If the empty set is a subset of every set, why write … ∪ ∅? The 2019 Stack Overflow Developer Survey Results Are In Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Is the void set (∅) a proper subset of every set?Direct proof of empty set being subset of every setIf the empty set is a subset of every set, why isn't $emptyset,a=a$?Why $ emptyset $ is not a subset of $ emptyset $A set $X$ is called 'complete' if every element of $X$ is subset of $X$.Why "to every set and to every statement p(x), there exists p(x)$?What subset am I missing from a set containing the empty set and a set with the empty set?Does an element of a set, that can't be in a list, make that set uncountable?Question about the empty setUnderstanding empty set, set with empty set and set with set of empty set.

What can I do if neighbor is blocking my solar panels intentionally?

Did God make two great lights or did He make the great light two?

How did passengers keep warm on sail ships?

Did the UK government pay "millions and millions of dollars" to try to snag Julian Assange?

Is this wall load bearing? Blueprints and photos attached

Make it rain characters

Typeface like Times New Roman but with "tied" percent sign

What was the last x86 CPU that did not have the x87 floating-point unit built in?

Why can't wing-mounted spoilers be used to steepen approaches?

How is simplicity better than precision and clarity in prose?

I could not break this equation. Please help me

How does this infinite series simplify to an integral?

Can a 1st-level character have an ability score above 18?

How to copy the contents of all files with a certain name into a new file?

Would an alien lifeform be able to achieve space travel if lacking in vision?

Are spiders unable to hurt humans, especially very small spiders?

How do you keep chess fun when your opponent constantly beats you?

Do warforged have souls?

How are presidential pardons supposed to be used?

Road tyres vs "Street" tyres for charity ride on MTB Tandem

How to politely respond to generic emails requesting a PhD/job in my lab? Without wasting too much time

Sort list of array linked objects by keys and values

Why did all the guest students take carriages to the Yule Ball?

Does the AirPods case need to be around while listening via an iOS Device?



If the empty set is a subset of every set, why write … ∪ ∅?



The 2019 Stack Overflow Developer Survey Results Are In
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Is the void set (∅) a proper subset of every set?Direct proof of empty set being subset of every setIf the empty set is a subset of every set, why isn't $emptyset,a=a$?Why $ emptyset $ is not a subset of $ emptyset $A set $X$ is called 'complete' if every element of $X$ is subset of $X$.Why "to every set and to every statement p(x), there exists $xin A ?What subset am I missing from a set containing the empty set and a set with the empty set?Does an element of a set, that can't be in a list, make that set uncountable?Question about the empty setUnderstanding empty set, set with empty set and set with set of empty set.










6












$begingroup$


I met the notation $ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $



I know $S$ is a family of subsets ,a set of intervals, and from set theory $emptyset$ is a subsets of every set then why in the notation :$ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $ appear $colorredcupemptyset$?










share|cite|improve this question











$endgroup$
















    6












    $begingroup$


    I met the notation $ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $



    I know $S$ is a family of subsets ,a set of intervals, and from set theory $emptyset$ is a subsets of every set then why in the notation :$ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $ appear $colorredcupemptyset$?










    share|cite|improve this question











    $endgroup$














      6












      6








      6





      $begingroup$


      I met the notation $ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $



      I know $S$ is a family of subsets ,a set of intervals, and from set theory $emptyset$ is a subsets of every set then why in the notation :$ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $ appear $colorredcupemptyset$?










      share|cite|improve this question











      $endgroup$




      I met the notation $ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $



      I know $S$ is a family of subsets ,a set of intervals, and from set theory $emptyset$ is a subsets of every set then why in the notation :$ S=(a,b] ; a,bin mathbb R,a<bcupemptyset $ appear $colorredcupemptyset$?







      measure-theory elementary-set-theory






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 11 mins ago









      LarsH

      555624




      555624










      asked 7 hours ago









      Ica SanduIca Sandu

      1329




      1329




















          4 Answers
          4






          active

          oldest

          votes


















          21












          $begingroup$

          It is because the emptyset $emptyset$ is a subset of every set, but not an element of every set.
          It is $emptysetin S$ and you might want that to show, that the elements of $S$ define a topology.



