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D. Van Dalen, ‘Logic and Structure’, Springer-Verlag 1980 (good for Chapter 4) 3. The empty set can be used to conveniently indicate that an equation has no solution. P!$� , 2. Any object which is in a set is called a member of the set. Multiple Quantifiers. 0000039153 00000 n IV. �t��Y�d}��8�� =�+< �UrP�� X��!�?L@[�e ��*%�F�₆�%P�׀�%�'�L@�5�֒A��آ-�W�f -=e vԽh%Q��H�e�f"�0�M��̝��W�,;P�`��RL�Lj"��|�A�Ac�kW�xf�=�V�� J2�k�}s"�0{N 0000075927 00000 n They are not guaran-teed to be comprehensive of the material covered in the course. 0000022861 00000 n %%EOF �壐�D;B��A��Ч�~:�{v��B��g�s��~/B~HW�>��C~�yڮ�2B~Ő9&�$F�� t� W?W�~��u[vJ%~��V5T�b��%@Q��QQX�ɠp7��%�W��`�/h2d��%s �� 1�_�m$=S��H �3��OA��x��"�bR3i��l�2��*�,�� 0000047249 00000 n 0000042018 00000 n 0000041632 00000 n Such a relation between sets is denoted by A ⊆ B. This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. 0000075834 00000 n 0000069809 00000 n 0000011497 00000 n 0000064488 00000 n 0000047470 00000 n For example, a deck of cards, every student enrolled in Math 103, the collection of all even integers, these are all examples of sets of things. 0000070486 00000 n 0000063492 00000 n Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. 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 ��oسYEN�ʧ"�5��z�&�� These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. 0000073034 00000 n Informal Proof. 0000072804 00000 n << /Filter /FlateDecode 0000010201 00000 n III. 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. 0000041801 00000 n Unique Existence. Universal and Existential Quantifiers. 0000055948 00000 n In 1874 Cantor had shown that there is a one-to-one correspondence between the natural numbers and the algebraic numbers. 0000065343 00000 n 0000041289 00000 n LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. 0000011807 00000 n Mathematical Induction. Then ffag;fa;bgg= ffag;fa;agg= ffag;fagg= ffagg Since ffagg= ffcg;fc;dggwe must have fag= fcgand fag= fc;dg. Set Theory Basics.doc 1.4. 0000003293 00000 n axiomatic set theory with urelements. Mathematics are constr ucted from the axi oms of logic and the axi oms of class and set t heory. 0000071716 00000 n >> 0000045614 00000 n %PDF-1.3 %�� 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 A. Hajnal & P. Hamburger, ‘Set Theory’, CUP 1999 (for cardinals and ordinals) 4. 0000000016 00000 n ��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 This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. 0000076098 00000 n 0000002534 00000 n !��}�&)�MO8�eL6uFoJ��:�#@�f�� N`�`��RK��yD�}c~��'�*n��E��Ij�Tl The subjects of register machines and random access machines have been dropped from Section 5.5 Chapter 5. 1592 65 14 Chapter 1 Sets and Probability Empty Set The empty set, written as /0or{}, is the set with no elements. The study of these topics is, in itself, a formidable task. 0000010830 00000 n EXAMPLE 1 Finding Subsets Find all the subsets of {a,b,c}. It only remains to de ne ha;biin terms of set theory. Clearly if a= cand b= dthen ha;bi= ffag;fa;bgg= ffcg;fc;dgg= hc;di 1. .6�⊫�Ţ1o�/A��F�\��6f=iE��i�K��Lٛ�[�n&]=�x�Wȥ��噅Ak5��z�I�� 3. Negation of Quantified Predicates. %�� 0000070198 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. 0000057132 00000 n 0000056119 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)]. 0000041931 00000 n constructive set theory was called by Hilbert), at least we should know what we are m1ssmg. x��A 0ð4�v\Gcw��z�C. %PDF-1.2 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$Y��!m>��uS��v��d(t�mXiP�l2��.T�q��~;ۗ30�A�V��̜"�F��.�i��^]$ Chapter 1 Set Theory 1.1 Basic deﬁnitions and notation A set is a collection of objects. Conditional Proof. 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. Formal Proof. 0000021855 00000 n /Length 2960 Methods of Proof. 0000023247 00000 n So a= c= d, in particular, a= cand b= d. 2. {]xKA}�a\0�;��O`�d�n��8n��%{׆P�;�PL�L>��бL�~ 0000039958 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. 0000038078 00000 n 0000038686 00000 n 0000011168 00000 n {=��N΁�FH�d�_JG�+�б�ߝ�I�D�3)��|y~��~�د��௫/�~�z~�lw��;�z��E[�}�~��m��wY�R�i��_�+a+o��,�]})��f�nvw��f��@-%��fJ(��t�i��b�� X�;�cU�і�4R�X%_)#�=��6젉^� 4. ha;bi= ffag;fa;bgg Theorem 1.5. ha;bi= hc;dii a= cand b= d. Proof. 0000080242 00000 n Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. 0000022533 00000 n Primitive Concepts. The Axiom of Pair, the Axiom of Union, and the Axiom of 0000078112 00000 n V. Naïve Set Theory. 0000063750 00000 n The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. 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 SECTION 1.4 ELEMENTARY OPERATIONS ON SETS 3 Proof. 0000042252 00000 n LOGIC AND SET THEORY 1.2 Relations between Statements Strictly speaking, relations between statements are not formal statements them-selves. 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)]. De nition 1.7 (Ordered Pair). 0000070658 00000 n (Caution: sometimes ⊂ is used the way we are using ⊆.) 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. xref 1594 0 obj<>stream H��WYo�~��0�# ��}HV��Y4(�KR� �ǧ��ʔ1�̩>ꮯ�U��ٟ�T��d啮B�ծ9?�9? For our purposes, it will sufce to approach basic logical concepts informally. They are meta-statements about some propositions. 1. t IExercise 7 (1.3.7). Basics of Set Theory and Logic S. F. Ellermeyer August 18, 2000 Set Theory Membership A setis a well-defined collection of objects. 0000055776 00000 n 0000056396 00000 n The Axiom of Pair, the Axiom of Union, and the Axiom of stream Subsets A set A is a subset of a set B iff every element of A is also an element of B. ��Xe�e�� 81��c�� ˷�孇f�0h_mw. 0000047721 00000 n Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. trailer Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. 0000055416 00000 n in set theory, one that is important for both mathematical and philosophical reasons. For example {x|xis real and x2 =−1}= 0/ By the deﬁnition of subset, given any set A, we must have 0/ ⊆A. If the object x is a member of the set A, then we write x A which is read as “ x is a … 1592 0 obj<> endobj Predicates. Complex issues arise in Set Theory more than any other area of pure mathematics; in particular, Mathematical Logic is used in a fundamental way.

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