Let $A$ and $B$ be two sets of real numbers. Define the distance from $A$ to $B$ by $$\rho (A,B) = \inf \{ |a-b| : a \in A, b \in B\} \;.$$ Give an example to show that the distance between two closed sets can be $0$ even if the two sets are disjoint.
Is given metric space $(M, d)$. Let $A\cap B = \emptyset; \,\,\text{dist}(A,B):=\inf\{d(x,y):x\in A, y\in B\}$. $A, B$ are both closed sets. Is it possible that $\text{dist}(A,B)=0$?  The first thought comes into mind is that obviously $\text{dist}(A,B)&gt;0$, but possibly there are some tricky $d$ and $A, B$ so that it's untrue.  Thanks in advance!
Cardinality of the Irrationals Possible Duplicate:          We previously proved that $\mathbb{Q}$, the set of rational numbers, is countable and $\mathbb{R}$, the set of real numbers, is uncountable. What can you say, then, about the cardinality of the irrational numbers?  I would say that the cardinality of irrational numbers is uncountable. That would be because the set $\mathbb{Q}$ is a subset of $\mathbb{R}$. The irrationals is also a subset of $\mathbb{R}$. Basically, my thought is that the continuum is composed of irrationals and rationals (only two subsets of reals). Because we can prove that the rationals are countable by Cantor's proof, then the remaining subsets of $\mathbb{R}$, the irrationals, has to be uncountable for $\mathbb{R}$ to be uncountable. My concern is how to go about the prove this (rigorous, intro level). This is the PATH variable without sudo:  $ echo 'echo $PATH' | sh  /opt/local/ruby/bin:/usr/bin:/bin   This is the PATH variable with sudo:  $ echo 'echo $PATH' | sudo sh /usr/local/sbin:/usr/local/bin:/usr/sbin:/usr/bin:/sbin:/bin:/usr/X11R6/bin   As far as I can tell, sudo is supposed to leave PATH untouched. What's going on? How do I change this? (This is on Ubuntu 8.04).  UPDATE: as far as I can see, none of the scripts started as root change PATH in any way.  From man sudo:      To prevent command spoofing, sudo   checks ``.'' and ``'' (both denoting   current directory) last when searching   for a command in the user's PATH (if   one or both are in the PATH). Note,   however, that the actual PATH   environment variable is not modified   and is passed unchanged to the program   that sudo executes. My son has been playing Minecraft on our home computer for a couple of years now.  He just got an iPad Mini 2 for his birthday. Is it possible to transfer his game and his saves onto his iPad Mini 2? When should one use "should" and when should one use "must"? I tend to use should when it's a suggestion I don't have a strong opinion on, i.e. it could be done in many other ways than the one I'm suggesting and it can still happen.     You should stop by that gas station to go to the bathroom.   On the other hand, I usually use must for things that have to happen in a certain way; any other way would make that thing not to happen. The problem is that some people find this usage harsh, as they receive it like if I'm giving them an order.      You must cross the street to get to that store.   To me, it's just a depiction of reality, not an order. But I'm neither a native speaker nor a regular person. :-)  So, what's the proper use for must and should? Does this use differ among the English dialects? Given two functions $f$ and $g$, is there a formula for the Fourier transform of $f \circ g$ in terms of the Fourier transforms of $f$ and $g$ individually?  I know you can do this for the sum, the product and the convolution of two functions. But I haven't seen a formula for the composition of two functions. This discussion informed our process for seven years, but eventually required updates and augmentation. For discussion regarding the successors to this process, please see:    Stack Exchange has some of the best Moderators around — seriously — and that is due in no small part to the communities who scrutinize and vote in our Moderator elections.  We have a rather formal process for electing moderators, but up until now  has been quite informal. On those rare, rare occasion when things just aren't working out, you can see why it becomes such a sticky business to remove someone from a democratically elected position. Our current process is one of &quot;we've got your back.&quot; The Community Team routinely monitors moderator activities, so if a moderator violates the  or fails to uphold the , we can and do intervene. If the disputes and problems persist without resolution, eventually we can ask someone to step down. That's not good enough. We need a much more transparent and prescribed process for when — and more importantly how — to intervene when on-going problems are going unresolved. In urgent situations, we'll step in to protect a site, but ideally the process should be impartial and not subject to accusations of Stack Exchange simply applying undue influence over a community when the whim strikes. The Question: Before we start bouncing around ideas internally, I'd like to hear what you think. What process should be in place to add some checks and balances to deal with higher issues like possible impeachment of a moderator? When and how would such a process come into play? How should such situations be resolved? How do we keep it from being abused? Have at it. This is a round table vetting of ideas. I have heard several pseudoscientific explanations about the  and find them hard to believe.  As a mathematician mainly focusing on functional analysis, I have a considerable interest in this matter. Although I still have no idea about it, I aim to truly understand it one day. For now, the question is:  Can the Heisenberg Uncertainty Principle be explained so that a non-scientist can at least get a correct idea of what it says and what not?  One of the most heard explanations is the one that says naïvely that while trying to determine the position of a particle you send energy (light) so you modify its speed, therefore making it impossible to determine the speed. If I understand well this should only be a some sort of metaphor to explain it, right? I need to get rid of the GNOME Desktop Environment which has been installed by default when I updated my pc from 15.10 to 16.04.  