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. 0000054768 00000 n Then ffag;fa;bgg= ffag;fa;agg= ffag;fagg= ffagg Since ffagg= ffcg;fc;dggwe must have fag= fcgand fag= fc;dg. >> 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 %%EOF 0000078112 00000 n 0000072804 00000 n P!\$� t IExercise 7 (1.3.7). SECTION 1.4 ELEMENTARY OPERATIONS ON SETS 3 Proof. IV. in set theory, one that is important for both mathematical and philosophical reasons. Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. The Axiom of Pair, the Axiom of Union, and the Axiom of 0000077155 00000 n 0000041931 00000 n 14 Chapter 1 Sets and Probability Empty Set The empty set, written as /0or{}, is the set with no elements. 0000039958 00000 n !�MI��N���}��aP����Hd�K�Y�(�o�_HZ�٥���&�2 \�- This proves that P.X/“X, and P.X/⁄Xby the Axiom of Extensionality. Informal Proof. They are not guaran-teed to be comprehensive of the material covered in the course. {=���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^� (Caution: sometimes ⊂ is used the way we are using ⊆.) 0000003562 00000 n Set theory is a basis of modern mathematics, and notions of set theory are used in all formal descriptions. 0000055776 00000 n 0000041887 00000 n In mathematics, the notion of a set is a primitive notion. 0000011497 00000 n Any object which is in a set is called a member of the set. They are meta-statements about some propositions. Primitive Concepts. 0000003293 00000 n 0000010201 00000 n The study of these topics is, in itself, a formidable task. SECTION 1.4 ELEMENTARY OPERATIONS ON SETS 3 Proof. 0000070658 00000 n 0000069809 00000 n We study two types of relations between statements, implication and equivalence. Conditional Proof. 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 1874 Cantor had shown that there is a one-to-one correspondence between the natural numbers and the algebraic numbers. 0 0000011807 00000 n LOGIC AND SET THEORY A rigorous analysis of set theory belongs to the foundations of mathematics and mathematical logic. 4. axiomatic set theory with urelements. In mathematics, the notion of a set is a primitive notion. lX�Å xref 1. 0000002324 00000 n Suppose a= b. �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���*�,�� and most books of set theory contain im portant parts of mathe matical logic. Primitive Concepts. D. Van Dalen, ‘Logic and Structure’, Springer-Verlag 1980 (good for Chapter 4) 3. 0000038686 00000 n Proof by Counter Example. 0000075834 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� Clearly if a= cand b= dthen ha;bi= ffag;fa;bgg= ffcg;fc;dgg= hc;di 1. 0000073034 00000 n %PDF-1.3 %���� Indirect Proof. 0000041289 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 Universal and Existential Quantifiers. EXAMPLE 1 Finding Subsets Find all the subsets of {a,b,c}. Subsets A set A is a subset of a set B iff every element of A is also an element of B. Multiple Quantifiers. �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 For example {x|xis real and x2 =−1}= 0/ By the deﬁnition of subset, given any set A, we must have 0/ ⊆A. 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. {]xKA}�a\0�;��O`�d�n��8n��%{׆P�;�PL�L>��бL�~ The subjects of register machines and random access machines have been dropped from Section 5.5 Chapter 5. x�b```b``_���� �� Ȁ ��@����� ��bT; �}a~��ǯ��EO��z0�XN^�t[ut�\$. 0000039153 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. Such a relation between sets is denoted by A ⊆ B. 0000079248 00000 n , 2. 2. 0000047721 00000 n 0000001631 00000 n 0000041801 00000 n /Length 2960 0000056396 00000 n 0000042018 00000 n 0000041632 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 2 0 obj 0000045997 00000 n endstream endobj 1656 0 obj<>/W[1 1 1]/Type/XRef/Index[78 1514]>>stream Introduction to Logic and Set Theory-2013-2014 General Course Notes December 2, 2013 These notes were prepared as an aid to the student. Formal Proof. 0000080242 00000 n 0000064013 00000 n Set Theory Basics.doc 1.4. The problem actually arose with the birth of set theory; indeed, in many respects it stimulated the birth of set theory. .6�⊫�Ţ1o�/A���F�\���6f=iE��i�K��Lٛ�[�n&]=�x�Wȥ��噅Ak5��z�I��� Set Theory and Logic: Fundamental Concepts (Notes by Dr. J. Santos) A.1. 0000072849 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. LOGIC AND SET THEORY 1.2 Relations between Statements Strictly speaking, relations between statements are not formal statements them-selves. 0000047249 00000 n Unique Existence. %PDF-1.2 1592 65 0000042252 00000 n Predicate Logic and Quantifiers. 3. !���}�&)�MO8�eL6uFoJ��:�#@�f�� �N`�`���RK���yD�}c~���'�*n��E��Ij�Tl 0000002654 00000 n For our purposes, it will sufce to approach basic logical concepts informally. t IExercise 7 (1.3.7). 0000063750 00000 n 0000021855 00000 n constructive set theory was called by Hilbert), at least we should know what we are m1ssmg. It only remains to de ne ha;biin terms of set theory. Negation of Quantified Predicates. Chapter 1 Set Theory 1.1 Basic deﬁnitions and notation A set is a collection of objects. 0000022533 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. 0000039789 00000 n 0000076098 00000 n The Axiom of Pair, the Axiom of Union, and the Axiom of 0000022861 00000 n Predicates. These notes were prepared using notes from the course taught by Uri Avraham, Assaf Hasson, and of course, Matti Rubin. 1594 0 obj<>stream 0000070486 00000 n 0000075927 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. Mathematical Induction. Methods of Proof. 0000023682 00000 n stream logic has now taken on a life of its own, and also thrives on many interactions with other areas of mathematics and computer science. 0000055948 00000 n 0000057132 00000 n A. Hajnal & P. Hamburger, ‘Set Theory’, CUP 1999 (for cardinals and ordinals) 4. The empty set can be used to conveniently indicate that an equation has no solution. 0000047470 00000 n startxref 0000065343 00000 n Basics of Set Theory and Logic S. F. Ellermeyer August 18, 2000 Set Theory Membership A setis a well-defined collection of objects. 0000000016 00000 n 0000075488 00000 n Set Theory and Logic Supplementary Materials Math 103: Contemporary Mathematics with Applications A. Calini, E. Jurisich, S. Shields c 2008. 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. V. Naïve Set Theory. 1592 0 obj<> endobj << 0000041460 00000 n 0000003046 00000 n x���A 0ð4�v\Gcw��������z�C. 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. 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)].