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Hauskrecht CS 441 Discrete Mathematics for CS Lecture 7 Milos Hauskrecht milos@cs.pitt.edu 5329 Sennott Square Sets and set operations CS 441 Discrete mathematics for CS M. Hauskrecht Basic discrete structures • Discrete math = – study of the discrete structures used to represent discrete objects Print out this PDF worksheet and use it as a math test to review at home or in the classroom. 121 0 obj endobj 0 21 0 obj endobj 44 0 obj /MediaBox [0 0 612 792] endobj Sets are one of the most fundamental concepts in mathematics. 49 0 obj endobj 96 0 obj 72 0 obj (Exercises 1.6) (Exponents ) 16 0 obj (�^��,a�>�����u�W�z�*j�! (Addition ) (Exercises 2.2) endobj stream endstream endobj startxref Set Theory Basic building block for types of objects in discrete mathematics. P�yC�������G��p�I���$y�'��pR��[z�eO endobj /D [138 0 R /XYZ 36 765.938 null] (Real Numbers) 116 0 obj << /S /GoTo /D (section.2.3) >> endobj << /S /GoTo /D (section.1.8) >> endobj repeat represent irrational numbers. 81 0 obj endobj (Exercises 1.9) (Exercises 1.2) Many different systems of axioms have been proposed. h�bbd``b`�$�C�`���@�+#��#1�Ɗ *� 105 0 obj 93 0 obj << /S /GoTo /D (subsection.1.5.1) >> 89 0 obj (Order of Operations) endobj endobj Zermelo-Fraenkel set theory (ZF) is standard. (Exercises 1.1) 92 0 obj 83 0 obj <>/Filter/FlateDecode/ID[<7699FE2A76498BA3504AB9257FEAFED9>]/Index[77 17]/Info 76 0 R/Length 53/Prev 67195/Root 78 0 R/Size 94/Type/XRef/W[1 2 1]>>stream The diagram below shows the relationship between the sets … 1. �P%��R��X���`�YC�R 12 0 obj 61 0 obj endobj endobj �]� >�a/��a�vw4x���g���w�O�nn�� ��IR�'���D܂����������2�s{&�r{�?��e��c޺�׵ �'��/�=Pˆ}��y;���.�i��>�g�t{��ۿ5x��������?���f����}2g��;��q?�6���|s���a��������/������o�{�}�=�����~��nݱ���G��|w�V ։&��]b���p�聯v�e����) D֎�9�2ͳg}NHh�>�sz�=w5{�y��\�6�s"n���3Pˆ}��y;���-be�g�:�О}��^y{���o�ytv�}�uo�D���4��TT�֋��H$�R/��ʅ��-�>P�>@X� >�i��S�Ő߇��~�*~���=�-�}*n����=T�{��y;����=�i���$�\~"��%������|����O~s+f��O��n{��l茶ߵH�$:����@bqǑ�c��]:�m]�|�j�| ��K~q��4��%G�%����@�q|���ک��1n���w1����}��|����z �v,��)0Ӭc|*(b�|�������!�_?�nNd��~������#������p��>���]��@�AZ}|��hC��>�T�� ޲v�n e�7 p�� ��� � 130 �� @�o!�� P�H~��y;����`&�6����3�!A�B\��;$�j�p�}�����^�M;�zm辿�]��{lZ$9m���ظPqFGJxWy���pW���]�|����Eo~����;Q��՜�;�������}qr���*.Lna��B��V�t���܂�;>n"�z �vL Eyr�4�D�(�$��A�p؇�H�;���p�1L����ֿ�b2B��T�țlp����P-���X��. 142 0 obj << 104 0 obj endobj *�1��'(�[P^#�����b�;_[ �:��(�JGh}=������]B���yT�[�PA��E��\���R���sa�ǘg*�M��cw���.�"M޻O��6����'Q`MY�0�Z:D{CtE�����)Jm3l9�>[�D���z-�Zn��l���������3R���ٽ�c̿ g\� 69 0 obj h�b```f``�d`b``Kg�e@ ^�3�Cr��N?_cN� � W���&����vn���W�}5���>�����������l��(���b E�l �B���f`x��Y���^F��^��cJ������4#w����Ϩ` <4� endobj xڍRKO�0��W�Hċ��I��x'zC��=�nL�c��x�P%ǵ�}�����-D( D��d=2b���Y��y"��%(;DP�?����`���o0 endobj �u�Q��y�V��|�_�G� ]x�P? endobj /ProcSet [ /PDF /Text ] 132 0 obj >> endobj endobj rb�L��`�Ρ��>���o�i�n6� endobj 65 0 obj 97 0 obj endobj /Parent 149 0 R 73 0 obj << /S /GoTo /D (subsection.1.4.1) >> "�Wk��αs�[[d�>7�����* !BP!����P�K*�8 �� ��..ؤȋ29�+MJR:��!�z2׉I 9�A�cZ� ��sIeІ�O5�Rz9+�U�͂�.�l���r8\���d�Vz ��-1���N�J�p�%�ZMn��͟�k����Z��Q����:�l �9���5�"d�|���#�MW���N�]�?�g;]�����.����t������g��ܺSj�ڲ��ܥ�5=�n|l�Ƥy��7���w?��dJ͖��%��ŽH�E1/�گ�u�߰�l?�WY�O��2�mZ�'O /Filter /FlateDecode 2. perform the set operations a. union of sets; b. intersection of sets. The set of all rational numbers together with the set of irrational numbers is called the set of real numbers. 152 0 obj << 128 0 obj 113 0 obj 140 0 obj << Basic Concepts of Set Theory. (Exercise 1.5) << /S /GoTo /D (section.1.5) >> 93 0 obj <>stream endstream endobj 78 0 obj <> endobj 79 0 obj <> endobj 80 0 obj <>stream 1.1. 138 0 obj << 64 0 obj Pre-requisite Concepts: Whole Numbers, definition of sets, Venn diagrams Objectives: In this lesson, you are expected to: 1. describe and define a. union of sets; b. intersection of sets. endobj endobj 29 0 obj Because the fundamentals of Set Theory are known to all mathemati-cians, basic problems in the subject seem elementary. 85 0 obj (Solutions to Exercises) << /S /GoTo /D (section.1.9) >> 9 0 obj 1. 124 0 obj endobj 125 0 obj endobj h��UM��6��W�Q* �_"��8�A}h-��E^[^k㵼��m~H�{3CR�� ����L��p�7�O����Z �5���@W'�DŽ�-%� For any two sets X and Y, either there is a one-to-one function from endobj 100 0 obj endobj endobj ]�0�yg �$y����@ڰ�,G�⿐� << /S /GoTo /D (section.1.3) >> endobj Primitive Concepts. A set is just a collection of things. 117 0 obj (Exercises 1.4) Set (mathematics) From Wikipedia, the free encyclopedia A set in mathematics is a collection of well defined and distinct objects, considered as an object in its own right. << /S /GoTo /D (section.1.7) >> !�w �0c�o5����s�K7f���� Mathematics has a superbly efficient language by means of which vast amounts of information can be elegantly expressed in a few formal definitions and theorems. endobj endobj << /S /GoTo /D (subsection.1.9.1) >> /Type /Page << /S /GoTo /D (chapter.1) >> 4 CS 441 Discrete mathematics for CS M. Hauskrecht Equality Definition: Two sets are equal if and only if they have the same elements. 37 0 obj 5 0 obj