Let $H$ be a subgroup of $G$ and let $a$ and $b$ belong to $G$. Then :  $aH=bH$ or $aH \bigcap bH =\emptyset$ ...... (1)  Gallian gives the following proof which i have a little trouble understanding :  By a previous result, we have : $aH = bH $ if and only if $a \in bH$ ....(2)  It says $(1) $ follows directly from $(2)$ for if there is an element c in $aH \bigcap bH$, then $cH = aH $ and $cH = bH$ .  I don't understand the above statement, so i have made the following attempt :   Attempt: If $aH \neq bH$ , then $a \notin bH$ by $(2)$  So , if $a \notin bH$, what can i say about $aH \bigcap bH$?   Thank you for the help
This is a question from the free Harvard online abstract algebra .  I'm posting my solutions here to get some feedback on them.  For a fuller explanation, see  post.  This problem is from assignment $5$.     Prove directly that distinct cosets do not overlap.   Let $H$ be a subgroup of $G$ and let $a$, $b$, and $c$ be elements of $G$ such that $b\not\in aH$ and $c$ is in both $aH$ and $bH$.  Then there are elements $h$ and $h^\prime$ in $H$ such that $c=ah$ and $c=bh^\prime$.  So $ah=bh^\prime$ and $b=ahh^{\prime -1}$.  But $hh^{\prime -1}\in H$ so $b\in aH$.  This contradicts our original assumption. Therefore, there can be no element in more than one distinct coset.  Again, I welcome any critique of my reasoning and/or my style as well as alternative solutions to the problem.  Thanks.
So one of this question's  answers got me curious. What is the maximum damage a 20th level character can do with a 9th level Magic Missile by the rules to a single target.  So my best I've been able to find is Aasimar or Goblin WLK 1/WIZ 19   9th Level slot (1D4+1)(11) [keep in mind you only roll once for damage not 11 times]  per  Empowered Evocation (+5)(11) (assuming 20 INT) Hexblade Curse (+6)(11) (Proficiency Bonus +6)    Radiant Soul/Fury of the Small +20 (Level 20 Aasimar/Goblin)   For a total of (D4+12)(11)+20 for (min-max)avg (163-196)179.5  Am I looking at this correctly? Are there other abilities out there that can increase this?  Feel free to include non legendary magic items. NO TOMES PLEASE  Only use things available on D&amp;D Beyond please.  Acceptable books are :   Basic Rules DMG, MM, PHB Sword Coast Adventure's Guide The Tortle Package Volo's Guide to Monsters Xanathar's Guide to Everything and any officially released Adventure Book 'Just' and 'only' carry a similar meaning, and while my feel for language usually helps me decide which one to use, there are times when I'm at a loss.  From my understanding, 'just' is used as a softener, to add flexibility and politeness. 'Only', on the other hand, is a relatively sharp exclusion. 'Just' also has a wider range of meanings, for instance to denote time.  While this sounds simple enough, it can get messy in practical usage. The sentence that made me stumble was "Not only airports are part of the target customer group, but also other large infrastructures".  Would the meaning of this sentence change when using 'just' instead? The way I see it, there is no difference in this case, except 'just' would make it sound a tad more colloquial. Evaluating $ \lim_{x \rightarrow 0}\left (\frac{\sin(x)}{x}\right)^\frac{1}{x^2}$ This is a canonical question and answer developed by the community to help address common questions. If you've been directed here, or your question has been closed as a duplicate, please look through the resources here and use them to shape more specific questions. To browse all canonical questions and answers, including more unit test resources, navigate to the  tag.  How do I unit test code that includes a SOAP or REST-based callout, or indirectly calls code that makes a callout? I'm getting an error saying that callouts are not supported. Why is $a^n - b^n$ divisible by $a-b$? Novel about interstellar war: Both sides win, both sides think they lost Is it okay to ask a professor from another university for class lecture notes of a course that he offered on that university? I find two very interesting courses that were offered in the Spring 2019 semester at ETH Zürich. Unfortunately, their class lecture notes are not shared publically. I wonder if it is appropriate to email the professors and ask them for class notes. Terminating Python scripts in Processing framework I'm writing a Python script to use it in Processing framework. I need to cancel script execution if certain conditions are met (to save user's time). I tried to use sys.exit(arguments) command. The issue is that not only the script, but QGIS itself shuts down too. I also tried quit() but result is the same. How should I terminate script in Processing framework? The following code is for reproduction purpose only. I don't need ad hoc solution for this particular case because I already have it. I need to know how to deal with this issue in general. ##Raster processing=group ##raster_1=raster ##raster_2=raster   from osgeo import gdal import sys from numpy import * import ntpath import re import platform from PyQt4 import QtGui   raster_1 = gdal.Open(raster_1) raster_2 = gdal.Open(raster_2)  rasters_list = [raster_1, raster_2]  # create a message for the case when CRSs of rasters do not match class WrongCRS(QtGui.QMessageBox):    def __init__(self):     super(WrongCRS, self).