6 0 obj [ 0 1 ] /Extend [ false false ] /Function 17 0 R >> 948 ód2��� �m�%�C���~�\^�Zoi��9�t�sJ{�N�m,�qG'�sM�sD��Yٵ�z��=81AKm Functions Deﬁnitions and Examples Functions Let Aand B be sets. %��������� In a perhaps unsympathetic view, the standard presenta-tions (and there are many )the material in the course is treated as a discrete collection of so many techniques that the students must master for further stud-ies in Computer Science. /MediaBox [0 0 612 792] Compute permutations and combinations of a set, and interpret the meaning in the context of the particular application. /Length 352 13 0 obj << A function from Ato B is an association to each element in Aof exactly one element in B. Explain with examples the basic terminology of functions, relations, and sets. Apply formal methods of symbolic propositional and predicate logic. << /Length 5 0 R /Filter /FlateDecode >> Solve a variety of basic recurrence equations. ���ެJf���E �;5n�O8`$�b��"1��&.���=-dUFhOv��y��&�KQ�Kp�H��1�6�Ԅ���*�|`'^�xGc��j�Z���A��_Թ}'�f����� `V��Ģ� �߫�Kn@2�����ej*���"���/�-d�Ҿhe��Ʒd;�L&o�@�#�4���5-mhg�%n���_��o��f�/�K?�g�P�0qW�7/&+��jy�z�aN���[`��H˯��j��]�y�5���wM�]�o�LS�A���#��c"0xq`ӑw/�CvPg�� \Ĥ�����Z�?�^�5��dr+Rҧ-x�����^��+"@�Q���_1�ϋ�����O��i��e������IeP�=O5�T�|ݒX~u�=R}�2���ϛ�i�/G�j��\K��J�o˚�4���%���_��=�Y߭�_�����1M�e /Font << /F16 6 0 R /F17 9 0 R >> Discrete mathematics is a required course in the undergraduate Computer Science curriculum. Demonstrate different traversal methods for trees and graphs. << /ColorSpace 16 0 R /ShadingType 2 /Coords [ 0 76200 9144000 76200 ] /Domain *���r����G����&��A��$�Ť���S;�v�-b4���,�p،��?��^fjg ��E���� Ρ��m��ݑ�ȝV�Ş�����d>��$��I�D�6���8��f����Co�Z[DG����M����"����E�I�yD>�M/����x�>*C�)�ݣ��3C�L~�ԑ��yd: Sci. Discrete Structures (DS) Discrete structures are foundational material for computer science. Toggle navigation. Set Theory for Computer Science Part IA Comp. Discrete structures include important material from such areas as set … Discrete Structures; Sets in Discrete Structures; Friday, 18 July 2014. This page was last modified 20:27, 15 August 2008. >> /Font << /TT4 15 0 R /TT2 13 0 R /TT3 14 0 R /TT1 12 0 R >> /XObject << By foundational we mean that relatively few computer scientists will be working primarily on discrete structures, but that many other areas of computer science require the ability to work with concepts from discrete structures. %PDF-1.3 Describe how formal tools of symbolic logic are used to model real-life situations, including those arising in computing contexts such as program correctness, database queries, and algorithms. x�U�?O�0��~���DL|�����P��A��J�b�I����s�#�S|�{�w�~�W��\g�0�Jfu�q!���d�Ye����;� G��\��؞�R��S3��R�b�6$������`�I��D?�-�V(i��M9�[ Use formal logic proofs and/or informal but rigorous logical reasoning to, for example, predict the behavior of software or to solve problems such as puzzles. Notions of implication, converse, inverse, contrapositive, negation, and contradiction. HK��_,��Z�^s�IW���Q~�XޥvCe �u-P7w�r?�{��(�x� Differentiate between dependent and independent events. Model problems in computer science using graphs and trees. Relate the ideas of mathematical induction to recursion and recursively defined structures. >> endobj ��������4܉�cf�����V�٘����Q��[Z�� V ^O�ol9�,^Em����ړ�)����}�Oz�-&��� �a�-ޓ���������6nq|�� f��A^�T�ruyt9��H-6�Hu�(z�P��T p�p��g61RA^�� T�p���s�8�cd���^8�0��0;�V�#\"��7T+�@����!e7~M��`�~�)v��92X%� =n��9v��fr|u ��v�����`��� ���t���L���:_f�����Mlx�?��"�A�R�H�$h(+�23nk��dmO) �ƶ���B3�i�Ь��p;`�4 �qg;�0t��0���Ц/.���1U��mR�����s�G|��W$�q�BI�����]cs�|L�Z0�����l�(�yw��QeȀ�_!�Ls@!$�%��mH^q�Mo�OR�����r�LKb�8��A���Q�z�W?�m�Oendstream Two sets are equal if and only if they have the same elements. Describe the importance and limitations of predicate logic. However, the decision about where to draw the line between this area and the Algorithms and Complexity area (AL) on the one hand, and topics left only as supporting mathematics on the other hand, was inevitably somewhat arbitrary. For example, an ability to create and understand a proof is important in virtually every area of computer science, including (to name just a few) formal specification, verification, databases, and cryptography. This page has been accessed 49,051 times. Copyright © 2008, ACM, Inc. and IEEE, Inc. SIGCSE Committee Report On the Implementation of a Discrete Mathematics Course, http://wiki.acm.org/cs2001/index.php?title=Discrete_structures, To give feedback on this area of revision, go to, Functions (surjections, injections, inverses, composition), Relations (reflexivity, symmetry, transitivity, equivalence relations), Sets (Venn diagrams, complements, Cartesian products, power sets). /Resources 1 0 R 2 0 obj << Identify the difference between mathematical and strong induction and give examples of the appropriate use of each. x�mTKo�0��W�h�*Z���n���(����!