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6o endstream endobj 39 0 obj <>stream HVmh[U~ν7Ƶh514ؖknMZY1m %ɶYT:n@TqT'2" 21q*ܛkdNAyo<{sPih}x`K[Zf=?>¢r ّ_>E{=텑CS'@T2OҞ5E4˴קF'.>4՜`Nhl2 Ll4[Cgl.=>lk1 C"\&u9BZaDPzq1:zDĤeX(?Tm6}\L*kV>SWѻl1OY q VV.SkRGL]I1},;Krc|NG2d&˝_`s[kkP 7Ls;4X2 SlRrIIg"9c÷b 91؈86ĸ7MOUolFZyЂ~ v!0[{jy^MimĖD r*;F\LPysrJܑ>RXT83Wg( V[+kECe:U endstream endobj 40 0 obj <>stream Hj```P0 endstream endobj 41 0 obj <>stream hXmo8 +X@_-lnk HN,JCg7n|I/Vjg岮}\&SCw*NYGʗۗM].˿V,pIN`Rg^Wj:/E oI2qsm3TΕͤIsnD&ȅLa:ͅf0)ra&@&ϭ1?;ݝ$m!FeTCjp5 jy!ywq dM^jrOU&u XEm.뇧f^sǺkpDΗX=1'}Hmܖ?gR$4R>Vzrr>[ݗyS`Z-MբX~;jL wW-ϗ*֛X>^+rgB56ޖ8H6dᷛûHFEy9_Q6!?u5#mJi_[6n-$l|QL+k7F |u;~>td^WloF)ƍzhX?8,߽ q 166Q6(rY#oԖk'C!>;&jM!L|[AnxD X2LG `q wEw^maĖ7} k]]5`!`1>p0y;6pB8t++7v/_O~3~5d)=$b$cl$/ݫɘ8&cu5  ven=Fkt`+Yj ^s?zu(ѠhஊEiM4Nw4XAu9NGͨKxpj>{=P  r(0P0)MM+~i: oNpczHv܉ك;gs8f-(0\X9tG.".݃ N+ipei6C÷PNLm`3x=ևfküMIs|0ϕp-olҷus[RTlB!|3)f]7Y+Vn PjoU䡊7u Usswz^.ԷT߮¥nvکpF| mʷcQ;^'Y*G>stream hބ]o0S| 1HU۪i[> 5S@w{} 0`}y HBcJ#b׷RH S$RPUm$ZZ'XLCkTIEnƍ:9pd0O O u¨XC+֝CnprLpQpR6`7mYqpg1ܹZ6yy|]3>͝K<|u8;q'E.BKлC+ F K9Wfbs+Tx/V 0 >vNe߲>շ&_(eLepZ\Z +} _ -WCbFR@/Ȏ;@xkWD|tWfx>AxٔGv&瀪aTPF5]Tb⑥7X[tJFzWؚ_4%ח⾱Qxyp'Zcy2aq]'\{.8`t.5SmGO|=ښCtԉ=%GWL-ALyTf]'\I=WC{o䞪AU`4a0^^pNf7Q@ 5t- %{t 8 +_Ԩ endstream endobj 43 0 obj <>stream 2019-07-01T08:55:35+08:00 2019-07-01T08:55:33+08:00 2019-07-01T08:55:35+08:00 Acrobat PDFMaker 11 Word 版 uuid:c03db8b3-dded-4430-be46-89ddac4e9b44 uuid:b53f3553-b20e-401d-8688-a215b12d380d 3 application/pdf Number Systems from a General Point of View This paper presents recent results of this author and Z. Kadelburg and also contains additional comments and remarks. These results complete the study of number systems which this author has published in OALib Journal in search of a ground for designing a simplified content assigned to primary teachers. This ground consisted of derivation of properties of the system N of natural numbers with 0 and of the use of these properties as a basis for the extension of N to the systems of positive rational numbers and the system of integers. The crucial step in our final research paper has been the selection of basic operative properties of the system N (properties of operations and the order relation). Using these properties and those deduced from them, the system Q+ of positive rational numbers with 0 has been constructed and its basic operative properties verified (being those of N, plus the property of existence of multiplicative inverse). Then, using properties of Q+, the system Q of rational numbers is constructed and its properties verified (being those of Q+, plus the existence of additive inverse). Taking the basic operative properties of N for the axioms, we define the algebraic structure {S, +, 脳, <}, where S is a non-empty set, 鈥?鈥?and 鈥溍椻€?two binary operations and 鈥?鈥?the order relation, which we call N-structure, The number systems N, Q+ and Q, as well as the system R of real numbers are examples of N-structure. Thus, they share all properties that the axioms express and all those deduced from them. Hence all these properties when transcribed with the letters denoting the corresponding variables are transferable from N to the systems Q+, Q and R. This is a precise formulation of the Peacock鈥檚 principle of permanence and a firm logical basis for its validity. Finally, we prove that each system satisfying the axioms of N-structure contains an isomorphicb copy of N. Thus, the natural numbers with 0 are the smallest system satisfying