/Users/andrea/_magisterarbeit/korpus/clean/testkorpus/30/file2.html NN ----------------------------------------- : Empty JJ set NN From IN Wikipedia NP , , the DT free JJ encyclopedia NN . SENT Jump VV to TO . SENT navigation NN , , search NN In IN mathematics NNS and CC more JJR specifically RB set VVN theory NN , , the DT empty JJ set NN is VBZ the DT unique JJ set NN which WDT contains VVZ no DT elements NNS . SENT In IN axiomatic JJ set NN theory NN it PP is VBZ postulated VVN to TO exist VV by IN the DT axiom NN of IN empty JJ set NN and CC all DT finite JJ sets NNS are VBP constructed VVN from IN it PP . SENT The DT empty JJ set NN is VBZ also RB sometimes RB called VVN the DT null JJ set NN , , but CC because IN null JJ set NN means VVZ something NN else RB in IN measure NN theory NN , , that DT term NN is VBZ generally RB avoided VVN in IN current JJ work NN . SENT Various JJ possible JJ properties NNS of IN sets NNS are VBP trivially RB true JJ for IN the DT empty JJ set NN . SENT Contents NNS 1 CD Notation NN 2 CD Properties NP 3 CD Common NP problems NNS 4 CD Axiomatic JJ set NN theory NN 5 CD Does VVZ it PP exist VVP or CC is VBZ it PP necessary JJ . SENT 6 CD Operations NPS on IN the DT empty JJ set NN 7 CD Bounds NP 8 CD The DT empty JJ set NN and CC zero CD 9 CD Category NN theory NN edit VV Notation NN The DT standard JJ notation NN for IN denoting VVG the DT empty JJ set NN is VBZ the DT symbol NN , , , , or CC , , introduced VVN by IN the DT Bourbaki NP group NN specifically RB Andr NP Weil NP in IN 1939 CD 1 CD . SENT This DT should MD not RB be VB confused VVN with IN the DT Greek JJ letter NN . SENT Another DT common JJ notation NN for IN the DT empty JJ set NN is VBZ . SENT edit VV Properties NP Here RB we PP use VVP mathematical JJ symbols NNS . SENT The DT empty JJ set NN is VBZ NOT RB 0 CD . SENT 0 CD For IN any DT set VVN A NP , , the DT empty JJ set NN is VBZ a DT subset NN of IN A DT . SENT A DT . SENT A DT For IN any DT set VVN A NP , , the DT union NN of IN A DT with IN the DT empty JJ set NN is VBZ A DT . SENT A DT . SENT A DT A NP For IN any DT set VVN A NP , , the DT intersection NN of IN A DT with IN the DT empty JJ set NN is VBZ the DT empty JJ set NN . SENT A DT . SENT A DT For IN any DT set VVN A NP , , the DT Cartesian JJ product NN of IN A DT and CC the DT empty JJ set NN is VBZ empty JJ . SENT A DT . SENT A DT The DT only JJ subset NN of IN the DT empty JJ set NN is VBZ the DT empty JJ set NN itself PP . SENT A DT . SENT A DT A NP The DT number NN of IN elements NNS of IN the DT empty JJ set NN that WDT is VBZ its PP$ cardinality NN is VBZ zero CD . SENT in IN particular JJ , , the DT empty JJ set NN is VBZ finite JJ . SENT 0 CD For IN any DT property NN . SENT for IN every DT element NN of IN the DT property NN holds VVZ vacuous JJ truth NN there EX is VBZ no DT element NN of IN for IN which WDT the DT property NN holds VVZ Conversely RB . SENT if IN , , for IN some DT property NN , , the DT following VVG two CD statements NNS hold VVP . SENT for IN every DT element NN of IN V NN the DT property NN holds VVZ there EX is VBZ no DT element NN of IN V NN for IN which WDT the DT property NN holds VVZ then RB V NN Mathematicians NNS speak VVP of IN the DT empty JJ set NN rather RB than IN an DT empty JJ set NN . SENT In IN set VVN theory NN , , two CD sets NNS are VBP equal JJ if IN they PP have VHP the DT same JJ elements NNS . SENT therefore RB there EX can MD be VB only RB one CD set NN with IN no DT elements NNS . SENT Considered VVN as IN a DT subset NN of IN the DT real JJ number NN line NN or CC