discrete mathematics 이재원 school of information technology sungshin w. university (special...

Discrete MathematicsDiscrete Mathematics

이재원School of Information Technology

Sungshin W. University(Special Thanks to the original author of this

lecture notes, Prof. 홍기형 )

Text and ReferencesText and References

• Text– Discrete Mathematics and Its Applications, 6th

Edition, Kenneth H. Rosen

Professor InfoProfessor Info

• 이재원– jwlee@sungshin.ac.kr–수정관 811 호

• 질문 및 상담–수업시간 전후– E-mail 로 신청

• Course Home– http://cs.sungshin.ac.kr/~jwlee/dm2/dm.htm

GradingGrading

• Mid : 35%• Final Exam : 35%• Home Work : 10%– Handwriting : word processor 하지 말 것 .

• 출석 : 20%

What is Mathematics, really?What is Mathematics, really?

• It’s not just about numbers!• Mathematics is much more than that:

• But, the concepts can relate to numbers, symbols, visual patterns, or anything!

Mathematics is, most generally, the study of any and all absolutely certain truths about any and all perfectly well-defined concepts.

So, what’s So, what’s thisthis class about? class about?What are “discrete structures” anyway?• “Discrete” ( “discreet”!) - Composed of distinct,

seperable parts. (Opposite of continuous.) discrete:continuous :: digital:analog

• “Structures” - objects built up from simpler objects according to a definite pattern.

• “Discrete Mathematics” - The study of discrete, mathematical objects and structures.

Discrete Structures We’ll StudyDiscrete Structures We’ll Study• Propositions• Predicates• Sets• (Discrete) Functions• Orders of Growth• Algorithms• Integers • Proofs

• Summations• Permutations • Combinations• Relations• Graphs• Trees• Boolean Algebra• Logic Circuits

Relationships Between StructuresRelationships Between Structures

• “→” :≡ “Can be defined in terms of”

Sets

Sequences

n-tuples

MatricesNaturalnumbers

Integers

Relations

Functions

GraphsReal numbers

Complex numbers

Strings

Propositions

ProofsTreesOperators

Programs

Infiniteordinals Vectors

Groups

Bits

Why Study Discrete Math?Why Study Discrete Math?• The basis of all of digital information

processing: Discrete manipulations of discrete structures represented in memory.

• It’s the basic language and conceptual foundation of all of computer science.

• Discrete concepts are also widely used throughout math, science, engineering, economics, biology, etc., …

• A generally useful tool for rational thought!

SSWU MIPS Lab. Ki-Hyung Hong

Uses for Discrete Math in Computer ScienceUses for Discrete Math in Computer Science• Advanced algorithms &

data structures• Programming language

compilers & interpreters.• Computer networks• Operating systems• Computer architecture

• Database management systems

• Cryptography• Error correction codes• Graphics & animation

algorithms, game engines• Just about everything!

Course Outline (as per Rosen)Course Outline (as per Rosen)1. Logic (§1.1-1.4)2. Proof methods (§1.5-1.7)3. Set theory (§2.1-2.2)4. Functions (§2.3)5. Sequences & Summations

(§2.4)6. Algorithms (§3.1)7. Orders of Growth &

Complexity (§3.2-3.3)8. Number Theory

(§3.4-3.8)9. Recursion(§4.1-4.4)

10. Counting (§5.1-5.3)11. Discrete Probability (§6.1)12. Recurrence Relations (§7.1,

7.3)13. Relations (§8.1-8.6)14. Graph (Tree) Theory (§9.1-

9.5, 10.1-10.3)15. Boolean Algebra (§11.1-11.4)

Symbol, Notation (Symbol, Notation ( 기호기호 , , 표기표기법법 ))• 1– What is this ?

• Think: your name, symbol ‘+’, ‘seoul’, …

– 2+2 ?– “Seoul” + “, Korea” ?

Basics for studying Math. and LanguagesBasics for studying Math. and Languages• Understanding the semantics of Symbols and

Notations– 1, 2, 3, …, 9, 0– Positional semantics : 912 vs. 219

• Both are consisting the same 3 symbols, but are different.

– x, y (variables)• Each of them can have a value from a designated domain.• x + y = 10, true when x=4, y=6. false when x=5, y=6. • x + y < x * y, true for positive integers x and y

– I am a student. Iamastudent.

– for (i=1; i<10; i++) x=x+1;

Some Notations We’ll LearnSome Notations We’ll Learn

)(deg][)|(),,;(

][)(

) (mod modlcmgcd,/|max min,,,

)(:

||

)}(|{,,},,{)(

)(

1

][T0

1

1

1

1

vaRFEpnnnCr

naaa

mbabaO

aaxgfxfBAf

AABASTS

SxxPxaaxPx

xPxqpqpqpqpp

Rm

nijbk

n

ii

S

n

ii

n

ABAA

RNZ

Ο