0000023682 00000 n 0000001631 00000 n So a= c= d, in particular, a= cand b= d. 2. 0000010201 00000 n 0000077155 00000 n For our purposes, it will sufce to approach basic logical concepts informally. startxref 0000039958 00000 n Methods of Proof. ha;bi= ffag;fa;bgg Theorem 1.5. ha;bi= hc;dii a= cand b= d. Proof. >> 0000055416 00000 n 0000041116 00000 n 0000056396 00000 n 0000064488 00000 n 4. IV. 0000038686 00000 n Predicate Logic and Quantifiers. 0000069809 00000 n 14 Chapter 1 Sets and Probability Empty Set The empty set, written as /0or{}, is the set with no elements. 0000075927 00000 n Mathematics are constr ucted from the axi oms of logic and the axi oms of class and set t heory. 1592 65 0000079248 00000 n xref 2. Primitive Concepts. H��WYo�~��0�# ��}HV����Y4(�KR� �ǧ����ʔ1�̩>ꮯ�U������ٟ�T������d啮B�ծ9?�9? Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. 0000002654 00000 n Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. 0000047721 00000 n Then ffag;fa;bgg= ffag;fa;agg= ffag;fagg= ffagg Since ffagg= ffcg;fc;dggwe must have fag= fcgand fag= fc;dg. The notion of set is taken as “undefined”, “primitive”, or “basic”, so we don’t try to define what a set is, but we can give an informal description, describe important properties of sets, and give examples. 0000038078 00000 n axiomatic set theory with urelements. %PDF-1.3 %���� 3. Predicates. 0000070198 00000 n 0000070658 00000 n V. Naïve Set Theory. That is, we admit, as a starting point, the existence of certain objects (which we call sets), which we won’t define, but which we assume satisfy some basic properties, which we express as axioms. 0000063750 00000 n Let Xbe an arbitrary set; then there exists a set Y Df u2 W – g. Obviously, Y X, so 2P.X/by the Axiom of Power Set.If , then we have Y2 if and only if – [SeeExercise 3(a)]. In the second half of the last century, logic as pursued by mathematicians gradually branched into four main areas: model theory, computability theory (or recursion theory), set theory, and proof theory. 2 0 obj In mathematics, the notion of a set is a primitive notion. 0000054768 00000 n Formal Proof. Subsets A set A is a subset of a set B iff every element of A is also an element of B. in set theory, one that is important for both mathematical and philosophical reasons. Such a relation between sets is denoted by A ⊆ B. Multiple Quantifiers. These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. In 1874 Cantor had shown that there is a one-to-one correspondence between the natural numbers and the algebraic numbers. An appendix on second-order logic will give the reader an idea of the advantages and limitations of the systems of first-order logic used in Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. 0000011497 00000 n 0000070486 00000 n constructive set theory was called by Hilbert), at least we should know what we are m1ssmg. 0000010830 00000 n P!$� 0000064013 00000 n 0000039153 00000 n 0000041460 00000 n stream <]>> /Length 2960 �����oسYEN�ʧ"�5��z�&���� 0000041801 00000 n logic has now taken on a life of its own, and also thrives on many interactions with other areas of mathematics and computer science. 0000041289 00000 n A. Hajnal & P. Hamburger, ‘Set Theory’, CUP 1999 (for cardinals and ordinals) 4. Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. We study two types of relations between statements, implication and equivalence. << 0000011807 00000 n lX�Å 0000003046 00000 n 0000071716 00000 n In Chapter 2, a section has been added on logic with empty domains, that is, on what happens when we allow interpretations with an empty domain. Chapter 1 Set Theory 1.1 Basic definitions and notation A set is a collection of objects. The Axiom of Pair, the Axiom of Union, and the Axiom of 0000072804 00000 n %���� Conditional Proof. ��r��* ����/���8x�[a�G�:�ln-97ߨ�k�R�s'&�㕁8W)���+>v��;�-���9��d��S�Z��-�&j�br�YI% �����ZE$��։(8x^[���0`ll��JJJ...iii2@ 8��� ����Vfcc�q�(((�OR���544,#����\��-G�5�2��S����� |��Qq�M���l�M�����(�0�)��@���!