more RBR generally RB any DT topological JJ space NN , , the DT empty JJ set NN is VBZ both DT closed JJ and CC open JJ . SENT All PDT its PP$ boundary NN points NNS of IN which WDT there EX are VBP none NN are VBP in IN the DT empty JJ set NN , , and CC the DT set NN is VBZ therefore RB closed VVN . SENT while IN all PDT its PP$ interior JJ points NNS of IN which WDT there EX are VBP again RB none NN are VBP in IN the DT empty JJ set NN , , and CC the DT set NN is VBZ therefore RB open JJ . SENT Moreover RB , , the DT empty JJ set NN is VBZ a DT compact JJ set NN by IN the DT fact NN that IN every DT finite JJ set NN is VBZ compact JJ . SENT The DT closure NN of IN the DT empty JJ set NN is VBZ empty JJ . SENT This DT is VBZ known VVN as IN preservation NN of IN nullary NN unions NNS . SENT edit VV Common JJ problems NNS The DT empty JJ set NN is VBZ not RB the DT same JJ thing NN as IN nothing NN . SENT it PP is VBZ a DT set NN with IN nothing NN inside IN it PP , , and CC a DT set NN is VBZ something NN . SENT This DT often RB causes VVZ difficulty NN among IN those DT who WP first RB encounter VV it PP . SENT It PP may MD be VB helpful JJ to TO think VV of IN a DT set NN as IN a DT bag NN containing VVG its PP$ elements NNS . SENT an DT empty JJ bag NN may MD be VB empty JJ , , but CC the DT bag NN itself PP certainly RB exists VVZ . SENT Some DT people NNS balk VVP at IN the DT first JJ property NN listed VVN above IN , , that IN the DT empty JJ set NN is VBZ a DT subset NN of IN any DT set VVN A NP . SENT By IN the DT definition NN of IN subset NN , , this DT claim NN means VVZ that IN for IN every DT element NN x SYM of IN , , x NN belongs VVZ to TO A DT . SENT If IN it PP is VBZ not RB true JJ that IN every DT element NN of IN is VBZ in IN A DT , , there EX must MD be VB at IN least JJS one CD element NN of IN that WDT is VBZ not RB present JJ in IN A DT . SENT Since IN there EX are VBP no DT elements NNS of IN at IN all DT , , there EX is VBZ no DT element NN of IN that WDT is VBZ not RB in IN A DT , , leading VVG us PP to TO conclude VV that IN every DT element NN of IN is VBZ in IN A DT and CC that DT is VBZ a DT subset NN of IN A DT . SENT Any DT statement NN that WDT begins VVZ for IN every DT element NN of IN is VBZ not RB making VVG any DT substantive JJ claim NN . SENT it PP is VBZ a DT vacuous JJ truth NN . SENT This DT is VBZ often RB paraphrased VVN as IN everything NN is VBZ true JJ of IN the DT elements NNS of IN the DT empty JJ set NN . SENT edit VV Axiomatic JJ set NN theory NN In IN the DT axiomatization NN of IN set NN theory NN known VVN as IN Zermelo NP Fraenkel NP set VVD theory NN , , the DT existence NN of IN the DT empty JJ set NN is VBZ assured VVN by IN the DT axiom NN of IN empty JJ set NN . SENT The DT uniqueness NN of IN the DT empty JJ set NN follows VVZ from IN the DT axiom NN of IN extensionality NN . SENT Any DT axiom NN that WDT states VVZ the DT existence NN of IN any DT set NN will MD imply VV the DT axiom NN of IN empty JJ set NN , , using VVG the DT axiom NN schema NN of IN separation NN . SENT For IN example NN , , if IN A DT is VBZ a DT set NN then RB the DT axiom NN schema NN of IN separation NN allows VVZ the DT construction NN of IN the DT set NN B NN x SYM in IN A NP x SYM x NN , , which WDT can MD be VB defined VVN to TO be VB the DT empty JJ set NN . SENT edit VV Does NP it PP exist VVP or CC is VBZ it PP necessary JJ . SENT While IN the DT empty JJ set NN is VBZ a DT standard JJ and CC universally RB accepted VVD concept NN in IN mathematics NN , , there EX are VBP those DT who WP still RB