my slides on discrete mathematics

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  • Diskrete Mathematik fr Informatik(SS 2018)

    Martin Held

    FB ComputerwissenschaftenUniversitt Salzburg

    A-5020 Salzburg, Austriaheld@cosy.sbg.ac.at

    9. Februar 2018

    Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    mailto:held@cosy.sbg.ac.at

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Personalia

    LVA-Leiter (VO+PS): Martin Held.Email-Adresse: held@cosy.sbg.ac.at.

    Basis-URL: https://www.cosy.sbg.ac.at/held.Bro: Universitt Salzburg, Computerwissenschaften, Zi. 1.20,

    Jakob-Haringer Str. 2, 5020 Salzburg-Itzling.Telefonnummer (Bro): (0662) 8044-6304.Telefonnummer (Sekr.): (0662) 8044-6328.

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 2

    mailto:held@cosy.sbg.ac.athttps://www.cosy.sbg.ac.at/~held

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Personalia

    LVA-Leiter (PS): Peter Palfrader.Email-Adresse: palfrader@cosy.sbg.ac.at.

    Bro: Universitt Salzburg, Computerwissenschaften, Zi. 0.28,Jakob-Haringer Str. 2, 5020 Salzburg-Itzling.

    Telefonnummer (Bro): (0662) 8044-6326.Telefonnummer (Sekr.): (0662) 8044-6328.

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 3

    mailto:palfrader@cosy.sbg.ac.at

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Formalia

    LVA-URL (VO+PS): Basis-URL/teaching/diskrete_mathematik/dm.html.Allg. Information: Basis-URL/for_students.html.

    Abhaltezeit der VO: Donnerstag 8001055, mit etwa 20 Minuten Pause.Abhalteort der VO: T01, Computerwissenschaften, Jakob-Haringer Str. 2.

    Abhaltezeit des PS: Freitag 800945.Abhalteort des PS: T01+T03, Computerwissenschaften, Jakob-Haringer Str. 2.

    Tutorium: TBA,Computerwissenschaften, Jakob-Haringer Str. 2.

    Achtung das Proseminar ist prfungsimmanent!

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 4

    https://www.cosy.sbg.ac.at/~held/teaching/diskrete_mathematik/dm.htmlhttps://www.cosy.sbg.ac.at/~held/for_students.html

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Electronic Slides and Online Material

    In addition to these slides, you are encouraged to consult the WWW home-page ofthis lecture:

    https://www.cosy.sbg.ac.at/held/teaching/diskrete_mathematik/dm.html.

    In particular, this WWW page contains up-to-date information on the course, plus linksto online notes, slides and (possibly) sample code.

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 5

    https://www.cosy.sbg.ac.at/~held/teaching/diskrete_mathematik/dm.html

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    A Few Words of Warning

    I hope that these slides will serve as a practice-minded introduction to various aspectsof discrete mathematics which are of importance for computer science. I would like towarn you explicitly not to regard these slides as the sole source of information on thetopics of my course. It may and will happen that Ill use the lecture for talking aboutsubtle details that need not be covered in these slides! In particular, the slides wontcontain all sample calculations, proofs of theorems, demonstrations of algorithms, orsolutions to problems posed during my lecture. That is, by making these slidesavailable to you I do not intend to encourage you to attend the lecture on an irregularbasis.

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 6

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Acknowledgments

    These slides are a revised and extended version of a draft prepared by KamranSafdar. Included is material written by Christian Alt, Caroline Atzl, Michael Burian,Peter Gintner, Bernhard Guillon, Yvonne Hller, Stefan Huber, Sandra Huemer,Christian Lercher, Sebastian Stenger, Alexander Zrinyi. I also benefited fromcomments and suggestions made by Stefan Huber and Peter Palfrader.

    This revision and extension was carried out by myself, and I am responsible for allerrors.

    Salzburg, February 2018 Martin Held

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 7

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Legal Fine Print and Disclaimer

    To the best of our knowledge, these slides do not violate or infringe upon somebodyelses copyrights. If copyrighted material appears in these slides then it wasconsidered to be available in a non-profit manner and as an educational tool forteaching at an academic institution, within the limits of the fair use policy. Forcopyrighted material we strive to give references to the copyright holders (if known).Of course, any trademarks mentioned in these slides are properties of their respectiveowners.