          Or to be more clear it is $1neq1,emptyset$. The set on the left has one element, the set on the right has two elements, with $emptysetin1,emptyset$






          share|cite|improve this answer











          $endgroup$




















            5












            $begingroup$

            Because the empty set $(emptyset)$ is one thing, but what you have there is $emptyset$, which is a different thing: it's a set with a single element (which happens to be the empty set).






            share|cite|improve this answer









            $endgroup$




















              5












              $begingroup$

              The answer is: the given definition uses $cupemptyset $, not $cupemptyset $, so it adds the empty set as an element, not a subset of $S $.






              share|cite|improve this answer











              $endgroup$




















                3












                $begingroup$

                It looks like $S$ is denoting subintervals of the real line that are open on the left and closed on the right with the convention that $emptyset$ is such a subinterval. In which case there is nothing to show, it's just a convention that $emptyset$ is a subinterval. The reason for using $emptyset$ is show you can write out the collection of all such subintervals in a nice form.



                As for the empty set is a subset of every set, well that's a vacuous truth. For all $ainemptyset$ if $X$ is a set it follows that $ain X.$ This is true, because there are no $ainemptyset.$






                share|cite|improve this answer









                $endgroup$













                  Your Answer








                  StackExchange.ready(function()
                  var channelOptions =
                  tags: "".split(" "),
                  id: "69"
                  ;
                  initTagRenderer("".split(" "), "".split(" "), channelOptions);

                  StackExchange.using("externalEditor", function()
                  // Have to fire editor after snippets, if snippets enabled
                  if (StackExchange.settings.snippets.snippetsEnabled)
                  StackExchange.using("snippets", function()
                  createEditor();
                  );

                  else
                  createEditor();

                  );

                  function createEditor()
                  StackExchange.prepareEditor(
                  heartbeatType: 'answer',
                  autoActivateHeartbeat: false,
                  convertImagesToLinks: true,
                  noModals: true,
                  showLowRepImageUploadWarning: true,
                  reputationToPostImages: 10,
                  bindNavPrevention: true,
                  postfix: "",
                  imageUploader:
                  brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
                  contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
                  allowUrls: true
                  ,
                  noCode: true, onDemand: true,
                  discardSelector: ".discard-answer"
                  ,immediatelyShowMarkdownHelp:true
                  );



                  );













                  draft saved

                  draft discarded


















                  StackExchange.ready(
                  function ()
                  StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3186480%2fif-the-empty-set-is-a-subset-of-every-set-why-write-%25e2%2588%25aa-%25e2%2588%2585%23new-answer', 'question_page');

                  );

                  Post as a guest















                  Required, but never shown

























                  4 Answers
                  4






                  active

                  oldest

                  votes








                  4 Answers
                  4






                  active

                  oldest

                  votes









                  active

                  oldest

                  votes






                  active

                  oldest

                  votes









                  21












                  $begingroup$

                  It is because the emptyset $emptyset$ is a subset of every set, but not an element of every set.
                  It is $emptysetin S$ and you might want that to show, that the elements of $S$ define a topology.



                  Or to be more clear it is $1neq1,emptyset$. The set on the left has one element, the set on the right has two elements, with $emptysetin1,emptyset$






                  share|cite|improve this answer











                  $endgroup$

















                    21












                    $begingroup$

                    It is because the emptyset $emptyset$ is a subset of every set, but not an element of every set.
                    It is $emptysetin S$ and you might want that to show, that the elements of $S$ define a topology.



                    Or to be more clear it is $1neq1,emptyset$. The set on the left has one element, the set on the right has two elements, with $emptysetin1,emptyset$






                    share|cite|improve this answer











                    $endgroup$















                      21












                      21








                      21





                      $begingroup$

                      It is because the emptyset $emptyset$ is a subset of every set, but not an element of every set.
                      It is $emptysetin S$ and you might want that to show, that the elements of $S$ define a topology.



                      Or to be more clear it is $1neq1,emptyset$. The set on the left has one element, the set on the right has two elements, with $emptysetin1,emptyset$






                      share|cite|improve this answer











                      $endgroup$



                      It is because the emptyset $emptyset$ is a subset of every set, but not an element of every set.
                      It is $emptysetin S$ and you might want that to show, that the elements of $S$ define a topology.