__init__()      self.initUI()    def initUI(self):      self.warning(self, 'Oops!',             &quot;Rasters must have the same CRS!\n\nExecution cancelled!&quot;, QtGui.QMessageBox.Ok)  # check CRS proj = None for raster in rasters_list:   new_proj = raster.GetProjection()   if proj is None:     proj = new_proj   else:     if proj != new_proj:       WrongCRS()       sys.exit('CRSs do not match!')     else:       continue "The antipodal map of $S^2$ is the isometry $\bar m$ sending each point $P$ to its opposite point $-P$" (from textbook)  Show that antipodal points remain antipodal under any isometry of $S^2$. In other words, for each isometry f of $S^2$, if $P$ and $P'$ are antipodal, then so are $f(P)$ and $f(P')$  I am unsure how I can define the $P$ and $P'$ so that they are opposites of each other in the antipodal map, and am I supposed to apply the traditional isometries (translation, reflection and rotation) to the points? I don't understand this problem. How to subtract X days from a date using Java calendar? Failed to mount Windows share: Invalid argument A fair coin is tossed until a head comes up for the first time. The probability of this happening on an odd number toss is?  How do i approach this problem ? I can't understand how the "cryptographic functions" are to be selected RSA + OAEP which are used in OAEP. How to choose these "cryptographic functions"(G and H)? How delete an application that is bundled with macOS? Backstory:   I am running macOS Sierra and use TextEdit a lot. The problem is that I hate the Versions save schema that causes the universal Save As shortcut of CmdShiftS to instead create a copy of the document. I also hate how Auto-Save makes permanent changes to documents I don't want to modify when I'm tentatively modifying a work-in-progress. Since the introduction of these nuisance features in Lion I have always backed up and then replaced TextEdit.app with the legacy 1.6 version from Snow Leopard.  The problem:  Today when I tried to delete TextEdit.app from my Applications folder using this Terminal command  sudo rm -rf /Applications/TextEdit.app   I got this error  rm: TextEdit.app: Operation not permitted   I was surprised to find that I can't kill it, even with sudo.  How can I actually get rid of an application that is bundled with macOS?  What I've tried so far:  I followed the instructions in , but it didn't work. I also read , which almost accomplishes the same solution by means of disabling Versions and Auto-Save, but then I don't get the benefit of legacy TextEdit's temporary backup saving feature. See the part in the answer stating "This effectively disables the old-style, pre-Lion autosaving completely".   So I think deleting the newer TextEdit v1.12 and replacing it with the legacy v1.6 is still the best option, however I can't get my system to delete it. Any help appreciated. According to this , if at least one of the nouns involved is plural then it should take the plural form of the verb. Otherwise, it should take the singular form of the verb. But in the last part it says not all grammars agree to those rules thus proximity rule applies. However, it doesn't explain when to apply the proximity rule and when not to use the former rules. What are the complexity guarantees of the standard containers? Why does the light side of the moon appear not to line up correctly with the evening sun? Function with limits only at irrational points This is the starred example 5.1.8 in Krantz' Real Analysis and Foundations.     Give an example of a function $f: \mathbb{R} \rightarrow \mathbb{R}$ so that $\lim_{x\rightarrow c} f(x)$ exists when $c$ is irrational but does not exist when $c$ is rational.   I'm aware of the Dirichlet and Thomae functions but this seems slightly different. I'm not even sure a solution exists. Thanks for any help! I've removed the poll style answers in favour of open discussion.  I realise now this isn't the best way to have a proper discussion about this issue - sorry to anyone who commented/voted.    This is mostly a followup on this question:    As I've answered, the reason this is happening is that the root database (in the production Stack Exchange database server) is still called StackExchange.Programmers - a database name that propagates to SEDE and through it to the data dump.  The question here is what to do about it - the current situation is somewhat broken for users of the data dump.  These are the options we have:   Do nothing Update the root database name (requires downtime - so very undesirable) Update the update scripts so the SEDE database name is the new name, change will flow to the data dump export (breaks any queries using the old name) Update the export to the data dump so only it gets the new name Something else we haven't thought about   I have added all of these options as answers below - please vote on the one you most prefer, we will try to accommodate the top voted answer. One image per overlay?