�E�ԗ�u��#�g�'� BLg�`�t�0�~��f'�q��L�6�1,Qc b�&``�(v�,� ���T��~�3ʛz���3�0{� p6ts��m�d��}"(�t��o�L��@���^�@� iQݿ Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more difficult and more interesting. ����sɞ .�;��7!0y�d�t����C��dL��e���Y��Y>����k���fs��u��H��yX�}�ލ��b�)B��h�@����V�⎆�>�)�'�'����m�����$ѱ�K�b�IO+1P���qPDs�E[R,��B����E��N]M�yP���S"��K������\��J����,��Y'���]V���Z����(`��O���U� 0000000016 00000 n ��Xe�e���� �81��c������ ˷�孇f�0h_mw. III. Any object which is in a set is called a member of the set. The major changes in this new edition are the following. Basics of Set Theory and Logic S. F. Ellermeyer August 18, 2000 Set Theory Membership A setis a well-defined collection of objects. 0000041887 00000 n This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. Set Theory and Logic is the result of a course of lectures for advanced undergraduates, developed at Oberlin College for the purpose of introducing students to the conceptual foundations of mathematics.Mathematics, specifically the real number system, is approached as a unity whose operations can be logically ordered through axioms. /Filter /FlateDecode 0000047249 00000 n Unique Existence. An Overview of Logic, Proofs, Set Theory, and Functions aBa Mbirika and Shanise Walker Contents 1 Numerical Sets and Other Preliminary Symbols3 2 Statements and Truth Tables5 3 Implications 9 4 Predicates and Quanti ers13 5 Writing Formal Proofs22 6 Mathematical Induction29 7 Quick Review of Set Theory & Set Theory Proofs33 0000056119 00000 n 0000063492 00000 n They are meta-statements about some propositions. (Caution: sometimes ⊂ is used the way we are using ⊆.) Primitive Concepts. If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. 0000065343 00000 n 0000011168 00000 n File Name: Logic And Set Theory With Applications 6th Edition Pdf.pdf Size: 6514 KB Type: PDF, ePub, eBook Category: Book Uploaded: 2020 Nov 20, 04:15 Rating: 4.6/5 from 914 votes. The empty set can be used to conveniently indicate that an equation has no solution. t IExercise 7 (1.3.7). Clearly if a= cand b= dthen ha;bi= ffag;fa;bgg= ffcg;fc;dgg= hc;di 1. 0000072849 00000 n 0000055776 00000 n 1594 0 obj<>stream {]xKA}�a\0�;��O`�d�n��8n��%{׆P�;�PL�L>��бL�~ 0000042018 00000 n 0000075834 00000 n SECTION 1.4 ELEMENTARY OPERATIONS ON SETS 3 Proof. P. T. Johnstone, ‘Notes on Logic & Set Theory’, CUP 1987 2. 0000021855 00000 n 1. Universal and Existential Quantifiers. cHy� ��#���P�gw��l���k-�l��X���&���"O�Q�L//f�n�?�Kh�B\�f˼�+h���Tg_�ssw������d����ڶ�5��^{Z���oDp��F��*O���T���(�l6tu15&c��~����zƖ�v�3c�����j�`~[/��X��j�AZrV]o�>���ׯ6��>�e>�p�Z���j���O�NHd|�n� i$Y�����!m>���uS��v���d(t�mXiP�l2��.T�q��~;ۗ30�A�V��̜"�F��.�i��^]$ 0000042252 00000 n Complex issues arise in Set Theory more than any other area of pure mathematics; in particular, Mathematical Logic is used in a fundamental way. 0000047470 00000 n 0000003562 00000 n Mathematical Induction. 0000076098 00000 n 0000041632 00000 n țP� {�~حM�Ъ1����s,B��s�)Sd_�+d�K��|+wb�Lc�@Ԡ���s �r��@ံPv�⚝��s����; ���xBeY���^��cvhnϳ�y�W��`�9BD���#)p���a4S��RA��z�k�'y���~h�)�I��O��N�:+��*��ㄯ��y��mAu 9� &��7�^19�>� �%OD+U�|��F�~|I�n���;=���p����e��~ ,�/�� w��-�Ȼ�v|�2 zy?�tq~�iq�q��0��0q��h=�y�F_ A 0P�T�����,��;@�ig�p��y�!��|n�v��P������b1%��� ���GLV.t�W[��Y�2��{N�Nw\=����Ԡ�2`q�#f��f��x�|X���|Z,�ns�f{��A���JU�en��ϛ����G�|��Eg-TX�2� ����"��0���=k3� �ʤu?-�P E��(���#�k����䐷��@�ˀ�A��a+���f�l2e��������5��3�#�:�t2Tf�@xӄ�m2����DL��1M�1|3t���3�ї"r5���_%$�Qr��eZ���#�cr��˔��)l���m�Ӿ=��f����k8�B,ࡩ�m �uz��>���'GZBy��u8��?�Bx��["����CӴsd_T����T@0#�1�?��I~:c+�Yxrfl{��Ŝ #�r}�:�(��R=��KN�N�K]4���қ�p혉��a�]x�X�˽� ���v !�MI��N���}��aP����Hd�K�Y�(�o�_HZ�٥���&�2 \�- D. Van Dalen, ‘Logic and Structure’, Springer-Verlag 1980 (good for Chapter 4) 3. endstream endobj 1656 0 obj<>/W[1 1 1]/Type/XRef/Index[78 1514]>>stream Negation of Quantified Predicates. Set Theory Basics.doc 1.4. 0000039789 00000 n 0000022861 00000 n They are not guaran-teed to be comprehensive of the material covered in the course. 0000057132 00000 n

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