entertain VV doubts NNS . SENT Jonathan NP Lowe NP has VHZ argued VVN that IN while IN the DT idea NN was VBD undoubtedly RB an DT important JJ landmark NN in IN the DT history NN of IN mathematics NN , , we PP should MD not RB assume VV that IN its PP$ utility NN in IN calculation NN is VBZ dependent JJ upon IN its PP$ actually RB denoting VVG some DT object NN . SENT It PP is VBZ not RB clear JJ that IN such PDT an DT idea NN makes VVZ sense NN . SENT All DT that IN we PP are VBP ever RB informed VVN about IN the DT empty JJ set NN is VBZ that IN it PP 1 CD is VBZ a DT set NN , , 2 CD has VHZ no DT members NNS , , and CC 3 CD is VBZ unique JJ amongst IN sets NNS in IN having VHG no DT members NNS . SENT However RB , , there EX are VBP very RB many JJ things NNS that WDT have VHP no DT members NNS , , in IN the DT set VVN theoretical JJ sense NN namely RB , , all DT non JJ sets NNS . SENT It PP is VBZ perfectly RB clear JJ why WRB these DT things NNS have VHP no DT members NNS , , for IN they PP are VBP not RB sets NNS . SENT What WP is VBZ unclear JJ is VBZ how WRB there EX can MD be VB , , uniquely RB amongst IN sets NNS , , a DT set NN which WDT has VHZ no DT members NNS . SENT We PP cannot NN conjure VV such PDT an DT entity NN into IN existence NN by IN mere JJ stipulation NN . SENT In IN To TO be VB is VBZ to TO be VB the DT value NN of IN a DT variable NN , , Journal NP of IN Philosophy NN , , 1984 CD reprinted VVN in IN his PP$ book NN Logic NN , , Logic NP and CC Logic NP , , the DT late JJ George NP Boolos NP has VHZ argued VVN that IN we PP can MD go VV a DT long JJ way NN just RB by IN quantifying VVG plurally RB over IN individuals NNS , , without IN reifying VVG sets NNS as IN singular JJ entities NNS having VHG other JJ entities NNS as IN members NNS . SENT In IN a DT recent JJ book NN Tom NP McKay NP has VHZ disparaged VVD the DT singularist NN assumption NN that IN natural JJ expressions NNS using VVG plurals NNS can MD be VB analysed VVN using VVG plural JJ surrogates NNS , , such JJ as IN signs NNS for IN sets NNS . SENT He PP argues VVZ for IN an DT anti JJ singularist NN theory NN which WDT differs VVZ from IN set VVN theory NN in IN that WDT there EX is VBZ no DT analogue NN of IN the DT empty JJ set NN , , and CC there EX is VBZ just RB one CD relation NN , , among IN , , that WDT is VBZ an DT analogue NN of IN both CC the DT membership NN and CC the DT subset NN relation NN . SENT edit VV Operations NP on IN the DT empty JJ set NN Operations NP performed VVD on IN the DT empty JJ set NN as IN a DT set NN of IN things NNS to TO be VB operated VVN upon RP can MD also RB be VB confusing JJ . SENT Such JJ operations NNS are VBP nullary NN operations NNS . SENT For IN example NN , , the DT sum NN of IN the DT elements NNS of IN the DT empty JJ set NN is VBZ zero CD , , but CC the DT product NN of IN the DT elements NNS of IN the DT empty JJ set NN is VBZ one PP see VVP empty JJ product NN . SENT This DT may MD seem VV odd JJ , , since IN there EX are VBP no DT elements NNS of IN the DT empty JJ set NN , , so RB how WRB could MD it PP matter VV whether IN they PP are VBP added VVN or CC multiplied VVN since IN they PP do VVP not RB exist VV . SENT Ultimately RB , , the DT results NNS of IN these DT operations NNS say VVP more JJR about IN the DT operation NN in IN question NN than IN about IN the DT empty JJ set NN . SENT For IN instance NN , , notice NN that IN zero CD is VBZ the DT identity NN element NN for IN addition NN , , and CC one PP is VBZ the DT identity NN element NN for IN multiplication NN . SENT edit VV