Chapter 1. The Basics of Counting
Our study of discrete mathematics begins with two basic rules of counting: the
sum rule and the product rule. The statements and initial applications of these two
rules are fairly simple. To analyze more complicated probl

Graph Theory
Part II/II
Outline
Definitions
Complete graph, bipartite graph
More on node degree
The handshaking theorem
Hamiltonian circuit
Graph isomorphism
Copyright belongs to the owner
Complete & Bipartite
A complete graph with n vertices is a s

MH1812 Tutorial 07
NTU, AY14/15 S2
Week 09, Date: 09-13/03/2015
Note: leave the questions with complex numbers for next tutorial session.
Q1: Compute the sum A + B of the matrices A and B, where A and A are as follows:
1.
A=
2
1
2
0
2
,B =
3
4 2
where

Set Theory
Part I/II
Outline
Definitions related to set:
Set, membership, empty set, set equality, subset,
cardinality, power set
Venn diagram
Union,
intersection
cardinality
xkcd
Set
A set is a collection of abstract objects
Examples: prime number

Graph Theory
Part I/II
Outline
Definitions
Vertex, edge, adjacent, incident
Simple graph, multigraph, directed (multi)graph
Euler path and Euler Theorem.
Definitions
A graph G = (V,E) is a structure consisting of a set V of
vertices (nodes) and a set

Predicate Logic
Part II/III
Outline
Negation of quantification
Truth value for quantifiers
Exhaustion
Case
Logic derivation
Conditional quantification
belongs to the cartoonist
Truth Value of Quantified Statements
Statement
xD,P(x)
When true
When f

Chapter 11
Graph Theory
The origins of graph theory are humble, even frivolous. (N.
Biggs, E. K. Lloyd, and R. J. Wilson)
Let us start with a formal definition of what is a graph.
Definition 72. A graph G = (V, E) is a structure consisting of a set V of
v

Relations
Part I/II
Outline
Binary relations
Definition
Inverse and composition
Graphical representation
Properties
Reflexivity
Symmetry
Transitivity
Binary Relations between Two Sets
Let A and B be sets. A binary relation R from A to B
is a subset o

Relations
Part II/II
Outline
Equivalence relations
Definition
Equivalence class
Partial order
Antisymmetry
Definition
Beyond binary relations
Equivalence Relation
A relation R on a set A is an equivalence relation if
1. R is reflective: ,
2. R is sy

Chapter 3
Predicate Logic
Logic will get you from A to B. Imagination will take you everywhere. A. Einstein
In the previous chapter, we studied propositional logic. This chapter is
dedicated to another type of logic, called predicate logic.
Let us start w

Example Class 2
Propositional Logic
Outline
Knights & Knaves
Find the murderers knife
The Island of Knights & Knaves
Knights never lie
Knaves always lie
Knave = a dishonest or
unscrupulous man,
in cards a jack.
Art work belongs to Michael Kutsche
Knight

Set Theory
Part II/II
Outline
Set identities
Prove set equalities.
belongs to the cartoonist
Set Identities
_
A B A B
_
A
B
B
_
B
A
B
Compare A B with A-B = cfw_x | xA xB
(not a formal proof)
Set Identities
_
A B A B
A-B
B
A
_
Consider A B A B
Apply

Chapter 6
Linear Recurrences
Everything goes, everything comes back; eternally rolls the wheel
of being. (Friedrich Nietzsche)
This chapter is dedicated to linear recurrences, a special type of equations
that defines a sequence, that is a series of terms

MH1812 Discrete Mathematics - Quiz 2
NTU, AY14/15 S2
Name: Guo Jian
02 April 2015, 11:30AM - 12:30PM, LT2A
Tutorial Group: not sure
NTU Email: guojian@ntu.edu.sg
Note: try ALL 3 questions (5 sub-questions), you can write on the back of paper if there is
n

Functions
Part I/II
Outline
Functions
Definition
Injectivity, surjectivity, bijectivity
Copyright belongs to author
Function
Let X and Y be sets. A function f from X to Y is a rule
that assigns every element x of X to a unique y in Y.
We write f: X Y a

Complex Numbers
Part I/I
Outline
Definition of i and imaginary
numbers
Complex numbers
Complex numbers operations
Polar coordinates
Euler Formula
Nth roots
cartoonist
Definition of i
There is no real number z such that
2 = 1.
Define an imaginary

Functions
Part II/II
Outline
Functions
Identity
Inverse
Composition of functions and their properties.
Pigeonhole principle
Image from wiki
Identity Function
The identity function on a set A is defined as:
iA:AA, iA(x) = x.
Example. Any identity func

264
CHAPTER 11. GRAPH THEORY
Exercises for Chapter 11
Exercise 96. Prove that if a connected graph G has exactly two vertices
which have odd degree, then it contains an Euler path.
Exercise 97. Draw a complete graph with 5 vertices.
Exercise 98. Show that

MH1812 Tutorial 03
NTU, AY14/15 S2
Week 04, Date: 02-06/02/2015
Q1: Show that this argument is valid:
p F ; p.
Q2: Show that this argument is valid, where C denotes a contradiction.
p C; p.
Q3: Determine whether the following argument is valid:
p r s
ts
u

Chapter 10
Functions
One of the most important concepts in all of mathematics is that
of function. (T.P. Dick and C.M. Patton)
Functions.finally a topic that most of you must be familiar with. However here, we will not study derivatives or integrals, but

Example Class 4
Linear Algebra
Outline
Examples from Cryptography
Examples from Data Storage
Copyright belongs to the artist
Modulo n
Recall
a b (mod n)
If a b (mod n), then a-b = qn and a=qn+b.
We represent integers mod n as cfw_0,1,n-1
(thanks to t

Chapter 5
Combinatorics
I think youre begging the question, said Haydock, and I can
see looming ahead one of those terrible exercises in probability
where six men have white hats and six men have black hats and
you have to work it out by mathematics how l

Predicate Logic
Part III/III
Outline
More important inference rules:
Universal instantiation
Universal generalization
Existential instantiation
Existential generalization.
Proof techniques
direct
induction
contradiction
belongs to the cartoonist
Uni

Chapter 4
Set Theory
A set is a Many that allows itself to be thought of as a One.
(Georg Cantor)
In the previous chapters, we have often encountered sets, for example,
prime numbers form a set, domains in predicate logic form sets as well.
Defining a set

Propositional Logic
Part III/III
Outline
Arguments
Valid arguments
Invalid arguments
Counter-Example
Fallacy
Inference rules
belongs to the cartoonist
Valid Argument
An argument is a sequence of statements. The last
statement is called the conclusion,

Example Class 1
Elementary Number Theory
Outline
Modulo n
Binary arithmetic
Applications of binary arithmetic
Copyright belongs to the artist
Modulo n
Recall
a b (mod n)
If a b (mod n), then a-b = qn and a=qn+b.
We represent integers mod n as cfw_0,

Chapter 8
Linear Algebra
Algebra is generous; she often gives more than is asked of her.
(Jean DAlembert)
This chapter is called linear algebra, but what we will really see is the
definition of a matrix, a few basic properties of matrices, and how to comp