0000070486 00000 n 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 xref A. Hajnal & P. Hamburger, ‘Set Theory’, CUP 1999 (for cardinals and ordinals) 4. startxref <]>> The Axiom of Pair, the Axiom of Union, and the Axiom of Negation of Quantified Predicates. 1. Mathematical Induction. 0000056396 00000 n Then ffag;fa;bgg= ffag;fa;agg= ffag;fagg= ffagg Since ffagg= ffcg;fc;dggwe must have fag= fcgand fag= fc;dg. %PDF-1.2 0000054768 00000 n The study of these topics is, in itself, a formidable task. t IExercise 7 (1.3.7). axiomatic set theory with urelements. Set Theory Basics.doc 1.4. 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)]. 0000045614 00000 n The Axiom of Pair, the Axiom of Union, and the Axiom of 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. ����sɞ .�;��7!0y�d�t����C��dL��e���Y��Y>����k���fs��u��H��yX�}�ލ��b�)B��h�@����V�⎆�>�)�'�'����m�����$ѱ�K�b�IO+1P���qPDs�E[R,��B����E��N]M�yP���S"��K������\��J����,��Y'���]V���Z����(`��O���U� 0000055948 00000 n Clearly if a= cand b= dthen ha;bi= ffag;fa;bgg= ffcg;fc;dgg= hc;di 1. 0000076098 00000 n ��r��* ����/���8x�[a�G�:�ln-97ߨ�k�R�s'&�㕁8W)���+>v��;�-���9��d��S�Z��-�&j�br�YI% �����ZE$��։(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 These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. 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$Y�����!m>���uS��v���d(t�mXiP�l2��.T�q��~;ۗ30�A�V��̜"�F��.�i��^]$ 0000070198 00000 n Universal and Existential Quantifiers. LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. 0000064013 00000 n The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. 0000075834 00000 n Mathematics are constr ucted from the axi oms of logic and the axi oms of class and set t heory. If the object x is a member of the set A, then we write x A which is read as “ x is a … 0000045997 00000 n 0000078112 00000 n Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. SECTION 1.4 ELEMENTARY OPERATIONS ON SETS 3 Proof. The major changes in this new edition are the following. EXAMPLE 1 Finding Subsets Find all the subsets of {a,b,c}. (Caution: sometimes ⊂ is used the way we are using ⊆.) {=���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^� III. IV. Chapter 1 Set Theory 1.1 Basic definitions and notation A set is a collection of objects. 0000047721 00000 n We study two types of relations between statements, implication and equivalence. 0000073034 00000 n Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. 0000002654 00000 n Informal Proof. Indirect Proof. P. T. Johnstone, ‘Notes on Logic & Set Theory’, CUP 1987 2. constructive set theory was called by Hilbert), at least we should know what we are m1ssmg. 0000011497 00000 n 0000023682 00000 n 1592 0 obj<> endobj They are meta-statements about some propositions. t IExercise 7 (1.3.7). Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more difficult and more interesting. V. Naïve Set Theory. 0000072849 00000 n 0000047470 00000 n 0000055776 00000 n Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. 0000080242 00000 n P!$� 0000041887 00000 n Any object which is in a set is called a member of the set. 0000000016 00000 n 0000041116 00000 n 0000042018 00000 n 0000039958 00000 n x���A 0ð4�v\Gcw��������z�C. ��Xe�e���� �81��c������ ˷�孇f�0h_mw. logic has now taken on a life of its own, and also thrives on many interactions with other areas of mathematics and computer science. x�b```b``_���� �� Ȁ ��@����� ��bT; �}a~��ǯ��EO��z0�XN^�t[ut�$. 0000002324 00000 n 0000055416 00000 n 0000039789 00000 n 0000001631 00000 n H��WYo�~��0�# ��}HV����Y4(�KR� �ǧ����ʔ1�̩>ꮯ�U������ٟ�T������d啮B�ծ9?�9? << 0000064488 00000 n An example of an implication meta-statement is the observation that “if the statement ‘Robert gradu-ated from Texas … 0000071716 00000 n In mathematics, the notion of a set is a primitive notion. .6�⊫�Ţ1o�/A���F�\���6f=iE��i�K��Lٛ�[�n&]=�x�Wȥ��噅Ak5��z�I��� This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. 0000041289 00000 n 0000041931 00000 n 0000047249 00000 n 14 Chapter 1 Sets and Probability Empty Set The empty set, written as /0or{}, is the set with no elements. in set theory, one that is important for both mathematical and philosophical reasons. 0000072804 00000 n This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. 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. 0000056119 00000 n 0000003046 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)].
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