Bounds NP Since IN the DT empty JJ set NN has VHZ no DT members NNS , , when WRB it PP is VBZ considered VVN as IN a DT subset NN of IN any DT ordered VVN set NN , , then RB any DT member NN of IN that DT set NN will MD be VB an DT upper JJ bound JJ and CC lower RBR bound VVN for IN the DT empty JJ set NN . SENT For IN example NN , , when WRB considered VVN as IN a DT subset NN of IN the DT real JJ numbers NNS , , with IN its PP$ usual JJ ordering VVG , , represented VVN by IN the DT real JJ number NN line NN , , every DT real JJ number NN is VBZ both CC an DT upper JJ and CC lower RBR bound VVN for IN the DT empty JJ set NN . SENT When WRB considered VVN as IN a DT subset NN of IN the DT extended JJ reals NNS formed VVN by IN adding VVG two CD numbers NNS or CC points NNS to TO the DT real JJ numbers NNS , , namely RB negative JJ infinity NN , , denoted VVN which WDT is VBZ defined VVN to TO be VB less JJR than IN every DT other JJ extended JJ real JJ number NN , , and CC positive JJ infinity NN , , denoted VVN which WDT is VBZ defined VVN to TO be VB greater JJR than IN every DT other JJ extended JJ real JJ number NN , , then RB . SENT and CC That DT is VBZ , , the DT least JJS upper JJ bound JJ sup NN or CC supremum NN of IN the DT empty JJ set NN is VBZ negative JJ infinity NN , , while IN the DT greatest JJS lower VV bound VVN inf NN or CC infimum NN is VBZ positive JJ infinity NN . SENT edit VV The DT empty JJ set NN and CC zero VV It PP was VBD mentioned VVN earlier RBR that IN the DT empty JJ set NN has VHZ zero CD elements NNS , , or CC that IN its PP$ cardinality NN is VBZ zero CD . SENT The DT connection NN between IN the DT two CD concepts NNS goes VVZ further RBR however RB . SENT in IN the DT standard NN set VVD theoretic JJ definition NN of IN natural JJ numbers NNS , , zero NN is VBZ defined VVN as IN the DT empty JJ set NN . SENT edit VV Category NN theory NN If IN A DT is VBZ a DT set NN , , then RB there RB exists VVZ precisely RB one CD function NN f SYM from IN to TO A DT , , the DT empty JJ function NN . SENT As IN a DT result NN , , the DT empty JJ set NN is VBZ the DT unique JJ initial JJ object NN of IN the DT category NN of IN sets NNS and CC functions NNS . SENT The DT empty JJ set NN can MD be VB turned VVN into IN a DT topological JJ space NN in IN just RB one CD way NN by IN defining VVG the DT empty JJ set NN to TO be VB open JJ . SENT this DT empty JJ topological JJ space NN is VBZ the DT unique JJ initial JJ object NN in IN the DT category NN of IN topological JJ spaces NNS with IN continuous JJ maps NNS . SENT Retrieved VVN from IN http NN . SENT en FW . SENT wikipedia NN . SENT org NP wiki NP Empty JJ set NN Category NN . SENT Set VVN theory NN Views NNS ArticleDiscussionEdit NN this DT pageHistory JJ Personal JJ tools NNS Sign VV in IN create VV account NN Navigation NP Main NP PageCommunity NP PortalCurrent NP eventsRecent JJ changesRandom NN articleHelpContact NN usDonations NNS Search NP Toolbox NP What WP links VVZ here RB Related JJ changes NNS Upload NN file VVP Special JJ pages NNS Printable JJ version NN Permanent JJ link NN In IN other JJ languages NNS esky NP Dansk NP Deutsch NP Eesti NP Espa NP ol JJ Fran NP ais NNS Italiano NP Nederlands NP Polski NP Sloven NN ina NP Svenska NP This DT page NN was VBD last RB modified VVN 18 CD . SENT 25 CD , , 25 CD October NP 2005 CD . SENT All DT text NN is VBZ available JJ under IN the DT terms NNS of IN the DT GNU NN Free JJ Documentation NN License NN see VVP Copyrights NNS for IN details NNS . SENT Privacy NN policy NN About IN Wikipedia NP Disclaimers NNS