Choice function without AC in special case I read the Jech, Set theory, and saw following proposition.  (☆) If S is a finite family of nonempty sets, existence of choice function of S can be proved without axiom of choice.  I tried to prove the proposition using axioms in ZF... But how can I pick the element of X in S explicitly despite I don't know information about X in S?  Edit: There seems to be confusions on what exactly I am asking.  I think ZFC Axioms instruct when we call a object set.(If object is not set, the object is called urelement or proper class) So I think we must show that nonemptyset has a element that is set. For example, if S is singleton, US is element of S and is set by axiom of union. That is, we can find a element of S that is set. Then using other axioms in ZF, we can construct a set we call choice function of S.  But, If S has two elements, how I can know the element of S is set??
Finite choice without AC Can anyone explain how we choose one sock from each of finitely many pairs without the axiom of choice?  I mean : To choose one sock from each of infinitely many pairs of socks requires the Axiom of Choice, but for shoes the Axiom is not needed.  The idea is that the two socks in a pair are identical in appearance, and so we must make an arbitrary choice if we wish to choose one of them. For shoes, we can use an explicit algorithm -- e.g., "always choose the left shoe." Why does Russell's statement mention infinitely many pairs? Well, if we only have finitely many pairs of socks, then AC is not needed -- we can choose one member of each pair using the definition of "nonempty," and we can repeat an operation finitely many times using the rules of formal logic (not discussed here).
What is the best file extension for an SEO friendly URL? I'm working for a website to convert the website URL to an SEO friendly URL. I plan to use this:  example.com/category-name/pageid-123-page-name  I looked at some similarly categorized, highly ranked websites. They have the same structure, except for one thing. In one case, the URL format was  example.com/category-name/pageid-123-page-name.html  Another was   example.com/category-name/pageid-123-page-name.php  Now I know the text in URLs help with SEO. Is it more helpful to have a file extension? If yes, which one is better?  Or if my current plan is okay, will it be better with a / at the end? Polynomial in $n$ variables $y_{2n}, y_{2n+1}$ and $y_{3n}$ all converge. What can we say about the sequence $ y_n$? My friend and I are currently debating the following question:      Let $y_n$ be a sequence in a metric space and assume that the subsequences    $y_{2n}$, $y_{2n + 1}$, and $y_{3n}$ all converge. What can we say about the sequence $y_n$?    To me it seems that all the subsequences will converge to a limit say $y$. Then any term in the original sequence will be in one of these subsequences, it follows that the entire sequence $y_n$ converges to $y$.  I let $\varepsilon &gt; 0$. Since $y_{2n}$ converges to $y$ there exists an index $2N$ such that $d(y_{2n},y) &lt; \varepsilon$ for $2n \geq 2N$, and since $(y_{2n+1})$ converges to $y$ there exists $2M + 1$ such that $d(y_{2n+1}, y) &lt; \varepsilon$ for $2n + 1 \geq 2M + 1$.  Let $m \geq \max\{2N,2M+1\}$. If  $m$ is even, since $m\geq 2N$ we have $d(y_m,y)&lt;\varepsilon$; if $m$ is odd, since $m\geq 2M+1$ we have $d(y_m,y)&lt;\varepsilon$. Thus $d(y_m,y)&lt;\varepsilon$ for any $m \geq \max\{2N,2M+1\}$, and we conclude that $(y_n)$ converges to $y$.   However, my friend does not think that this is completely correct because I didn't use the $y_{3n}$ in the proof. I think he is right, that I do in fact need it, but I am unsure of how to add it in. All in all, I think that since any term in the original sequence will be in one of these subsequences, it follows that the entire sequence $(y_n)$ converges to $y$.  $2n+1$ covers all the odd terms and $2n$ converse all the even terms, so is $3n$ even needed? I would like to use it since the proof has it in the question, but I am unsure of what to do. Could someone kindly help me construct this proof in a better manner,and is my line of thinking correct? Thank you! Handling large numbers in C++? I'm currently reading a little deeper into the Axiom of Choice, and I'm pleasantly surprised to find it makes the arithmetic of infinite cardinals seem easy. With AC follows the Absorption Law of Cardinal Arithmetic, which states that for $\kappa$ and $\lambda$ cardinal numbers, the larger infinite and the smaller nonzero, then $\kappa+\lambda=\kappa\cdot\lambda=\max(\kappa,\lambda)$.  I was playing around with the equation $k+\aleph_0=\mathfrak{c}$ for some cardinal $k$. From the above, it follows that $\mathfrak{c}=k+\aleph_0=\max(k,\aleph_0)$, which implies $k=\mathfrak{c}$.  