                      Or to be more clear it is $1neq1,emptyset$. The set on the left has one element, the set on the right has two elements, with $emptysetin1,emptyset$







                      share|cite|improve this answer














                      share|cite|improve this answer



                      share|cite|improve this answer








                      edited 5 hours ago

























                      answered 7 hours ago









                      CornmanCornman

                      3,68321231




                      3,68321231





















                          5












                          $begingroup$

                          Because the empty set $(emptyset)$ is one thing, but what you have there is $emptyset$, which is a different thing: it's a set with a single element (which happens to be the empty set).






                          share|cite|improve this answer









                          $endgroup$

















                            5












                            $begingroup$

                            Because the empty set $(emptyset)$ is one thing, but what you have there is $emptyset$, which is a different thing: it's a set with a single element (which happens to be the empty set).






                            share|cite|improve this answer









                            $endgroup$















                              5












                              5








                              5





                              $begingroup$

                              Because the empty set $(emptyset)$ is one thing, but what you have there is $emptyset$, which is a different thing: it's a set with a single element (which happens to be the empty set).






                              share|cite|improve this answer









                              $endgroup$



                              Because the empty set $(emptyset)$ is one thing, but what you have there is $emptyset$, which is a different thing: it's a set with a single element (which happens to be the empty set).







                              share|cite|improve this answer












                              share|cite|improve this answer



                              share|cite|improve this answer










                              answered 7 hours ago









                              José Carlos SantosJosé Carlos Santos

                              174k23134243




                              174k23134243





















                                  5












                                  $begingroup$

                                  The answer is: the given definition uses $cupemptyset $, not $cupemptyset $, so it adds the empty set as an element, not a subset of $S $.






                                  share|cite|improve this answer











                                  $endgroup$

















                                    5












                                    $begingroup$

                                    The answer is: the given definition uses $cupemptyset $, not $cupemptyset $, so it adds the empty set as an element, not a subset of $S $.






                                    share|cite|improve this answer











                                    $endgroup$















                                      5












                                      5








                                      5





                                      $begingroup$

                                      The answer is: the given definition uses $cupemptyset $, not $cupemptyset $, so it adds the empty set as an element, not a subset of $S $.






                                      share|cite|improve this answer











                                      $endgroup$



                                      The answer is: the given definition uses $cupemptyset $, not $cupemptyset $, so it adds the empty set as an element, not a subset of $S $.







                                      share|cite|improve this answer














                                      share|cite|improve this answer



                                      share|cite|improve this answer








                                      edited 4 hours ago

























                                      answered 5 hours ago









                                      CiaPanCiaPan

                                      10.3k11248




                                      10.3k11248





















                                          3












                                          $begingroup$

                                          It looks like $S$ is denoting subintervals of the real line that are open on the left and closed on the right with the convention that $emptyset$ is such a subinterval. In which case there is nothing to show, it's just a convention that $emptyset$ is a subinterval. The reason for using $emptyset$ is show you can write out the collection of all such subintervals in a nice form.



                                          As for the empty set is a subset of every set, well that's a vacuous truth. For all $ainemptyset$ if $X$ is a set it follows that $ain X.$ This is true, because there are no $ainemptyset.$






                                          share|cite|improve this answer









                                          $endgroup$

















                                            3












                                            $begingroup$

                                            It looks like $S$ is denoting subintervals of the real line that are open on the left and closed on the right with the convention that $emptyset$ is such a subinterval. In which case there is nothing to show, it's just a convention that $emptyset$ is a subinterval. The reason for using $emptyset$ is show you can write out the collection of all such subintervals in a nice form.



                                            As for the empty set is a subset of every set, well that's a vacuous truth. For all $ainemptyset$ if $X$ is a set it follows that $ain X.$ This is true, because there are no $ainemptyset.$






                                            share|cite|improve this answer









                                            $endgroup$















                                              3












                                              3








                                              3





                                              $begingroup$

                                              It looks like $S$ is denoting subintervals of the real line that are open on the left and closed on the right with the convention that $emptyset$ is such a subinterval. In which case there is nothing to show, it's just a convention that $emptyset$ is a subinterval. The reason for using $emptyset$ is show you can write out the collection of all such subintervals in a nice form.



