Stable Matching
Lecture 7: Oct 3
Matching
1
2
3
4
A
B
C
D
5
Boys
Girls
E
Todays goal: to match the boys and the girls in a good way.
Matching
Todays goal: to match the boys and the girls in a good way.
What is a matching?
Each boy is matched to at most o
Number Sequences
?
overhang
Lecture 17: Nov 14
This Lecture
We will study some simple number sequences and their properties.
The topics include:
Representation of a sequence
Sum of a sequence
Arithmetic sequence
Geometric sequence
Applications
Harmonic se
Solving Recurrence
Lecture 19: Nov 25
Some Recursive Programming (Optional?)
Test
int hello(int n)
cfw_
if (n=0)
else
return 0;
printf(Hello World 0\n,n);
hello(n-1);
1. What would the program do if I call hello(10)?
2. What if I call hello(-1)?
3. What
Functions, Pigeonhole Principle
f( ) =
A
B
Lecture 14: Nov 4
This Lecture
We will define what is a function formally, and then
in the next lecture we will use this concept in counting.
We will also study the pigeonhole principle and its applications.
Exa
Binomial Coefficients,
Inclusion-exclusion principle
A
B
C
D
Lecture 13: Oct 31
1
Plan
Binomial coefficients, combinatorial proof
Inclusion-exclusion principle
2
Binomial Theorem
We can compute the coefficients ci by counting arguments.
e.g.
(expanding
Recursion
Lecture 18: Nov 18
Plan
Recursion is one of the most important techniques in computer science.
The main idea is to reduce a problem into the same but smaller problems.
Setting up recurrences
Fibonacci recurrence
Problem solving recurrences
C
More Counting by Mapping
Lecture 16: Nov 7
1
This Lecture
Division rule
Catalan number
2
Division Rule
If a function from A to B is k-to-1,
meaning that each element in B is mapped by exactly k elements in A
then
A k B
(This generalizes the Bijection Ru
Counting by Mapping
B
A
f
Lecture 15: Nov 4
1
Plan
We will study how to define mappings to count.
There will be many examples shown.
Bijection rule (this set of slides)
Division rule (next set of slides)
More mapping (next set of slides)
2
Counting Rul
Sets
A
B
C
Lecture 11: Oct 24
This Lecture
We will first introduce some basic set theory before we do counting.
Basic Definitions
Operations on Sets
Set Identities
Russells Paradox
Defining Sets
Definition: A set is an unordered collection of objects.
Basic Counting
Lecture 12: Oct 28
This Lecture
We will study some basic rules for counting.
Sum rule, product rule, generalized product rule
Permutations, combinations, poker hands
Sum Rule
|S|: the number of elements in a set S.
A
B
If sets A and B are
Bipartite Matching
Lecture 8: Oct 7
This Lecture
Graph matching is an important problem in graph theory.
It has many applications and is the basis of more advanced problems.
In the last lecture we consider the stable matching problem.
Today we will study
Planar Graphs
Lecture 10: Oct 21
This Lecture
Today we will talk about planar graphs,
and how to color a map using 6 colors.
Planar graphs
Eulers formula
6-coloring
Map Colouring
Colour the map using minimum number of colours so that
two countries shar
Propositional Logic
Lecture 1: Sep 2
Content
1. Mathematical proof (what and why)
2. Logic, basic operators
3. Using simple operators to construct any operator
4. Logical equivalence, DeMorgans law
5. Conditional statement (if, if and only if)
6. Argument
Mathematical Induction I
Lecture 4: Sep 16
This Lecture
Last time we have discussed different proof techniques.
This time we will focus on probably the most important one
mathematical induction.
This lectures plan is to go through the following:
The ide
Graph Colouring
L09: Oct 10
This Lecture
Graph coloring is another important problem in graph theory.
It also has many applications, including the famous 4-color problem.
Graph Colouring
Applications
Some Positive Results
Graph Colouring
Graph Colourin
Introduction to Graphs
Lecture 6: Sep 26
This Lecture
In this part we will study some basic graph theory.
Graph is a useful concept to model many problems in computer science.
Seven bridges of Konigsberg
Graphs, degrees
Isomorphism
Path, cycle, connec
First Order Logic
Lecture 2: Sep 9
This Lecture
Last time we talked about propositional logic, a logic on simple statements.
This time we will talk about first order logic, a logic on quantified statements.
First order logic is much more expressive than p
Mathematical Induction II
1
2
3
4
1
2
3
4
5
6
7
8
5
6
7
8
9 10 11 12
9 10 11 12
13 14 15
13 15 14
Lecture 5: Sep 19
This Lecture
We will continue our discussions on mathematical induction.
The new elements in this lecture are some variants of induction:
S
About the Course
A
B
C
Lecture 0: Sep 2
Plan
Course Information and Arrangement
Course Requirement
Topics and objectives of this course
Basic Information
Course homepage: http:/www.cse.cuhk.edu.hk/~chi/csc2110/
Instructor: Lau, Lap Chi
Office hour:
Methods of Proof
Lecture 3: Sep 9
This Lecture
Now we have learnt the basics in logic.
We are going to apply the logical rules in proving mathematical theorems.
Direct proof
Contrapositive
Proof by contradiction
Proof by cases
Basic Definitions
An int