set theory and logic pdf
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0000039789 00000 n 0000047470 00000 n LOGIC AND SET THEORY 1.2 Relations between Statements Strictly speaking, relations between statements are not formal statements them-selves. 0000010201 00000 n 0000064013 00000 n stream 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���_%$�Qr��eZ���#�cr��˔��)l���m�Ӿ=��f����k8�B,ࡩ�m �uz��>���'GZBy��u8��?�Bx��["����CӴsd_T����T@0#�1�?��I~:c+�Yxrfl{��Ŝ #�r}�:�(��R=��KN�N�K]4���қ�p혉��a�]x�X�˽� ���v 0000045997 00000 n 0000057132 00000 n 0000002534 00000 n Let Xbe an arbitrary set; then there exists a set Y Df u2 W – g. Obviously, Y X, so 2P.X/by the Axiom of Power Set.If , then we have Y2 if and only if – [SeeExercise 3(a)]. in set theory, one that is important for both mathematical and philosophical reasons. 4. Predicate Logic and Quantifiers. Primitive Concepts. 0000072804 00000 n %PDF-1.2 0000038686 00000 n 0000055948 00000 n Negation of Quantified Predicates. 0000022533 00000 n {]xKA}�a\0�;��O`�d�n��8n��%{׆P�;�PL�L>��бL�~ This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. Universal and Existential Quantifiers. An example of an implication meta-statement is the observation that “if the statement ‘Robert gradu-ated from Texas … axiomatic set theory with urelements. 0000041460 00000 n They are meta-statements about some propositions. endstream endobj 1656 0 obj<>/W[1 1 1]/Type/XRef/Index[78 1514]>>stream x���A 0ð4�v\Gcw��������z�C. 0000002654 00000 n ����sɞ .�;��7!0y�d�t����C��dL��e���Y��Y>����k���fs��u��H��yX�}�ލ��b�)B��h�@����V�⎆�>�)�'�'����m�����$ѱ�K�b�IO+1P���qPDs�E[R,��B����E��N]M�yP���S"��K������\��J����,��Y'���]V���Z����(`��O���U� 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. �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���*�,�� Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. 0000045614 00000 n Multiple Quantifiers. 0000021855 00000 n %���� (Caution: sometimes ⊂ is used the way we are using ⊆.) SECTION 1.4 ELEMENTARY OPERATIONS ON SETS 3 Proof. !���}�&)�MO8�eL6uFoJ��:�#@�f�� �N`�`���RK���yD�}c~���'�*n��E��Ij�Tl 0000077155 00000 n 0000011168 00000 n 0000063750 00000 n Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. Although Elementary Set Theory is well-known and straightforward, the modern subject, Axiomatic Set Theory, is both conceptually more difficult and more interesting. 0000001631 00000 n LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. 0000065343 00000 n 0000041931 00000 n 0000041632 00000 n 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. 0000075834 00000 n 0000042018 00000 n 0000002324 00000 n Formal Proof. 0000078112 00000 n ��Xe�e���� �81��c������ ˷�孇f�0h_mw. lX�Å Proof by Counter Example. Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. 0000047721 00000 n 0000003562 00000 n 0000022861 00000 n 0000041116 00000 n Mathematical Induction. %%EOF IV. V. Naïve Set Theory. 0000063492 00000 n Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. III. 0000075488 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)]. 2 0 obj The subjects of register machines and random access machines have been dropped from Section 5.5 Chapter 5. {=���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^� 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. x�b```b``_���� �� Ȁ ��@����� ��bT; �}a~��ǯ��EO��z0�XN^�t[ut�$. 0000047249 00000 n 0000073034 00000 n 0000023682 00000 n 0000003293 00000 n 0000075927 00000 n We study two types of relations between statements, implication and equivalence. 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 0000041887 00000 n 0000056396 00000 n A. Hajnal & P. Hamburger, ‘Set Theory’, CUP 1999 (for cardinals and ordinals) 4. 3. 0000079248 00000 n Mathematics are constr ucted from the axi oms of logic and the axi oms of class and set t heory. 1. startxref Clearly if a= cand b= dthen ha;bi= ffag;fa;bgg= ffcg;fc;dgg= hc;di 1. 1592 0 obj<> endobj The Axiom of Pair, the Axiom of Union, and the Axiom of <]>> Conditional Proof. 0000070486 00000 n Then ffag;fa;bgg= ffag;fa;agg= ffag;fagg= ffagg Since ffagg= ffcg;fc;dggwe must have fag= fcgand fag= fc;dg.
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