Let us assume $A,B\subseteq \Bbb R^+ $ are disjoint sets and $A\cup B=\Bbb R^+ $. Furthermore, $ \forall_{x,y\in A} \ x+y \in B \ \text{and}  \ \ \forall_ {x,y \in B} \ x+y \in A.$  Is it possible to provide an example for such $A$ and $B$?  If not, I would like to know why, and also prove the existence of such $A$, $B$ using Zorn's lemma. However, I couldn't figure out how to do it.  Please do not provide full proof as I am only looking for hints.
Let $\Bbb R^+$ be the set of positive real numbers. Use Zorn's Lemma to show that $\Bbb R^+$ is the union of two disjoint, non-empty subsets, each closed under addition.
How to calculate hash value of a file in Java? Please add TeX rendering in titles on the iOS app How to install packages to external drive (and run them)? I am running Xenial 16.04 and looking to install packages to an external drive - my computer has a 32gb SSD so it would be very useful to me to be able to use an external drive.  In a quick search I found a framework called  which seems to be designed to package all dependencies for a program (chromium, GIMP, some python packages, etc), but haven't had time to look into it.   seems to be looking for the same solution I am.   Thanks, Chris I installed Opera 12.16 from a .deb for reasons. Just assume that I need this specific browser of this specific version and that there’s no alternative.  However, that deb depends on packages (such as the gstreamer0.10 series) which are not in my distribution anymore (Debian testing). This makes apt fail on every operation except apt remove opera with dependency errors:  # apt install cli-common Reading package lists... Done Building dependency tree        Reading state information... Done You might want to run 'apt --fix-broken install' to correct these. The following packages have unmet dependencies:  opera : Depends: gstreamer0.10-plugins-good but it is not installable          Recommends: flashplugin-nonfree but it is not going to be installed E: Unmet dependencies. Try 'apt --fix-broken install' with no packages (or specify a solution).   apt --fix-broken install will just propose to remove opera:  # apt --fix-broken install Reading package lists... Done Building dependency tree        Reading state information... Done Correcting dependencies... Done The following packages will be REMOVED:   opera 0 upgraded, 0 newly installed, 1 to remove and 92 not upgraded. 1 not fully installed or removed. After this operation, 46.6 MB disk space will be freed. Do you want to continue? [Y/n]   Currently, my workaround is to install Opera when I need it, and remove it as soon as anything else needs to be done with apt. This is annoying.  Any suggestions? Ideally, I’d like to make apt ignore the dependencies of opera forever, since it works well-enough for my purposes. Minimum Length Regular Expression How to change boot priority? I have a laptop with Ubuntu installed as the only OS. I also have a USB stick with a Windows 7 installation on it. I want to install Windows from the USB stick on my laptop but when I try to restart the laptop to get access to BIOS, I can't seem to find the command for it, its like it skips everything and just says Ubuntu and then takes me to the login screen.  How do I change the boot priority to USB in Ubuntu? Question:     Let $f$ be differentiable on $[0,+\infty)$, such as$$\lim_{x\to+\infty}\left([f'(x)]^2+f^3(x)\right)=0$$show that $$\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$$   I think this problem is similar to  and (the  background)  For my problem I guess that if  $$\lim_{x\to+\infty}\left([f'(x)]^{2m}+f^{2n+1}(x)\right)=0,m,n\in N^{+},n,m&gt;1$$ then we have $\lim_{x\to\infty}f(x)=\lim_{x\to\infty}f'(x)=0$. Arduino Negative Voltage Reading with Positive Ground I am trying to measure voltage &amp; current off a solar charge controller from panel, battery &amp; load simultaneously with my arduino. I have managed to achieve this by tying all the negatives together and using it as GND, however, when I do that the charge controller doesn't function properly and when I switch to off-mode the loads still run. After checking the instruction manual further, I found out that the positives are shared common between Pan, Bat, Load. This was confirmed with a multimeter.   