the axioms of N-structure, This is a characterization of the system N which is analogous to that of Q which is the Milosav M. Marjanović Adobe PDF Library 11.0 Number Systems, Logical Foundation for Peacock’s Principle of Permanence, Characterization of the System of Natural Numbers D:20190701005529 Microsoft This paper presents recent results of this author and Z. Kadelburg and also contains additional comments and remarks. These results complete the study of number systems which this author has published in OALib Journal in search of a ground for designing a simplified content assigned to primary teachers. This ground consisted of derivation of properties of the system N of natural numbers with 0 and of the use of these properties as a basis for the extension of N to the systems of positive rational numbers and the system of integers. The crucial step in our final research paper has been the selection of basic operative properties of the system N (properties of operations and the order relation). Using these properties and those deduced from them, the system Q+ of positive rational numbers with 0 has been constructed and its basic operative properties verified (being those of N, plus the property of existence of multiplicative inverse). Then, using properties of Q+, the system Q of rational numbers is constructed and its properties verified (being those of Q+, plus the existence of additive inverse). Taking the basic operative properties of N for the axioms, we define the algebraic structure {S, +, è—³, <}, where S is a non-empty set, é‹¥?é‹¥?and 鋥溓椻⇬?two binary operations and é‹¥?é‹¥?the order relation, which we call N-structure, The number systems N, Q+ and Q, as well as the system R of real numbers are examples of N-structure. Thus, they share all properties that the axioms express and all those deduced from them. Hence all these properties when transcribed with the letters denoting the corresponding variables are transferable from N to the systems Q+, Q and R. This is a precise formulation of the Peacocké‹¥æªı principle of permanence and a firm logical basis for its validity. Finally, we prove that each system satisfying the axioms of N-structure contains an isomorphicb copy of N. Thus, the natural numbers with 0 are the smallest system satisfying the axioms of N-structure, This is a characterization of the system N which is analogous to that of Q which is the endstream endobj 44 0 obj <>stream h24P0Pw/+QL)645)I0i1PfLATXʂTb;;` endstream endobj 45 0 obj <>stream h޴X_o#57O<;F$W*P*' uZzm!H|>#0d6f^=|2/l" z踮h鵎r{qIf5%U?x,yҪ3x.k~dQ.@,~Ęq /k f3%,g)И \$Ҡ۸@~?J^S400?o\&0^V+Y;!fL`Lg24oXoħ} ߿]5IxOtYs/VO:(gs҆1$cj Xy{}TtCv5su ,R,.P~?~KE;Q~O/~G|?!;oK'"6XiSz+}9^VcQR \=9HIAfƼ8>mԖ+UeXH/! f Y@*u<+G|ȢЫg+}| xүg}tL" amy|N;/qQEkgK[2.ʖdSoU(Z5߭tUVd1tEw1>7;uR<u*7UBrKi5k+ <);9T5p[V>/Filter/FlateDecode/ID[<685D39D557B5EE41AFB3B084A2875187>]/Info 149 0 R/Length 216/Root 151 0 R/Size 150/Type/XRef/W[1 3 1]>>stream hbb&F} L RV RdHE`ZxT ! mXV,f`H2@1J%`]E`}m  &3A1lDo2@"Y]l; A$ޫ``qU`/P ;,N]?хL}#ڮ_"o~'>jdzv@r!