Title: Bogazici University Department of Computer Engineering CmpE 220 Discrete Mathematics Overview Fall 2
1Bogazici UniversityDepartment of Computer
EngineeringCmpE 220 Discrete MathematicsOvervie
wFall 2005Haluk Bingöl
2About CmpE 220
3CmpE 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.
4CmpE 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
5CmpE 220 in This Semester
6CmpE 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
7CmpE 220 Discrete Computational Structures
(300) 3 Bingol - 2005 Fall
- Grading20 Midterm 120 Midterm 210
Quizzes10 Home works 40 Final
8About Slides
9Michael 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
10Course Overview
11Module 0Course Overview
12What 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)
14So, 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.
15Discrete 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
16Densely 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
18Discrete Structures Well Study
- Propositions
- Predicates
- Proofs
- Sets
- Functions
- (Orders of Growth)
- (Algorithms)
- Integers
- (Summations)
- (Sequences)
- Strings
- Permutations
- Combinations
- Relations
- Graphs
- Trees
- (Logic Circuits)
- (Automata)
19Some Notations Well Learn
20Uses 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
21Discrete 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.
22Some 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!
23Course 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))
24Topics 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
25Course 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.
26Have Fun!
- Many people find Discrete Mathematics more
enjoyable than, for example, Analysis - Applicable to just about anything
- Some nice puzzles
- Highly varied