So now what I am thinking of doing is using the positive from the battery as the GND reference and then measuring all other voltages (through voltage dividers with respect to the GND). The voltages should then range between -17V to 0V.   However, I have read that the analog input pins are not able to read negative voltages, and with respect to the positive GND, the Analog inputs will all be negative.   EDIT: I was thinking of using an amplifier to invert positive ground to a negative ground, meaning I could measure all the positive sides of the voltages. But after getting answers from you guys it seems I may not have to use an amplifier and a voltage divider may suffice. I have currently included a 2 resistor voltage divider between each analog input, +5V source, Vmeasure which you can see in the diagram below.  According to my calculations, if this works, @-17V I should see 0.6V @ analog input. @0V I should see 4V at analog input.   I'd appreciate feedback on this circuit, and below in the second image I've also included a piece of another method suggested by @jonk (thanks), and would be interested in knowing what the benefits are of method B) over method A)  Thanks!  Ps. the ameters are there in place of ACS712 current sensors which also go into analog input pins.   Method A)        &ndash; Schematic created using   Method B (EDIT: This method has worked, I am still yet to test method A) thanks @Jonk)) Why the first moment is standardized before computing higher moments, but higher moments are not? Is ∅ equivalent to {∅}? Execution of AfterInsert caused by: System.StringException: Invalid id: () - where have I gone wrong? We've had a community deployment and my brain is fried and I just whipped up a quick class that is throwing the below error. What have I done wrong here?     Error: CANNOT_INSERT_UPDATE_ACTIVATE_ENTITY, MSI_UserTrigger:   execution of AfterInsert caused by: System.StringException: Invalid   id: () : [] () : []: ()   What I am trying to accomplish is:   When a new user is created and is active, if they are linked to a ContactId and part of a specific profile (by name), add their ContactId to a Set&lt;ID> Then loop through contacts where the Id is in the set, and update the Community_User__c field to TRUE      Trigger:   trigger MSI_UserTrigger on User (after insert) {       if(Trigger.isAfter){          if(Trigger.isInsert){              MSIUserTriggerHandler.executeAfterInsert(Trigger.new);          }      } }      Class:   public with sharing class MSIUserTriggerHandler {     private static final String COMMUNITY_PROFILE = 'Community Knowledge User';     private MSIUserTriggerHandler() {}      public static void executeAfterInsert(List&lt;User&gt; newList){         setCreatedCommunityUsers(newList);     }      private static void setCreatedCommunityUsers(List&lt;User&gt; newList){         List&lt;User&gt; usersWithContacts = new List&lt;User&gt;();         List&lt;Contact&gt; contactsToUpdate = new List&lt;Contact&gt;();         Set&lt;ID&gt; contactIds = new Set&lt;ID&gt;();          for(User u: newList){             if(u.ContactId == null &amp;&amp; u.ContactId == ''){                 continue;             }             if((u.ContactId != null &amp;&amp; u.ContactId != '') &amp;&amp; u.Profile.Name == COMMUNITY_PROFILE &amp;&amp; u.isActive == true){                 contactIds.add(u.ContactId);             }         }          for(Contact c : [Select Id,Name,Community_User_Created__c from Contact where Id in :contactIds]){             if(!c.Community_User_Created__c){                 c.Community_User_Created__c = true;                 contactsToUpdate.add(c);             }         }         update contactsToUpdate;     } }   Am I not adding the ID correctly to the Set&lt;ID>, or am I querying wrong in my for(Contacts) loop? Time-ordered operator in Srednicki , "Note that the operators are in time order...we can insert $T$ without changing anything". This I agree with. But then on the next paragraph he states "The time order operator $T$, moves all...where they annihilate...". How correct is this paragraph? Is equation (5.14) always valid? Is equation (5.14) the reason why the time-ordering symbol $T$ appears all over the place in QFT?  