Introduction to Discrete Mathematics
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Transcript of Introduction to Discrete Mathematics
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Introduction to Discrete Mathematics
Lecture 1: Sep 1
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Basic Information
Course homepage: http://www.cse.cuhk.edu.hk/~chi/csc2110/
Newsgroup: cuhk.cse.csc2110. at news.cse.cuhk.edu.hk
Instructor: Lau, Lap Chi
Office hour: W7 (SHB 911)
Lectures: M7-8 (ERB LT), W6 (TYW LT)
Tutors: Chan Yuk Hei, Tom Fung Wai Shing, Isaac
Yung Chun Kong, Darek Zhang Zixi, Jesse
Tutorials: H5 (ERB 404) or H6 (EGB 404)
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Course Material
Textbook: Discrete Mathematics with Applications (DMA)
Author: Susanna S. Epp
Publisher:
Reference: Course notes from “mathematics for computer science”
http://courses.csail.mit.edu/6.042/spring07/
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Course Requirements
Homework, 20%
Midterm, 30%
Course project, 10%
Final Exam, 40%
Midterm: Oct 27 (Monday), M7-8
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Course Project
4 students in a group
Pick an interesting mathematical topic,
write a report from 5 to 10 pages.
Can use any references, but cite them.
Choose 3 groups to present, up to 5% bonus
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Tell an interesting story related to mathematics.
More about good topic and nice presentation, than mathematical difficulty.
A Project
Interesting or curious problems, interesting history
Surprising or elegant solutions
Nice presentation, easy to understand
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Checker
x=0
Start with any configuration with all men on or below the x-axis.
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Checker
x=0
Move: jump through your adjacent neighbour, but then your neighbour will disappear.
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Checker
x=0
Move: jump through your adjacent neighbour, but then your neighbour will disappear.
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Checker
x=0
Goal: Find an initial configuration with least number of men to jump up to level k.
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K=1
x=0
2 men.
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K=2
x=0
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K=2
x=0
4 men.
Now we have reduced to the k=1 configuration, but one level higher.
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K=3
x=0
This is the configuration for k=2, so jump two level higher.
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K=3
x=0
8 men.
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K=4
x=0
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K=4
x=0
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K=4
x=0
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K=4
x=0
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K=4
x=0
Now we have reduced to the k=3 configuration, but one level higher
20 men!
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K=5
a. 39 or below
b. 40-50 men
c. 51-70 men
d. 71- 100 men
e. 101 – 1000 men
f. 1001 or above
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Example 1
How to play Rubik Cube?
http://blog.sciencenews.org/mathtrek/2007/08/cracking_the_cube.html
Google: Rubik cube in 26 steps
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Example 2
The mathematics of paper folding
http://erikdemaine.org/foldcut/
http://www.ushistory.org/betsy/flagstar.html
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http://128.100.68.6/~drorbn/papers/PDI/
Example 3
3D-images
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Magic tricks
More games, more paper folding, etc
Famous paradoxes
Prime numbers
Game theory
Project Ideas
Deadline: November 17.
http://www.cse.cuhk.edu.hk/~chi/csc2110/project.html
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Why Mathematics?
Design efficient computer systems.
•How did Google manage to build a fast search engine?
•What is the foundation of internet security?
algorithms, data structures, database,
parallel computing, distributed systems,
cryptography, computer networks…
Logic, number theory, counting, graph theory…
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Topic 1: Logic and Proofs
Logic: propositional logic, first order logic
Proof: induction, contradiction
How do computers think?
Artificial intelligence, database, circuit, algorithms
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Topic 2: Number Theory
• Number sequence
• Euclidean algorithm
• Prime number
• Modular arithmetic
Cryptography, coding theory, data structures
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Topic 3: Counting
• Sets
• Combinations, Permutations, Binomial theorem
• Functions
• Counting by mapping, pigeonhole principle
• Recursions, generating functions
Probability, algorithms, data structures
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Topic 3: Counting
How many steps are needed to sort n numbers?
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Topic 4: Graph Theory
• Relations, graphs• Degree sequence, isomorphism, Eulerian graphs• Trees
Computer networks, circuit design, data structures
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What is discrete mathematics?
logic, sets, functions, relations, etc
Logic: artificial intelligence (AI), database, circuit design
Number theory: cryptography, coding theory
Counting: probability, analysis of algorithm
Graph theory: computer network, data structures
CSC 2100, ERG 2040, CSC 3130, CSC 3160
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2 2 2a b c
Familiar?
Obvious?
cb
a
Pythagorean theorem
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cb
a
Rearrange into: (i) a cc square, and then (ii) an aa & a bb square
Good Proof
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c
cc
a b
c
b-a
Good Proof
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cb
a b-a
b-a
Good Proof
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ba
a
ab-a
74 proofs in http://www.cut-the-knot.org/pythagoras/index.shtml
b
Good Proof
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Bad Proof
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Statement (Proposition)
Statement is either True or False
Examples:
Non-examples:
Hello.
How are you?
True
False
2 + 2 = 4
3 x 3 = 8
787009911 is a prime
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Logic Operators
P QQP
NOT::
AND::
P QQP
OR::
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Compound Statement
p = “it is hot” q = “it is sunny”
It is hot and sunny
It is not hot but sunny
It is neither hot nor sunny
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Exclusive-Or
coffee “or” tea exclusive-or
How to construct a compound statement for exclusive-or?
p q p q
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Check by Truth Table
p q
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Logical Equivalence
Two statements have the same truth table
De Morgan’s Law
De Morgan’s Law
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Simplifying Statement
A tautology is a statement that is always true.
A contradiction is a statement that is always false.
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Class Photos! Identify your face and send us your name and nicknames
Two Important Things
http://appsrv.cse.cuhk.edu.hk/~acmprog/web2008/