According to set theory, are $\{\}$ and $\{\{\}\}$ equal?
In naive set theory, I believe ∅ = {∅} = {{∅}} is correct, but just wanted to make sure that I understood this correctly.   ∅ is an empty set, so having an empty set as an element of a set that contains nothing else is pretty much the same thing as e.g. marking the number 2 as 2.000000, right? Doesn't the symbol ∅ intrinsically imply that this is an empty set which contains it self?
I'm not a native English speaker. I am having difficulty with some words. Could anyone please tell me if there's any difference between the above mentioned short sentences, as I'm a little confused. Recommendations for Numerical Analysis texts? Why, with Springer's  svmono template, do the letters written with the \mathcal font change? It seems that \mathcal becomes \mathscr.  Edit: Look at this image:    With the svmono template \mathcal returns the first alphabet and not the (classical) latter.  Here the packages I use in my document:  \documentclass[envcountsame,envcountchap]{svmono}  \usepackage{subfiles} \usepackage[utf8]{inputenc}  \usepackage{mathptmx} \usepackage[bottom]{footmisc} \usepackage{microtype} \usepackage{stmaryrd}    \usepackage{amsmath} \usepackage{courier} \usepackage{mathrsfs}  \usepackage{amssymb}  \usepackage[all,cmtip]{xy} \usepackage{faktor} \usepackage{graphicx} \usepackage{mathdots} \usepackage[colorlinks={true},linkcolor={blue},citecolor={blue}]{hyperref} \usepackage[frame,width=15truecm,height=23.5truecm,center]{crop}   With this settings I have the above strange behavior of \mathcal. Looking for the title of an old sci-fi book, pretty sure it is a Baen or Del Rey imprint. Basically, mankind has reached the stars and expanded throughout the galaxy. Suddenly, an unknown race appears and begins a war of extermination against man. Finally, the galaxy is desolate, and one man remains. He is in a starship with an AI and becomes injured in the battle.  The ship's AI takes over and speeds out into the unknown, followed by the last remaining member of the unknown alien race. As all speeds are at tremendous multiples of the speed of light, the man cannot get away, but the alien cannot catch up.  Finally, communication is opened, at which point the man asks "why?" The alien tells him that man had discovered immortality, but that it required the death of a member of a race known as the Flower People for every human who achieved immortality. The alien says that Man would have found the answer eventually, but that genocide was so intolerable that the aliens came from another dimension, to which they could never return, to stop Man, which resulted in the war of extermination.  Finally, the light speeds become so great that the two remaining living beings in the universe approach the Big Crunch. As the Man begins to dive into the singularity, with the alien hot on his heels, he says that maybe there will be room in the next universe for both races to live in peace, and the final line is "not with a whimper, but with a roar of triumph"! Wifi losing connection, weak signal, Intel 7260 adapter I yesterday upgraded from Windows to Ubuntu 15.04 but I have problem with my wireless adapter Intel Corporation Wireless 7260 (rev 73). It randomly loses internet connection. Restarting adapter always helps. (same it did with Windows, except that sometimes restarting adapter didn't help).   Additionally, I have weaker connection then I had with Windows.  Thank you! Hi is it possible to exclude traffic from a certain country in google analytics?  If yes, how do I do it? How can I install Windows after I've installed Ubuntu? Classifying raw LAS data as bare earth in ArcGIS for Desktop? I have a raw LAS data that has only one return (no second/ third.. return) and not classified yet. The only information I have is intensity and elevation. Is there any way that I can classify bare earth in ArcGIS? (vendor does not have more info than that). I tried classifying by selection from the profile view, but I know it is impossible to apply to all data. LEFT OUTER JOIN in LINQ how many semantically different boolean functions are there for n boolean variables? In short, this is an assignment question for a course I am taking - the exact wording is this:  "Given n Boolean variables, how many 'semantically' different Boolean functions can you construct?"  Now, I had a crack at this myself - and got pretty stuck. The question doesnt state how many boolean operators there are (and, or, xor, nand, nor, iff, implies, not) nor does it state whether brackets should be used, i.e. a ^ (b v c) is different from (a ^ b) v c.  So, my question for you is - is this question possible given the limited information available?   Is it going to be something like ${n^x}$ where x is the number of boolean operators.  Any direction here would be greatly appreciated. Symbols of derivatives How find this limit $\lim\limits_{x\to 0^{+}}\frac{\sin{(\tan{x})}-\tan{(\sin{x})}}{x^7}$ When to use an interface instead of an abstract class and vice versa? UPDATE statement to re-assign a column value according to a numeric formula Leverage browser caching for Facebook, Google+, or other 3rd party resource files I tested my website with Google PageSpeed Insights.  I did cache optimization in my .htaccess, but it didn't affect resource files that are from Facebook and Google+  Leverage browser caching      http://connect.facebook.net/fr_FR/all.js (20 minutes)      https://apis.google.com/js/plusone.js (30 minutes)      https://oauth.googleusercontent.com/…e:rpc:shindig.random:shindig.sha1.js?c=2 (60 minutes)   What can I do to solve this? Respawning the Ender Dragon Why does `md5sum` not give the same hash as the Internet does? I want to find the md5 hash of the string "a", but running echo "a" | md5sum gives me another hash than what I get if I search the internet (for example using  or ).  Running echo "a" | md5sum gives me "60b725f10c9c85c70d97880dfe8191b3", but it should be "0cc175b9c0f1b6a831c399e269772661".  If I make a reverse hash lookup for "60b725f10c9c85c70d97880dfe8191b3", I do however get "a". I want to apply a macro to all the cells of a column with the cell content as the input to the macro. Is this possible?  For example if the cell contents have the word TEST, the macro should take TEST as its argument. How do I URL encode a string Determining direction of longest segment of polygon using ArcGIS Desktop? I have a shapefile with 9000 agriculture fields and what I like to do is to determine for each field (=each polygon) the direction of the longest segment.  For instance, the selected polygon in the screenshot has its longest segment from west to east so I would like to mark give the polygon the attribute W (for west) or either E (for east).   Is there any solution to solve my problem in a fast but generalized way?   I do not need very detailed results. It's just for the interpretation of a question.   I'm seeking an answer using ArcGIS 10.2 for Desktop.  To be more precise on what I'm looking for I try to explain more practical: I'm doing winderosion analysis for a specific area with 9000 fields. As result I have a potential susceptibility for each field. I like to check, if the fields with high susceptibility are correlated to a certain orientation of the fields, e.g. check if most of the fields with high risk are oriented in the main winddirection or not.   The search for the longest segment is not really important for me. It was just an idea I had to determine the direction. So forget about the longest segment if there is a way to determine the direction with another method ;)