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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– [email protected]–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
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