I'm curious, can we still show $k=\mathfrak{c}$ without the Axiom of Choice? Is it maybe possible to bound $\mathfrak{c}-\aleph_0$ above and below by $\mathfrak{c}$? But then I'm not quite sure such algebraic manipulations even mean anything, or work like that here. Certainly normal arithmetic does not! Thanks. Optimal bcrypt work factor Install Java JDK and JRE on Debian 8 View/edit ID3 data for MP3 files What's a quick and easy way to view and edit ID3 tags (artist, album, etc.) using C#? past simple or present perfect Please have a look at this sentence:     We haven't gotten a lot of Finnish groups that we liked here at Bananas but I'm pretty happy with this four tracker from Helsinki, apparently recorded in the same place where the Ramones played in the 70's.     The first verb is in present perfect (up to now) and the second is in simple past tense. I don't understand the simple past for the second verb.  Why "like" is not in its present perfect form as the first verb is? Today is the release of Ubuntu 21.04 (Hirsute Hippo). And I can't update to this version. And also I selected ‘Notify me of a new Ubuntu version’ and changed it to ‘For any new version' on the Updates tab on Software &amp; Update, but it's still not working. In addition, in the terminal, I've written: sudo do-release-upgrade -c  And output was: Checking for a new Ubuntu release No new release found.  What should I do? I just started playing Overwatch (played a few games during open beta, but wasn't able to play much). I'm enjoying Mercy, but who (in a group situation) should be getting the damage buff? Additionally, when my team splits up, who should I follow to support? In Book 1, Artemis et al escaped from the time stop by drugging themselves to sleep.  In the same time stop, Julius Root "accidentally" plugged Briar Cudgeon with a tranquilizer dart.  Cudgeon fell asleep, but did not disappear from the time stop.  Why not? I recieve the below error when I am trying to call a Custom Lightning Component in a Visualforce page that is used in Communities.    Unrecognized Content-Security-Policy directive 'reflected-xss'.   logo180.png Failed to load resource: the server responded with a status of 404 (Not Found)   CommunityHomePage:1 Uncaught SyntaxError: Unexpected token w in JSON at position 0     at JSON.parse ()     at XMLHttpRequest.xhr.onreadystatechange (lightning/lightning.out.delegate.js?v=ZCuyRUYzK_kAf4cKCIeS2A:143)   My Component  &lt;aura:component controller="LeadController" implements="forceCommunity:availableForAllPageTypes"&gt;     &lt;aura:attribute name="theLead"          type="Lead"          default="{ 'sobjectType': 'Lead',                    'Rating__c': 2,                    'Company': '',                    'FirstName': '',                    'LastName': '',                    'Email': '' }"/&gt;     &lt;form&gt;          &lt;ui:inputText value="{!v.theLead.Company}" label="Company" class="input"/&gt;         &lt;ui:inputText value="{!v.theLead.FirstName}" label="First" class="input"/&gt;         &lt;ui:inputText value="{!v.theLead.LastName}" label="Last" class="input" /&gt;         &lt;ui:inputEmail value="{!v.theLead.Email}" label="Email" class="input" /&gt;         &lt;ui:button label="Save" press="{!c.save}"/&gt;     &lt;/form&gt; &lt;/aura:component&gt;   My Lightning App is   &lt;aura:application access="GLOBAL" extends="ltng:outApp" implements="forceCommunity:availableForAllPageTypes"&gt;     &lt;aura:dependency resource="c:CommunityChatter"/&gt; &lt;/aura:application&gt;   My VF Page is  &lt;apex:page standardController="Lead"&gt;     &lt;apex:includeScript value="{!$Site.BaseUrl}/lightning/lightning.out.js"/&gt;     &lt;script&gt;         $Lightning.use("c:MyLightApp", function() {             $Lightning.createComponent(                 "c:CommunityChatter",                 {},                 "theLead",                 function(cmp) {                     console.log("Component created!");                     console.log(cmp);                 });             });     &lt;/script&gt;      &lt;div id="theLead"&gt;&lt;/div&gt; &lt;/apex:page&gt; Where to get QGIS toolbar icons? Does anybody know where to download the QGIS toolbar icons? How do I get a straight quote instead of curly quotes? I’m not inside verb or listing or any other special environments, and I’m not using XeLaTeX or anything—just normal LaTeX. I don’t want to change all quotes, just get the occasional straight quote. For example, I’d like a way to get something like this in the TeX:  ... some examples of these glyphs are curly quotes (``''), straight quotes ('"), and angled quotes ($'$$"$) ...   to look like this in the output:  ... some examples of these glyphs are curly quotes (“”), straight quotes ('"), and angled quotes (′″) ...   