Introduction to Discrete Mathematics

Lecture 1: Sep 1

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)

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/

Course Requirements

Homework, 20%

Midterm, 30%

Course project, 10%

Final Exam, 40%

Midterm: Oct 27 (Monday), M7-8

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

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

Checker

x=0

Start with any configuration with all men on or below the x-axis.

Checker

x=0

Move: jump through your adjacent neighbour, but then your neighbour will disappear.

Checker

x=0

Move: jump through your adjacent neighbour, but then your neighbour will disappear.

Checker

x=0

Goal: Find an initial configuration with least number of men to jump up to level k.

K=1

x=0

2 men.

K=2

x=0

K=2

x=0

4 men.

Now we have reduced to the k=1 configuration, but one level higher.

K=3

x=0

This is the configuration for k=2, so jump two level higher.

K=3

x=0

8 men.

K=4

x=0

K=4

x=0

K=4

x=0

K=4

x=0

K=4

x=0

Now we have reduced to the k=3 configuration, but one level higher

20 men!

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

Example 1

How to play Rubik Cube?

http://blog.sciencenews.org/mathtrek/2007/08/cracking_the_cube.html

Google: Rubik cube in 26 steps

Example 2

The mathematics of paper folding

http://erikdemaine.org/foldcut/

http://www.ushistory.org/betsy/flagstar.html

http://128.100.68.6/~drorbn/papers/PDI/

Example 3

3D-images

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

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…

Topic 1: Logic and Proofs

Logic: propositional logic, first order logic

Proof: induction, contradiction

How do computers think?

Artificial intelligence, database, circuit, algorithms

Topic 2: Number Theory

• Number sequence

• Euclidean algorithm

• Prime number

• Modular arithmetic

Cryptography, coding theory, data structures

Topic 3: Counting

• Sets

• Combinations, Permutations, Binomial theorem

• Functions

• Counting by mapping, pigeonhole principle

• Recursions, generating functions

Probability, algorithms, data structures

Topic 3: Counting

How many steps are needed to sort n numbers?

Topic 4: Graph Theory

• Relations, graphs• Degree sequence, isomorphism, Eulerian graphs• Trees

Computer networks, circuit design, data structures

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

2 2 2a b c

Familiar?

Obvious?

cb

a

Pythagorean theorem

cb

a

Rearrange into: (i) a cc square, and then (ii) an aa & a bb square

Good Proof

c

cc

a b

c

b-a

Good Proof

cb

a b-a

b-a

Good Proof

ba

a

ab-a

74 proofs in http://www.cut-the-knot.org/pythagoras/index.shtml

b

Good Proof

Bad Proof

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

Logic Operators

P QQP

NOT::

AND::

P QQP

OR::

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

Exclusive-Or

coffee “or” tea exclusive-or

How to construct a compound statement for exclusive-or?

p q p q

Check by Truth Table

p q

Logical Equivalence

Two statements have the same truth table

De Morgan’s Law

De Morgan’s Law

Simplifying Statement

A tautology is a statement that is always true.

A contradiction is a statement that is always false.

Class Photos! Identify your face and send us your name and nicknames

Two Important Things

http://appsrv.cse.cuhk.edu.hk/~acmprog/web2008/