                                              As for the empty set is a subset of every set, well that's a vacuous truth. For all $ainemptyset$ if $X$ is a set it follows that $ain X.$ This is true, because there are no $ainemptyset.$






                                              share|cite|improve this answer









                                              $endgroup$



                                              It looks like $S$ is denoting subintervals of the real line that are open on the left and closed on the right with the convention that $emptyset$ is such a subinterval. In which case there is nothing to show, it's just a convention that $emptyset$ is a subinterval. The reason for using $emptyset$ is show you can write out the collection of all such subintervals in a nice form.



                                              As for the empty set is a subset of every set, well that's a vacuous truth. For all $ainemptyset$ if $X$ is a set it follows that $ain X.$ This is true, because there are no $ainemptyset.$







                                              share|cite|improve this answer












                                              share|cite|improve this answer



                                              share|cite|improve this answer










                                              answered 7 hours ago









                                              MelodyMelody

                                              1,21312




                                              1,21312



























                                                  draft saved

                                                  draft discarded
















































                                                  Thanks for contributing an answer to Mathematics Stack Exchange!


                                                  • Please be sure to answer the question. Provide details and share your research!

                                                  But avoid


                                                  • Asking for help, clarification, or responding to other answers.

                                                  • Making statements based on opinion; back them up with references or personal experience.

                                                  Use MathJax to format equations. MathJax reference.


                                                  To learn more, see our tips on writing great answers.




                                                  draft saved


                                                  draft discarded














                                                  StackExchange.ready(
                                                  function ()
                                                  StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3186480%2fif-the-empty-set-is-a-subset-of-every-set-why-write-%25e2%2588%25aa-%25e2%2588%2585%23new-answer', 'question_page');

                                                  );

                                                  Post as a guest















                                                  Required, but never shown





















































                                                  Required, but never shown














                                                  Required, but never shown












                                                  Required, but never shown







                                                  Required, but never shown

































                                                  Required, but never shown














                                                  Required, but never shown












                                                  Required, but never shown







                                                  Required, but never shown







                                                  Popular posts from this blog

                                                  Can not update quote_id field of “quote_item” table magento 2Magento 2.1 - We can't remove the item. (Shopping Cart doesnt allow us to remove items before becomes empty)Add value for custom quote item attribute using REST apiREST API endpoint v1/carts/cartId/items always returns error messageCorrect way to save entries to databaseHow to remove all associated quote objects of a customer completelyMagento 2 - Save value from custom input field to quote_itemGet quote_item data using quote id and product id filter in Magento 2How to set additional data to quote_item table from controller in Magento 2?What is the purpose of additional_data column in quote_item table in magento2Set Custom Price to Quote item magento2 from controller

                                                  How to solve knockout JS error in Magento 2 Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?(Magento2) knockout.js:3012 Uncaught ReferenceError: Unable to process bindingUnable to process binding Knockout.js magento 2Cannot read property `scopeLabel` of undefined on Product Detail PageCan't get Customer Data on frontend in Magento 2Magento2 Order Summary - unable to process bindingKO templates are not loading in Magento 2.1 applicationgetting knockout js error magento 2Product grid not load -— Unable to process binding Knockout.js magento 2Product form not loaded in magento2Uncaught ReferenceError: Unable to process binding “if: function()return (isShowLegend()) ” magento 2

                                                  Nissan Patrol Зміст Перше покоління — 4W60 (1951-1960) | Друге покоління — 60 series (1960-1980) | Третє покоління (1980–2002) | Четверте покоління — Y60 (1987–1998) | П'яте покоління — Y61 (1997–2013) | Шосте покоління — Y62 (2010- ) | Посилання | Зноски | Навігаційне менюОфіційний український сайтТест-драйв Nissan Patrol 2010 7-го поколінняNissan PatrolКак мы тестировали Nissan Patrol 2016рвиправивши або дописавши її