As I’m using an academic-publisher-provided template that does its own required font setup, I’m extremely wary of changing fonts or font encodings. I think the template uses the times package, and the formatting instructions say all fonts must be Type 1. Getting a File's MD5 Checksum in Java I am looking to use Java to get the MD5 checksum of a file.  I was really surprised but I haven't been able to find anything that shows how to get the MD5 checksum of a file.  How is it done? How to make Arduino (Snap) access serial port /dev/ttyUSB0 (or /dev/ttyACM0)? I have Ubuntu 18.04 LTS installed.  I installed Arduino IDE v1.8.5 as a snap with  snap install arduino-mhall119   I have added my user to dial-out group and rebooted:  sudo usermod -a -G dialout $USER  $ groups  username adm dialout cdrom sudo dip plugdev lpadmin sambashare   I connected my Arduino Nano to the laptop, it is detected on port /dev/ttyUSB0 (ch341-uart).  $ ls /dev/ttyUSB0 -al crw-rw---- 1 root dialout 188, 0 авг 13 11:59 /dev/ttyUSB0   I tried to upload the simpliest Blink sketch and get the following errors in the terminal:  $ arduino-mhall119.arduino    avrdude: ser_open(): can't open device "/dev/ttyUSB0": Permission denied   Problem uploading to board.  See http://www.arduino.cc/en/Guide/Troubleshooting#upload for suggestions.**   Also I have installed arduino 1.8.5 with umake - it can access serial port in usual way.  Note: we already have  here, but the solution was to use , not fix for the Snap.  So my question is: How to make Arduino (Snap) access serial port /dev/ttyUSB0 (or /dev/ttyACM0)?  Update: connecting serial-port to the snap fails with the following message  $ snap connect arduino-mhall119:serial-port  error: snap "core" has no "serial-port" interface slots   Current interfaces are:  $ snap interfaces | grep arduino :home                            arduino-mhall119,gedit,organize-my-files,vlc :network                         arduino-mhall119,gedit,organize-my-files,pulsemixer,vlc :unity7                          arduino-mhall119,gedit,organize-my-files,vlc :x11                             arduino-mhall119,organize-my-files,pulsemixer,vlc -                                arduino-mhall119:serial-port   Changing permissions of serial port to 777 with sudo chmod 777 /dev/ttyUSB0 does not help.  The following do not work too (because of syntax errors):  $ sudo snap connect arduino-mhall119:arduino error: snap "arduino-mhall119" has no plug named "arduino"  $ snap connect arduino-mhall119:/dev/ttyUSB0 error: snap "arduino-mhall119" has no plug named "/dev/ttyUSB0"  $ snap connect arduino-mhall119 error: cannot resolve connection, plug snap name is empty   Information about YAML-manifest:  $ ls -l /snap/arduino-mhall119/current/meta/snap.yaml -rw-r--r-- 1 root root 353 июл 28 00:20 /snap/arduino-mhall119/current/meta/snap.yaml  $ cat /snap/arduino-mhall119/current/meta/snap.yaml |grep -A 12 slots  $ cat /snap/arduino-mhall119/current/meta/snap.yaml |grep -A 5 plug     plugs:     - x11     - unity7     - home     - network     - serial-port How did this question get a 'linked' question that isn't mentioned in comments or answers? Looking at  which has been closed as a duplicate (quite right), we see three 'linked' questions in the sidebar; however, only two of those questions are given as duplicate-close-targets.  which to my mind is the actual 'exact duplicate', isn't mentioned anywhere else except in the 'linked questions' sidebar itself! The  says:  The Linked sidebar... gathers up any links to this question provided by the community via comments, answers, or questions  So how did  get into the linked questions sidebar of ? (There's no explicit link in the other direction, in case you were wondering.) Editing a question with only a JSFiddle link does not work I can't edit a question that has a jsfiddle link because it has a lack of code. Can we get privileged users able to bypass restriction? The question had some really poor formatting and linking and I couldn't fix it. :  Html is not taking the css code Some of my html tags are not reacting by the css which i gaved them for example is offerbox not reaction to the html part. all html parts down from offerbox are not taking the styles that i gaved the enter code here http://jsfiddle.net/8Bw7M/  pic from the html and css  I feel that as a privileged user, I should be able to make the changes needed. This is related, but I'm not adding any new links. I did a fresh install of trusty 14.04, but a lot of icons are missing. I installed from DVD which was painfully slow and I hope to avoid repeating the process, especially since I started customizing stuff.  Here's my settings window:    I've tried:  sudo apt-get install --reinstall gnome-control-center sudo apt-get install --reinstall gnome-settings-daemon-schemas   None of which changed anything.  Any idea? I wanna edit preferences and none of them are in System Settings, neither in Unity Dash.