\ endstream endobj startxref 116 %%EOF 43 0 obj <>stream 2019-07-01T08:55:53+08:00 2019-07-01T08:55:33+08:00 2019-07-01T08:55:53+08:00 Acrobat PDFMaker 11 Word 版 uuid:c03db8b3-dded-4430-be46-89ddac4e9b44 uuid:19523323-4b07-4d65-9892-2323e1a4c31b 3 application/pdf Number Systems from a General Point of View This paper presents recent results of this author and Z. Kadelburg and also contains additional comments and remarks. These results complete the study of number systems which this author has published in OALib Journal in search of a ground for designing a simplified content assigned to primary teachers. This ground consisted of derivation of properties of the system N of natural numbers with 0 and of the use of these properties as a basis for the extension of N to the systems of positive rational numbers and the system of integers. The crucial step in our final research paper has been the selection of basic operative properties of the system N (properties of operations and the order relation). Using these properties and those deduced from them, the system Q+ of positive rational numbers with 0 has been constructed and its basic operative properties verified (being those of N, plus the property of existence of multiplicative inverse). Then, using properties ... Milosav M. Marjanović Number Systems, Logical Foundation for Peacock’s Principle of Permanence, Characterization of the System of Natural Numbers Adobe PDF Library 11.0 "Number Systems, Logical Foundation for Peacock’s Principle of Permanence, Characterization of the System of Natural Numbers" D:20190701005529 Microsoft This paper presents recent results of this author and Z. Kadelburg and also contains additional comments and remarks. These results complete the study of number systems which this author has published in OALib Journal in search of a ground for designing a simplified content assigned to primary teachers. This ground consisted of derivation of properties of the system N of natural numbers with 0 and of the use of these properties as a basis for the extension of N to the systems of positive rational numbers and the system of integers. The crucial step in our final research paper has been the selection of basic operative properties of the system N (properties of operations and the order relation). Using these properties and those deduced from them, the system Q+ of positive rational numbers with 0 has been constructed and its basic operative properties verified (being those of N, plus the property of existence of multiplicative inverse). Then, using properties of Q+, the system Q of rational numbers is constructed and its properties verified (being those of Q+, plus the existence of additive inverse). Taking the basic operative properties of N for the axioms, we define the algebraic structure {S, +, è—³, <}, where S is a non-empty set, é‹¥?é‹¥?and 鋥溓椻⇬?two binary operations and é‹¥?é‹¥?the order relation, which we call N-structure, The number systems N, Q+ and Q, as well as the system R of real numbers are examples of N-structure. Thus, they share all properties that the axioms express and all those deduced from them. Hence all these properties when transcribed with the letters denoting the corresponding variables are transferable from N to the systems Q+, Q and R. This is a precise formulation of the Peacocké‹¥æªı principle of permanence and a firm logical basis for its validity. Finally, we prove that each system satisfying the axioms of N-structure contains an isomorphicb copy of N. Thus, the natural numbers with 0 are the smallest system satisfying the axioms of N-structure, This is a characterization of the system N which is analogous to that of Q which is the endstream endobj 221 0 obj <> endobj 222 0 obj <>/Filter/FlateDecode/ID[<685D39D557B5EE41AFB3B084A2875187>]/Index[43 1 221 2]/Info 221 0 R/Length 24/Prev 116/Root 151 0 R/Size 223/Type/XRef/W[1 3 0]>>stream hbbdIab`KlG  endstream endobj startxref 296825 %%EOF