    Please note that these slides are copyrighted. The copyright holder(s) grant you theright to download and print it for your personal use. Any other use, including non-profitinstructional use and re-distribution in electronic or printed form of significant portionsof it, beyond the limits of fair use, requires the explicit permission of the copyrightholder(s). All rights reserved.

    These slides are made available without warrant of any kind, either express orimplied, including but not limited to the implied warranties of merchantability andfitness for a particular purpose. In no event shall the copyright holder(s) and/or theirrespective employers be liable for any special, indirect or consequential damages orany damages whatsoever resulting from loss of use, data or profits, arising out of or inconnection with the use of information provided in these slides.

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 8

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Recommended Textbooks I

    S. Maurer, A. Ralston.Discrete Algorithmic MathematicsA.K. Peters, 3rd edition, Jan 2005; ISBN 978-1-56881-166-6

    K. Rosen.Discrete Mathematics and Its ApplicationsMcGraw-Hill, 7th edition, June 2011; ISBN 9780073383095

    B. Kolmann, R.C. Busby, S.C. Ross.Discrete Mathematical StructuresPearson India, 6th edition, 2015; ISBN 978-9332549593.

    K.A. Ross, C.R.B. Wright.Discrete MathematicsPearson Prentice Hall, 5th edition, Aug 2002; ISBN 9780130652478

    K. Bogart, C. Stein, R.L.S. Drysdale.Discrete Mathematics for Computer ScienceAddison-Wesley, March 2010; ISBN 978-0132122719.

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 9

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Recommended Textbooks II

    J. ODonnell, C. Hall, R. Page.Discrete Mathematics Using a ComputerSpringer, 2nd edition, 2006; ISBN 978-1-84628-241-6

    N.L. Biggs.Discrete MathematicsOxford University Press, 2nd edition, Feb 2003, reprinted (with corrections) 2008;ISBN 978-0-19-850717-8

    M. Smid.Discrete Structures for Computer Science: Counting, Recursion, and Probabilityhttp://glab.ca/~michiel/DiscreteStructures/DiscreteStructures.pdf, 2017

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 10

    http://glab.ca/~michiel/DiscreteStructures/DiscreteStructures.pdfhttp://glab.ca/~michiel/DiscreteStructures/DiscreteStructures.pdf

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Table of Content

    1 Introduction

    2 Formalism: Definitions and Theorem Proving

    3 Numbers and Basics of Number Theory

    4 Principles of Elementary Counting and Combinatorics

    5 Complexity Analysis and Recurrence Relations

    6 Graph Theory

    7 Cryptography

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 11

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    1 IntroductionWhat is Discrete Mathematics?Motivation

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 12

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    1 IntroductionWhat is Discrete Mathematics?Motivation

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 13

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    What is Discrete Mathematics?

    No universally accepted definition of the scope of DM exists . . .

    Typically, objects studied in DM can only assume discrete, separate values ratherthan values out of a continuum; sets of such objects are countable.

    The term discrete is often used in contrast to continuous, where the values ofobjects may vary smoothly.

    Calculus does not belong to DM, but to continuous mathematics.

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 14

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    What is Discrete Mathematics?

    No universally accepted definition of the scope of DM exists . . .

    Typically, objects studied in DM can only assume discrete, separate values ratherthan values out of a continuum; sets of such objects are countable.

    The term discrete is often used in contrast to continuous, where the values ofobjects may vary smoothly.

    Calculus does not belong to DM, but to continuous mathematics.

    c M. Held (Univ. Salzburg) Diskrete Mathematik (SS 2018) 14

  • Computational Geometry and Applications Lab

    UNIVERSITAT SALZBURG

    Typical Topics of Discrete Mathematics

    Depending on what is covered in other courses a variety of topics tends to bestudied within a course on DM:

    Logic and Boolean algebra,Mathematical language,Algebraic structures,Set theory,Functions and relations,Formal languages,Automata theory;

    Number theory,Proofs and mathematical reasoning,Counting and elementary combinatorics,Graph theory,Complexity theory,Encoding and cryptograph

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