from zero to infinity x over the quantity x cubed plus one dx. I checked on wolfram alpha and the answer is that the indefinite integral is this:  $$\int \frac{x}{1+x^3} dx = \frac{1}{6}\left(\log(x^2-x+1)-2 \log(x+1)+2 \sqrt{3} \arctan((2 x-1)/\sqrt{3})\right)+\text{constant}$$  and the definite integral is this:  $$\int_0^\infty \frac{x}{1+x^3} dx = \frac{2 \pi}{3 \sqrt{3}}\approx 1.2092$$  I am trying to figure out all the steps in between. I see that there are logs, which are equivalent to the ln variant that i am more familiar with, which means it was integrating $1/x$ at some point; I also see an inverse tangent in there.  I started with long division to simplify and I got (which could be wrong because I am very tired right now) $x/(x^3+1) = x^2+ 1/(x^3+1)$ which seems to be a step in the right direction. Wolfram Alpha thinks I definitely did that step wrong. The two equations do not evaluate as equal.  Then I cheated and took the wolfram alpha factorization of $(x^3+1) = (x+1)(x^2-x+1)$...I probably should have known that but didn't offhand. Now it is looking like $x^2$ plus the partial fraction decomposition $1/(x^3+1) = A/(x+1)+(Bx+C)/(x^2-x+1)$. Am I heading in the right direction with this? At this point do I just plug in and crunch? Edit file based on existence of a string I have an input file, say, file1, which appears as follows:  WRITE=2 START=1 SMEAR=0 MIN=6 MAX=100   My problem is to search for the string SMEAR and if the string exists replace the entire line with SMEAR=-5. However, if the SMEAR does not exist, then I need to insert a new line SMEAR=-5. Ultimately the line SMEAR=-5 should appear only once. I can replace the string using sed:   sed -i '/SMEAR/c\SMEAR=-5' file1   Or I can insert the string as a new line    sed -i '10i SMEAR=-5' file1   But how can I combine them based on the existence of the string in the file?  EDIT: Also it would be helpful if I can pass the value of SMEAR as a variable. The  gives the recoil force, $\mathbf{F_{rad}}$, back on a charged particle $q$ when it emits electromagnetic radiation. It is given by:  $$\mathbf{F_{rad}} = \frac{q^2}{6\pi \epsilon_0 c^3}\mathbf{\dot{a}},$$  where $\mathbf{\dot{a}}$ is the rate of change of acceleration.  If a particle has a constant acceleration, $\mathbf{\dot{a}}=0$, then there is no reaction force acting on it. Therefore the particle does not lose energy.   Does this mean that a constantly accelerating charged particle does not emit electromagnetic radiation? Setup or "set up"? The word setup (as a noun), I feel, has become an acceptable part of the English language, at least insofar as technical manuals or computer textbooks are concerned. However I find that the word is used very often in verb form like so:     "You can setup the program quite easily ..."   when in fact my understanding is that the actual sentence should be:     "You can set up the program quite easily ..."   I would like to know if setup can be used as a verb. The situation seems be very similar to the word output (as a noun). Output is a purely technical word and its use is restricted solely to technical texts or at least in a technical context within non-technical books. However it is also used in verb form now like so:     "The system will output the signal ..."   I would be very grateful if this question is answered. Thank you Here is what it looks like in Blender Render (Not Cycles).   This is how I set it up.   Basically, I'm trying to export it, but it's not including the &quot;Freestyle&quot; lines. When I open up the .dae file in Finder, it looks like this:   What is wrong here? I want it to be exported (COLLIDA .dae) with the black outline. I'm seeming to think that maybe the &quot;Freestyle&quot; is only for the Blender Render. Am I right? How would I go about getting the black outline included in my exported model? Is it an export setting I have to tick? Thanks for any helpful answers. Django model with 2 foreign keys from the same table How do I compare binary files in Linux? I need to compare two binary files and get the output in the form:  &lt;fileoffset-hex> &lt;file1-byte-hex> &lt;file2-byte-hex>  for every different byte. So if file1.bin is    00 90 00 11   in binary form and file2.bin is    00 91 00 10   I want to get something like    00000001 90 91   00000003 11 10   Is there a way to do this in Linux? I know about cmp -l but it uses a decimal system for offsets and octal for bytes which I would like to avoid. I would like a way to mark a question in the Unanswered "my tags" page as something I'm not interested in or able to answer The /unanswered/tagged?tab=mytags page gives a list of questions from my favorite tags that are currently considered "unanswered" - they have no up voted or accepted answers. I think it's good to put them in front of me easily, but even though they have one or more tags that I'm interested in, I may not have the knowledge, ability, or interest to write an answer to them. However, they will remain on this page until someone writes an answer.  I would like to be able to dismiss questions from views, like this one, that show me interesting questions. I may choose to share them on my social media to get them views and answers, but I don't want to continually see them. Instead, I want to be able to use views like unanswered/tagged?tab=mytags as a queue of things that could use my attention. Closing Reject Suggested Edit dialogue freezes review process If you go to review a suggested edit and close the Reject dialogue box, the button at the top-right will continue to say "Rejecting...". You will not be able to continue with the review process.  I have refreshed the page, and tried clearing my cache. If I refresh the page, I can continue reviewing the edit.  I'm using Chrome Version 31.0.1650.57 m. How To Show Node Comment Count in node.tpl? I have had the bright idea of changing my user permissions so as guests cannot see the comments on a node. I have no problems with Views around my site, as they can clearly see there are comments on nodes, and they will be more inclined to register. The problem seems to be when they arrive on my site directly to a node, they don't see any comment, and think "that's a bit crap; let's go."  Is there a way I could show the amount of comments below a node?