%PDF-1.3 %���� Proof by Counter Example. 0000002534 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. in set theory, one that is important for both mathematical and philosophical reasons. (Caution: sometimes ⊂ is used the way we are using ⊆.) 1592 0 obj<> endobj 0000041116 00000 n 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. 0000011807 00000 n 0000041289 00000 n We study two types of relations between statements, implication and equivalence. 0000047249 00000 n 0000047470 00000 n The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. They are not guaran-teed to be comprehensive of the material covered in the course. Multiple Quantifiers. 0000056119 00000 n The Axiom of Pair, the Axiom of Union, and the Axiom of 1592 65 0000003562 00000 n �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���*�,�� If the object x is a member of the set A, then we write x A which is read as “ x is a … Formal Proof. III. 0000021855 00000 n 0000075488 00000 n Negation of Quantified Predicates. 0000055776 00000 n %���� If A ⊆ B and A ≠ B we call A a proper subset of B and write A ⊂ B. 0000064488 00000 n 0000003293 00000 n t IExercise 7 (1.3.7). /Length 2960 An example of an implication meta-statement is the observation that “if the statement ‘Robert gradu-ated from Texas … ��Xe�e���� �81��c������ ˷�孇f�0h_mw. >> 0000038078 00000 n 2 0 obj 0000063750 00000 n Conditional Proof. 0000010830 00000 n 0000039958 00000 n Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. The major changes in this new edition are the following. 0000054768 00000 n SECTION 1.4 ELEMENTARY OPERATIONS ON SETS 3 Proof. Set Theory Basics.doc 1.4. De nition 1.7 (Ordered Pair). P!\$� 0000041460 00000 n 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. Then ffag;fa;bgg= ffag;fa;agg= ffag;fagg= ffagg Since ffagg= ffcg;fc;dggwe must have fag= fcgand fag= fc;dg. , 2. 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)]. 0000070658 00000 n 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. Informal Proof. 0000042018 00000 n 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 This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. {]xKA}�a\0�;��O`�d�n��8n��%{׆P�;�PL�L>��бL�~ P. T. Johnstone, ‘Notes on Logic & Set Theory’, CUP 1987 2. startxref 0000079248 00000 n The Axiom of Pair, the Axiom of Union, and the Axiom of 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 4. 0000041632 00000 n trailer 0000011497 00000 n In 1874 Cantor had shown that there is a one-to-one correspondence between the natural numbers and the algebraic numbers. For example {x|xis real and x2 =−1}= 0/ By the deﬁnition of subset, given any set A, we must have 0/ ⊆A. {=���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^� 0 and most books of set theory contain im portant parts of mathe matical logic. In mathematics, the notion of a set is a primitive notion. 0000070486 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.