��5Xbg����~$���Q�ȏ���E��]yd��ճ(. For example, an ability to create and understand a proof—either a formal symbolic proof or a less formal but still mathematically rigorous argument—is essential in formal specification, in verification, in databases, and in cryptography. /Parent 10 0 R �v�6�0|J�e�@Y��I8d�N�@��pJ�mЬc��BQG Q���i:�Ȁ'���b�9v��.��F:۹@�n��!�C$H{1�q�N�1��e�@�3�L����U�q�M|^�TM�*��ү�,Uc�My�@5�Wg59�.җ$zY���E��J:^:�|��-�b�nn� C:�#������S�)�)�M-�x ���i� ����X}�c�P����f^$����'�oZ��X@W^�p��`��T��Ľ��}-���%�8`��\`�Q�s,��>ۢt3{���#W����� �h�T=�c00��i���)��? 4 0 obj Inﬁnite Sets George Voutsadakis (LSSU) Discrete Structures for Computer Science August 2018 2 / 66. of Computer Science, Lund University 2 axiomatic vs naïve set theory Zermelo-Fraenkel Set Theory w/Choice (ZFC) extensionality regularity specification union replacement infinity power set choice This course will be about “naïve” set theory. A set is said to contain its elements. /Im1 10 0 R >> /Shading << /Sh1 9 0 R >> >> Many common data structures used in computer science have recursive definitions Example: Full binary trees Base step: A single root node r is a full binary tree Recursive step:If T 1and T 2are disjoint full binary trees with roots r 1and r 2, then introducing a new root r connected to r 1and r … Normal forms (conjunctive and disjunctive). Outline the basic structure of and give examples of each proof technique described in this unit. stream Discrete Structures in Computer Science 1: Sets Jörn W. Janneck, Dept. ���� �nD�������h��֒�'�QF���Z��ǧ;~�Ԙ�{F@��ÿ�����F[+��/`I)��g�����z�%����~0ҍxjl��F��� !�*���g�iژ��we�?2y�� v�������In�"���m���J��2Ó�`��H��^O%W�{�u���5�G�PuUc4tT�`��5 Unknown . Apply the binomial theorem to independent events and Bayes’ theorem to dependent events. Perform the operations associated with sets, functions, and relations. Discrete Structures for Computer Science George Voutsadakis1 1Mathematics and Computer Science Lake Superior State University ... Let Aand B be sets. stream A function from Ato B is an association to each element in … Lecture Notes Glynn Winskel c 2005, 2006 Glynn Winskel June 26, 2006 . 9 0 obj 3 0 obj << 10 0 obj e!��ཎ÷�D���(j�r6���OL��9x3�K�I�$=������A��a72d�B���R�K� Apply the tools of probability to solve problems in areas such as the Monte Carlo method, the average case analysis of algorithms, and hashing. /ProcSet [ /PDF /Text ] endobj endobj Demonstrate basic counting principles, including uses of diagonalization and the pigeonhole principle. 21:59 No comments: Definition : Sets. The objects in a set are called the elements, or members,of the set. In April 2007, the SIGCSE Committee on the Implementation of a Discrete Mathematics Course released a report detailing three models for a one-semester discrete mathematics course to meet the criteria articulated in CC2001; these models remain applicable under the slightly revised suggestions in this interim report. Here you can access and discuss Multiple choice questions and answers for various compitative exams and interviews. Relate graphs and trees to data structures, algorithms, and counting. x� ��8��{O���S�����d�1Cwʬc�2v�rXn����g���D`"0�L&���D`"0�L&���D`"0�L&���D`"0�L&���D��D�����;��TgV���# ���߿�������}v���$9^8��D`"����z. stream >> endobj As the field of computer science matures, more and more sophisticated analysis techniques are being brought to bear on practical problems. << /Type /Page /Parent 3 0 R /Resources 6 0 R /Contents 4 0 R >> The material in discrete structures is pervasive in the areas of data structures and algorithms but appears elsewhere in computer science as well. The material in discrete mathematics is pervasive in the areas of data structures and algorithms but appears elsewhere in computer science as well. h��O�!%7. endobj Finally, we note that while areas often have somewhat fuzzy boundaries, this is especially true for discrete structures. endobj We remind readers that there are vital topics from those two areas that some schools will include in courses with titles like "discrete structures" and "discrete mathematics"; some will require one course, others two. Functions Deﬁnitions and Examples Subsection 1 Deﬁnitions and Examples George Voutsadakis (LSSU) Discrete Structures for Computer Science August 2018 3 / 66.

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