Introduction to Discrete Mathematics and computer science Theory

Introduction
to
Discrete
Mathematics
Lecture 1: Sep 7
A B
C
a = qb+r gcd(a,b) = gcd(b,r)

Basic Information
▪ Instructor: Dr Asif Taj
▪ Office hour: (SHB 911)
▪ Lectures: M7-8 (TYW LT), W6 (LSB)
▪ Tutorials: H5 (ERB 404) or H6 (EGB 404)

Course Material
▪ Textbook: Discrete Mathematics with Applications
Author: Susanna S. Epp
Publisher: Brooks/Cole
▪ Reference: Course notes from “mathematics for computer science”
http://courses.csail.mit.edu/6.042/spring07/

Course Requirements
▪ Homework, 20%
▪ Midterm, 25%
▪ Course project, 10%
▪ Final Exam, 45%
Midterm: Oct 28 (Wednesday), 7-9pm

Course Project
4 students in a group
Pick an interesting mathematical topic,
write a report of about 10 pages.
Can use any references, but cite them.
Choose 1-3 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
Goal: Find an initial configuration with least number of men to jump up to level k.

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=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?
Google: Rubik cube in 26 steps

Example 2
The mathematics of paper folding
http://www.ushistory.org/betsy/flagstar.html

▪ Magic tricks
▪ More games, more paper folding, etc
▪ Logic paradoxes
▪ Prime numbers
▪ Game theory
Project Ideas
Deadline: November 16.

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
• (Extended) Euclidean algorithm
• Prime number, modular arithmetic, Chinese remainder theorem
• Cryptography, RSA protocol
Cryptography, coding theory, data structures

Topic 3: Counting
• Sets and Functions
• Combinations, Permutations, Binomial theorem
• Counting by mapping, pigeonhole principle
• Recursions
Probability, algorithms, data structures
A B
C

Topic 3: Counting
How many steps are needed to sort n numbers?
Algorithm 1 (Bubble Sort):
Every iteration moves the i-th smallest number to the i-th position
Algorithm 2 (Merge Sort):
Which algorithm runs faster?

Topic 4: Graph Theory
• Graphs, Relations
• Degree sequence, Eulerian graphs, isomorphism
• Trees
• Matching
• Coloring
Computer networks, circuit design, data structures

Topic 4: Graph Theory
How to color a map?
How to send data efficiently?

Objectives of This Course
CSC 2100, ERG 2040, CSC 3130, CSC 3160
•To learn basic mathematical concepts, e.g. sets, functions, graphs
•To be familiar with formal mathematical reasoning, e.g. logic, proofs
•To improve problem solving skills
•To see the connections between discrete mathematics and computer science
Practice, Practice and Practice.

2 2 2
a b c
+ =
Familiar?
Obvious?
c
b
a
Pythagorean theorem

c
b
a
(i) a cc square, and then
(ii) an aa & a bb square
Good Proof
b-a
We will show that these five pieces can be rearranged into:
b-a
And then we can conclude that

c
c
c
a b
c
Good Proof
The five pieces can be rearranged into:
(i) a cc square

c
b
a b-a
b-a
Good Proof
How to rearrange them into an axa square and a bxb square?

Bad Proof
A similar rearrangement technique shows that 65=64…
What’s wrong with the proof?

Mathematical Proof
To prove mathematical theorems, we need a more rigorous system.
http://en.wikipedia.org/wiki/Pythagorean_theorem
Euclid’s proof of Pythagorean’s theorem
The standard procedure for proving mathematical theorems is invented by
Euclid in 300BC. First he started with five axioms (the truth of these
statements are taken for granted). Then he uses logic to deduce the truth
of other statements.
1.It is possible to draw a straight line from any point to any other point.
2.It is possible to produce a finite straight line continuously in a straight line.
3.It is possible to describe a circle with any center and any radius.
4.It is true that all right angles are equal to one another.
5.("Parallel postulate") It is true that, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles,
the two straight lines, if produced indefinitely, intersect on that side on which are the angles less than the two right angles.
See page 18 of the notes for the ZFC axioms that we now use.

Statement (Proposition)
A Statement is a sentence that is either True or False
Examples:
Non-examples: x+y>0
x2+y2=z2
True
False
2 + 2 = 4
3 x 3 = 8
787009911 is a prime
They are true for some values of x and y
but are false for some other values of x and y.

Logic Operators
F
F
F
T
P Q
F
F
T
F
F
T
T
T
Q
P
AND
::=

F
T
T
T
P Q
F
F
T
F
F
T
T
T
Q
P
OR
::=

NOT
::=
 ~p is true if p is false

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
T T F
T F T
F T T
F F F
Idea 1: Look at the true rows
Idea 2: Look at the false rows
Idea 3: Guess and check

Logical Equivalence
p q
T T F T F F
T F T T T T
F T T T T T
F F F F T F
Logical equivalence: Two statements have the same truth table

Writing Logical Formula for a Truth Table
p q r output
T T T F
T T F T
T F T T
T F F F
F T T T
F T F T
F F T T
F F F F
Given a truth table, how to write a logical formula with the same function?
First write down a small formula for each row, so that the formula
is true if the inputs are exactly the same as the row.
Then use idea 1 or idea 2. Idea 1: Look at the true rows
and take the “or”.
The formula is true iff the
input is one of the true rows.

Writing Logical Formula for a Truth Table
Digital logic:
Idea 2: Look at the false rows,
negate and take the “and”.
The formula is true iff the input
is not one of the false row.
p q r output
T T T F
T T F T
T F T T
T F F F
F T T T
F T F T
F F T T
F F F F
can be simplified further

DeMorgan’s Laws
T T F F
T F T T
F T T T
F F T T
De Morgan’s Law
De Morgan’s Law

DeMorgan’s Laws
De Morgan’s Law
De Morgan’s Law
Statement: Tom is in the football team and the basketball team.
Negation: Tom is not in the football team or not in the basketball team.
Statement: The number 783477841 is divisible by 7 or 11.
Negation: The number 783477841 is not divisible by 7 and not divisible by 11.

Simplifying Statement
See textbook for more identities.
DeMorgan
Distributive

Tautology, Contradiction
A tautology is a statement that is always true.
A contradiction is a statement that is always false. (negation of a tautology)
In general it is “difficult” to tell whether a statement is a contradiction.
It is one of the most important problems in CS – the satisfiability problem.

Class Photos! Identify your face and send us your name and nicknames
Two Important Things
Tutorial hours

Introduction to Discrete Mathematics and computer science Theory

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