After a while, I found myself not only with GNOME DE, but also with Cinnamon alongside (in the login screen, I could choose at least five or six DE!)  Now I am using Unity, and I'm happy with that, and I successfully uninstalled Cinnamon, but now I'm afraid that after uninstalling GNOME I will mess up the Unity DE, since I see many packages running in unity with the packagename starting with gnome-... Would someone mind to help me? Thx in advance a function or a factor to scale a list of real numbers from one range to another How can I String.Format a TimeSpan object with a custom format in .NET? What is the recommended way of formatting TimeSpan objects into  a string with a custom format? Dual Canadian and British citizen - healthcare coverage in the UK? Irreducible polynomial of degree $8$ Are there areas of mathematics (current or future) that cannot be formalized in set theory? I often read that ZFC can formalize "most" of everyday mathematics, but I could never find an example which it cannot. The closest I got is differential geometry (DF), where some article mentions that the translation from concepts of DF into set theory would render it almost "meaningless", in the sense that any intuitions one could have would be lost in the translation. So, if I didn't misunderstand, at least in theory, the translation could be done, so that a future AI machine could prove theorems of differential geometry using set theory. On the other hand, the Stanford Encyclopedia of  Philosophy states that "all of mathematics can be reduced to a formal system of set theory"().   So, the questions are:    Is there any current area of mathematics that could be not formalized within set theory? If so, which ones? and  Is there any reason to think that every math of the future will be always formalizable in set theory? any references to read on the subject are welcomed!   Clarification: by set theory I am not restricting to ZFC, but to any flavor of it, including all additional axioms you want (about large cardinals or any kind) What's the difference between the snap folder in /home/UserName and the snap folder in root (/)? I want to transfer my current system hard drive to a larger hard drive. Doing so, I thought of creating an image of my system hard drive which is 320 GB in size.   Would I be able to restore this image on a larger hard drive? I plan buying a 1TB hard drive.  If yes, how? I prefer no data loss.  I am using Windows 7 When I open the game, the screen stays at the Mojang sign and then crashes. It has something to do with my texture pack, but I can't start the game to change it and it won't let me delete the texture pack folder alone.  I've already deleted the bin and resources folders in the minecraft folder and downloaded the game again. I opened it but the same problem was still there.  How do I properly perform a clean uninstall of the game? I'm trying to crack a cypher that I believe is Vigenère encrypted and I'm currently stuck.  I calculated the key length by finding repeated sequences in the cypher and calculating the the common factors.  Word    distance factors agzd    16       2,4,8,16        zna     92       2,4,23,46,92 zdc     54       2,3,6,9,18,27,54 csio    44       2,4,11,22,44 omy     46       2,23,46 zab     5        5   I assume the key-length is 4 as these are the once that are common factors, but from here on trying to map the correct caesar shiffre for each position I'm stuck.  Can somebody confirm that my key-length calculation has no errors which would explain why I can't solve the cypher.  Here is the cypher:  jD 2GSA 7HNA ZNA w.m. V7KXA 36HBM MKE5 /H SGP9 +R GKKJ 2I A4OTC KDBCUR. 19+R 8Y TKJ JG4 IGO6 6U4T OJ hH/BIK. 196 qFKT+9 BNJK ZD6+Q AGZD. kJ'R EaOP6 4NCR ZK 36 / AGZD6C/HOIE2D. x0ZNAC2S8IO8DI HB xX8D4D 0XK O66M 0Y MN62S 8TZABBD2Za8BI. lBJ HA+DF 6UU/ 2J /FOZDC6S8I OJ kDC8G OO 2BLCYZ O66M 0Y G 925F4 UL J2JHCTGH FHH3K How to check whether a JavaScript variable defined in cross-browser way?  I ran into this problem when writing some JavaScript utilizing FireBug logging. I wrote some code like below:  function profileRun(f) {     // f: functions to be profiled     console.profile(f.constructor);     f();     console.profileEnd(f.constructor); }   It works fine in FireFox/FireBug, but it reports error in IE8 RC1. So, I'd like to do some checking whether console variable exists in the execution environment.  Below code works fine in FireFox, but not in IE8 RC1.   function profileRun(f) {     if (console != undefined) {         console.profile(f.constructor);     }      f();      if (console != undefined) {         console.profileEnd(f.constructor);     } }   However, if I do it this way. It works in IE8 RC1. Why?   function profileRun(f) {     if (window.console != undefined) {         console.profile(f.constructor);     }      f();      if (window.console != undefined) {         console.profileEnd(f.constructor);     } }   Is there any cross-browser way to check it? Let $\{X_\alpha\}_{\alpha\in J}$ be a family of topological spaces and let $X=\prod_{\alpha\in J}X_\alpha$ be endowed with the product topology. Show that $X$ is path connected if and only if each $X_\alpha$ is path connected.  I know this is true if path coennected is changed by connection but I am very confused with this problem, I have tried to do the following:  $\Rightarrow $ let $x,y \in X_\alpha$ be then $(x_\alpha), (y_\alpha)\in \prod X_\alpha$ where $x_\alpha=x, y_\alpha=y$ and $x_\beta=0, y_\beta=0$ for any $\beta\neq\alpha$, and since P is path connected then there is continuous $f:[0,1]\rightarrow \prod X_\alpha$ such that $f(0)=(x_\alpha)$ and $f(1)=(y_\alpha)$, then if $\pi_\alpha:\prod X_\alpha \rightarrow X_\alpha$ is the projection, which is continuous, we have that $\pi_\alpha f:[0,1]\rightarrow X_\alpha$ is continuous and $\pi_\alpha (f(0))=x$ and $\pi_\alpha (f(1))=y$.  $\Leftarrow$ I do not know how to do this address, could someone help me please? Thank you very much. How to write permutations as product of disjoint cycles and transpositions