The equations I'm referring to are (braket package available?):   $$\langle f|i\rangle = \langle0|~a_{1'}(+\infty)a_{2'}(+\infty)a_{1}^\dagger(-\infty)a_{2}^\dagger(-\infty)~|0\rangle \tag{5.13} $$  and since time goes from right to left inside the braket we put the $T$-symbol  $$\langle f|i\rangle = \langle 0|T ~a_{1'}(+\infty)a_{2'}(+\infty)a_{1}^\dagger(-\infty)a_{2}^\dagger(-\infty)~|0\rangle \tag{5.14} $$   without changing anything.   After this, the $T$-symbol appears everywhere! $n$ people attend the same meeting, what is the chance that two people share the same birthday? Given the first $b$ birthdays, the probability the next person doesn't share a birthday with any that went before is $(365-b)/365$. The probability that none share the same birthday is the following: $\Pi_{0}^{n-1}\frac{365-b}{365}$. How many people would have to attend a meeting so that there is at least a $50$% chance that two people share a birthday?   So I set $\Pi_{0}^{n-1}\frac{365-b}{365}=.5$ and from there I manipulated some algebra to get $\frac{364!}{(364-n)!365^{n}}=.5\iff (364-n)!365^{n}=364!/.5=.....$   There has to be an easier way of simplifying this. Importance of series resistor at output of Rectifier WMS and WFS queries needs SRS and BBOX parameters.  http://example.com/geoserver/wfs?   service=WFS&amp;   version=2.0.0&amp;   request=GetFeature&amp;   typeNames=namespace:featuretype&amp;   srsName=CRS&amp;   bbox=a1,b1,a2,b2&amp;   For example I have GeoServer to publish WFS. My data SRS is EPSG:4326 and my map SRS is EPSG:3857. So I send a WFS query.  http://example.com/geoserver/wfs?   service=WFS&amp;   version=2.0.0&amp;   request=GetFeature&amp;   typeNames=namespace:featuretype&amp;   srsName=EPSG:3857&amp;   bbox=978393.9621,2680799.4560,-430493.3433,3326539.4710&amp;   Does GeoServer or ArcGIS server give an error? Because my data has EPSG:4326 . Or does GeoServer convert these coordinates internally and find data and send me as EPSG:3857 I have an alpha version of Ubuntu.  When the stable version is released do I have to upgrade or is it already stable from the installed updates? Show that $\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$ Show that   $$\int_{0}^{\pi} xf(\sin(x))\text{d}x = \frac{\pi}{2}\int_{0}^{\pi}f(\sin(x))\text{d}x$$  this is so confusing. i have no idea how to even start. im thinking integration by parts but that doesnt seem right.  hope you guys can help!! Mixed DML operation error I am facing Mixed DML error in the following case:  static testmethod void method1() {     User u1 = DataFactory.createUser('testUser','test@user.com','permissionSetAPIname');        Obj1 rec1 = DataFactory.createrec1('Record1');     insert rec1;      Obj2 rec2 = DataFactory.createrec2('Rec2',rec1,system.today()-10, system.today()+10 );     insert rec2;       Obj3 rec3 = DataFactory.createrec3(rec2,'Tester',u1,'US');     insert Role;      test.starttest();         //testing code      test.stoptest();  }   However the following is working fine  static testmethod void method1() {      User u1 = new User(Alias = 'testRole',                   Email             = 'role@test.com',                   EmailEncodingKey  = 'UTF-8',                   LastName          = 'user',                   FirstName         = 'test',                   LanguageLocaleKey = 'en_US',                   LocaleSidKey      = 'en_US',                   ProfileId         = [select Id from Profile where Name = 'Basic User' limit 1].Id,                   TimeZoneSidKey    = 'America/Chicago',                   Username          = 'stakeholder@test' + Math.round(Math.random() * 10000) + '.com',                   UserRoleId        = null,                   Country           = 'Great Britain');     insert u1;         Obj1 rec1 = DataFactory.createrec1('Record1');     insert rec1;      Obj2 rec2 = DataFactory.createrec2('Rec2',rec1,system.today()-10, system.today()+10 );     insert rec2;       Obj3 rec3 = DataFactory.createrec3(rec2,'Tester',u1,'US');     insert Role;      test.starttest();         //testing code      test.stoptest();  }   Please explain the difference. As my understanding is that transaction remains the same in both the cases. Is Lambert Academic Publishing a reputable company? I was contacted by the Acquisition editor of Lambert Academic Publishing (LAP) to publish my bachelor's thesis as a printed book. Is LAP any good? I have no idea about it; I have just heard that it's a freelance publishing house. How to measure the component value on a PCB?