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Bogazici University Department of Computer Engineering CmpE 220 Discrete Mathematics Overview Fall 2

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Title: Bogazici University Department of Computer Engineering CmpE 220 Discrete Mathematics Overview Fall 2


1
Bogazici UniversityDepartment of Computer
EngineeringCmpE 220 Discrete MathematicsOvervie
wFall 2005Haluk Bingöl
2
About CmpE 220
3
CmpE 220 Discrete Computational Structures
(300) 3
  • Catalog DataPropositional Logic and Proofs. Set
    Theory. Relations and Functions. Algebraic
    Structures. Groups and Semi-Groups. Graphs,
    Lattices, and Boolean Algebra. Algorithms and
    Turing Machines.

4
CmpE 220 Discrete Computational Structures
(300) 3
  • Course OutlineA course in discrete mathematics
    should teach students how to work with discrete
    (meaning consisting of distinct or unconnected
    elements as opposed to continuous) structures
    used to represent discrete objects and
    relationships between these objects. These
    discrete structures include sets, relations,
    graphs, trees, and finite-state machines. Topics
  • Logic, Sets, and Functions
  • Methods of Proof
  • Recurrence Relations
  • Binary Relations
  • Graphs
  • Trees
  • Algebraic Structures
  • Introduction to Languages and Grammars

5
CmpE 220 in This Semester
6
CmpE 220 Discrete Computational Structures
(300) 3 Bingol - 2005 Fall
  • InstructorDr. Haluk Bingöl, bingol_at_boun.edu.tr,
    x7121, ETA 308
  • Assistant Evrim Itir Karaç, itir.karac_at_boun.edu.t
    r, x7183, ETA 203
  • Albert Ali Salah, salah_at_boun.edu.tr, x4490, ETA
    412
  • Web pagehttp//www.cmpe.boun.edu.tr/courses/cmpe2
    20/fall2005
  • Time/RoomWFF 523 ETA Z04
  • Text BookDiscrete Mathematics and Its
    Applications, 5e Rosen McGrawHill, 2003,
    QA39.3 R67

7
CmpE 220 Discrete Computational Structures
(300) 3 Bingol - 2005 Fall
  • Grading20 Midterm 120 Midterm 210
    Quizzes10 Home works 40 Final

8
About Slides
9
Michael Franks slides adapted
  • Were not using all his lectures
  • Various changes in those that we use
  • Possibly some new lectures
  • Your key resources
  • Courses web page
  • Ken Rosens book
  • http//www.mhhe.com/math/advmath/rosen/r5

10
Course Overview
11
Module 0Course Overview
12
What is Mathematics, really?
  • Its not just about numbers!
  • Mathematics is much more than that
  • But, these concepts can be about numbers,
    symbols, objects, images, sounds, anything!

Mathematics is, most generally, the study of any
and all certain truthsabout any and all
well-defined concepts.
13
(No Transcript)
14
So, whats this class about?
  • What are discrete structures anyway?
  • Discrete (? discreet!) - Composed of
    distinct, separable parts. (Opposite of
    continuous.) discretecontinuous
    digitalanalog
  • Structures - Objects built up from simpler
    objects according to some definite pattern.
  • Discrete Mathematics - The mathematical study
    of discrete objects and structures.

15
Discrete Mathematics
  • When using numbers, were much more likely to use
    N (natural numbers) and Z (whole numbers) than Q
    (fractions) and R (real numbers).
  • Reason Q and R are densely ordered
  • This notion can be defined precisely

16
Densely Ordered
  • ? Q,lt ? is densely ordered because?x? Q ?y? Q
    (x?y ? ?z (xltz zlty) )
  • Opposite of densely ordereddiscretely ordered

17
  • Yet, Q and R can be defined in terms of discrete
    concepts (as we have seen)
  • This means that Discrete Mathematics has no exact
    borders
  • Different books and courses treat slightly
    different topics

18
Discrete Structures Well Study
  • Propositions
  • Predicates
  • Proofs
  • Sets
  • Functions
  • (Orders of Growth)
  • (Algorithms)
  • Integers
  • (Summations)
  • (Sequences)
  • Strings
  • Permutations
  • Combinations
  • Relations
  • Graphs
  • Trees
  • (Logic Circuits)
  • (Automata)

19
Some Notations Well Learn

20
Uses of Discrete Math
  • Starting from simple structures of logic and set
    theory, theories are constructed that capture
    aspects of reality
  • Physics (see diagram)
  • Biology (DNA)
  • Common-sense reasoning (logic)
  • Natural Language (trees, sets, functions, ..)
  • Anything that we want to describe precisely

21
Discrete Math for Computing
  • The basis of all of computing isDiscrete
    manipulations of discrete structures represented
    in memory.
  • Discrete Math is the basic language and
    conceptual foundation for all of computer science.

22
Some Examples
  • Algorithms data structures
  • Compilers interpreters.
  • Formal specification verification
  • Computer architecture
  • Databases
  • Cryptography
  • Error correction codes
  • Graphics animation algorithms, game engines,
    etc.
  • DM is relevant for all aspects of computing!

23
Course Outline (as per Rosen)
  • Logic (1.1-4)
  • Proof methods (1.5)
  • Set theory (1.6-7)
  • Functions (1.8)
  • (Algorithms (2.1))
  • (Orders of Growth (2.2))
  • (Complexity (2.3))
  • Number theory (2.4-5)
  • Number theory apps. (2.6)
  • (Matrices (2.7))
  • Proof strategy (3.1)
  • (Sequences (3.2))
  • (Summations (3.2))
  • (Countability (3.2))
  • Inductive Proofs (3.3)
  • Recursion (3.4-5)
  • Program verification (3.6)
  • Combinatorics (ch. 4)
  • Probability (ch. 5)
  • (Recurrences (6.1-3))
  • Relations (ch. 7)
  • Graph Theory (chs. 89)
  • Boolean Algebra (ch. 10)
  • (Computing Theory (ch.11))

24
Topics Not Covered
  • Other topics we might not get to this term
  • Boolean circuits (ch. 10)
  • - You could learn this in more depth in a
    digital logic course.
  • Models of computing (ch. 11)
  • - Many of these are obsolete for engineering
    purposes now anyway
  • Linear algebra (not in Rosen, see Math dept.)
  • - Advanced matrix algebra, general linear
    algebraic systems

25
Course Objectives
  • Upon completion of this course, the student
    should be able to
  • Check validity of simple logical arguments
    (proofs).
  • Check the correctness of simple algorithms.
  • Creatively construct simple instances of valid
    logical arguments and correct algorithms.
  • Describe the definitions and properties of a
    variety of specific types of discrete structures.
  • Correctly read, represent and analyze various
    types of discrete structures using standard
    notations.

26
Have Fun!
  • Many people find Discrete Mathematics more
    enjoyable than, for example, Analysis
  • Applicable to just about anything
  • Some nice puzzles
  • Highly varied
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