Discrete Mathematics

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Discrete Mathematics

• ???
• School of Information Technology
• Sungshin W. University
• (Special Thanks to the original author of this
• lecture notes, Prof. ???)

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Text and References

• Text
• Discrete Mathematics and Its Applications, 6th
  Edition, Kenneth H. Rosen

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Professor Info

• ???
• jwlee_at_sungshin.ac.kr
• ??? 811?
• ?? ? ??
• ???? ??
• E-mail? ??
• Course Home
• http//cs.sungshin.ac.kr/jwlee/dm2/dm.htm

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Grading

• Mid 35
• Final Exam 35
• Home Work 10
• Handwriting word processor ?? ? ?.
• ?? 20

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What is Mathematics, really?

• Its 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.
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So, whats this class about?

• What are discrete structures anyway?
• Discrete (? discreet!) - Composed of
  distinct, seperable parts. (Opposite of
  continuous.) discretecontinuous
  digitalanalog
• Structures - objects built up from simpler
  objects according to a definite pattern.
• Discrete Mathematics - The study of discrete,
  mathematical objects and structures.

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Discrete Structures Well Study

• Propositions
• Predicates
• Sets
• (Discrete) Functions
• Orders of Growth
• Algorithms
• Integers
• Proofs

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

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Relationships Between Structures

• ? Can be defined in terms of

Programs
Proofs
Groups
Trees
Operators
Propositions
Complex numbers
Graphs
Real numbers
Strings
Functions
Integers
Naturalnumbers
Matrices
Relations
Sequences
Infiniteordinals
Bits
n-tuples
Vectors
Sets
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Why Study Discrete Math?

• The basis of all of digital information
  processing Discrete manipulations of discrete
  structures represented in memory.
• Its 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!

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Uses 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!

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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)

1. Counting (5.1-5.3)
2. Discrete Probability (6.1)
3. Recurrence Relations (7.1, 7.3)
4. Relations (8.1-8.6)
5. Graph (Tree) Theory (9.1-9.5, 10.1-10.3)
6. Boolean Algebra (11.1-11.4)

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Symbol, Notation (??, ???)

• 1
• What is this ?
• Think your name, symbol , seoul,
• 22 ?
• Seoul , Korea ?

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Basics 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 x4, y6. false when x5,
  y6.
• x y lt x y, true for positive integers x and
  y
• I am a student. Iamastudent.
• for (i1 ilt10 i) xx1

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Some Notations Well Learn