Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 1/18 Module #0 - Overview Bogazici University Department of Computer Engineering CmpE 220 Discrete Mathematics Overview Fall 2005 Haluk Bingöl

2/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 2/18 Module #0 - Overview About CmpE 220

3/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 3/18 Module #0 - Overview CmpE 220 Discrete Computational Structures (3+0+0) 3 Catalog Data Propositional Logic and Proofs. Set Theory. Relations and Functions. Algebraic Structures. Groups and Semi-Groups. Graphs, Lattices, and Boolean Algebra. Algorithms and Turing Machines.

4/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 4/18 Module #0 - Overview CmpE 220 Discrete Computational Structures (3+0+0) 3 Course Outline A 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

5/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 5/18 Module #0 - Overview CmpE 220 in This Semester

6/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 6/18 Module #0 - Overview CmpE 220 Discrete Computational Structures (3+0+0) 3 Bingol Fall Instructor Dr. Haluk Bingöl, x7121, ETA 308 Dr. Haluk Bingöl Dr. Haluk Bingöl Assistant Evrim Itır Karaç, x7183, ETA 203 Evrim Itır Karaç Evrim Itır Karaç Albert Ali Salah, x4490, ETA 412 Web page Time/Room WFF 523 ETA Z04 Text Book Discrete Mathematics and Its Applications, 5e Rosen McGrawHill, 2003, [QA39.3 R67]

7/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 7/18 Module #0 - Overview CmpE 220 Discrete Computational Structures (3+0+0) 3 Bingol Fall Grading 20% Midterm #1 20% Midterm #2 10% Quizzes 10% Home works 40% Final

8/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 8/18 Module #0 - Overview About Slides

9/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 9/18 Module #0 - Overview Michael Frank’s slides adapted We’re not using all his lectures Various changes in those that we use Possibly some new lectures Your key resources Course’s web pageCourse’s web page Ken Rosen’s bookKen Rosen’s book

10/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 10/18 Module #0 - Overview Course Overview

Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 11/18 Module #0 - Overview Module #0: Course Overview

12/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 12/18 Module #0 - Overview What is Mathematics, really? It’s 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 truths about any and all well-defined concepts.

13/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 13/18 Module #0 - Overview

14/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 14/18 Module #0 - Overview So, what’s this class about? What are “discrete structures” anyway? “Discrete” (  “discreet”!) - Composed of distinct, separable parts. (Opposite of continuous.) discrete:continuous :: digital:analog “Structures” - Objects built up from simpler objects according to some definite pattern. “Discrete Mathematics” - The mathematical study of discrete objects and structures.

15/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 15/18 Module #0 - Overview Discrete Mathematics When using numbers, we’re 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

16/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 16/18 Module #0 - Overview Densely Ordered  ℚ,<  is densely ordered because  x  ℚ  y  ℚ (x≠y   z (x<z & z<y) ) Opposite of densely ordered: discretely ordered

17/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 17/18 Module #0 - Overview 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

18/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 18/18 Module #0 - Overview Discrete Structures We’ll Study PropositionsPropositions PredicatesPredicates ProofsProofs SetsSets FunctionsFunctions (Orders of Growth)(Orders of Growth) (Algorithms)(Algorithms) IntegersIntegers (Summations)(Summations) (Sequences)(Sequences) StringsStrings PermutationsPermutations CombinationsCombinations RelationsRelations GraphsGraphs TreesTrees (Logic Circuits)(Logic Circuits) (Automata)(Automata)

19/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 19/18 Module #0 - Overview Some Notations We’ll Learn

20/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 20/18 Module #0 - Overview 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

21/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 21/18 Module #0 - Overview Discrete Math for Computing The basis of all of computing is: Discrete manipulations of discrete structures represented in memory. Discrete Math is the basic language and conceptual foundation for all of computer science.

22/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 22/18 Module #0 - Overview Some Examples Algorithms & data structuresAlgorithms & data structures Compilers & interpreters.Compilers & interpreters. Formal specification & verificationFormal specification & verification Computer architectureComputer architecture DatabasesDatabases CryptographyCryptography Error correction codesError correction codes Graphics & animation algorithms, game engines, etc.…Graphics & animation algorithms, game engines, etc.… DM is relevant for all aspects of computing!DM is relevant for all aspects of computing!

23/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 23/18 Module #0 - Overview Course Outline (as per Rosen) 1.Logic (§1.1-4) 2.Proof methods (§1.5) 3.Set theory (§1.6-7) 4.Functions (§1.8) 5.(Algorithms (§2.1)) 6.(Orders of Growth (§2.2)) 7.(Complexity (§2.3)) 8.Number theory (§2.4-5) 9.Number theory apps. (§2.6) 10.(Matrices (§2.7)) 11.Proof strategy (§3.1) 12.(Sequences (§3.2)) 13.(Summations (§3.2)) 14.(Countability (§3.2)) 15.Inductive Proofs (§3.3) 16.Recursion (§3.4-5) 17.Program verification (§3.6) 18.Combinatorics (ch. 4) 19.Probability (ch. 5) 20.(Recurrences (§6.1-3)) 21.Relations (ch. 7) 22.Graph Theory (chs. 8+9) 23.Boolean Algebra (ch. 10) 24.(Computing Theory (ch.11))

24/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 24/18 Module #0 - Overview Topics Not Covered Other topics we might not get to this term: Boolean circuits (ch. 10)Boolean circuits (ch. 10) - You could learn this in more depth in a digital logic course. Models of computing (ch. 11)Models of computing (ch. 11) - Many of these are obsolete for engineering purposes now anyway Linear algebra (not in Rosen, see Math dept.)Linear algebra (not in Rosen, see Math dept.) - Advanced matrix algebra, general linear algebraic systems

25/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 25/18 Module #0 - Overview 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.

26/18 Based on Rosen, Discrete Mathematics & Its Applications, 5e (c) Michael P. Frank Modified by (c) Haluk Bingöl 26/18 Module #0 - Overview Have Fun! Many people find Discrete Mathematics more enjoyable than, for example, Analysis: Applicable to just about anything Some nice puzzles Highly varied