Discrete Structures
Beifang Chen
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Contents
Chapter 1.
Set Theory
5
1.1.
Sets and Subsets
5
1.2.
Set Operations
6
1.3.
Functions
9
1.4.
Injection, Surjection, and Bijection
11
1.5.
Infinite Sets
13
1.6.
Permutations
15
Chapter 2.
Number Theory
19
2.1.
Divisibility
19
2.2.
Greatest Common Divisor
20
2.3.
Least Common Multiple
23
2.4.
Modulo Integers
24
2.5.
RSA Cryptography System
26
Chapter 3.
Propositional Logic
29
3.1.
Statements
29
3.2.
Connectives
29
3.3.
Tautology
31
3.4.
Methods of Proof
33
3.5.
Mathematical Induction
35
3.6.
Boolean Functions
35
Chapter 4.
Combinatorics
39
4.1.
Counting Principle
39
4.2.
Permutations
39
4.3.
Combination
42
4.4.
Combination with Repetition
45
4.5.
Combinatorial Proof
46
4.6.
Pigeonhole Principle
50
4.7.
Relation to Probability
51
4.8.
InclusionExclusion Principle
53
4.9.
More Examples
57
4.10.
Generalized InclusionExclusion Formula
59
Chapter 5.
Recurrence Relations
63
5.1.
Infinite Sequences
63
5.2.
Homogeneous Recurrence Relations
64
5.3.
Higher Order Homogeneous Recurrence Relations
67
5.4.
Nonhomogeneous Equations
68
3
4
CONTENTS
5.5.
DivideandConquer Method
69
5.6.
Searching and Sorting
75
5.7.
Growth of Functions
75
Chapter 6.
Binary Relations
77
6.1.
Binary Relations
77
6.2.
Representation of Relations
79
6.3.
Composition of Relations
80
6.4.
Special Relations
82
6.5.
Equivalence Relations and Partitions
84
CHAPTER 1
Set Theory
1.1. Sets and Subsets
A
set
is a collection of objects satisfying certain properties; the objects in the
collection are called
elements
(or
objects
or
members
). A set is considered to be
a whole entity and is different from its elements. Given a set
A
; we write “
x
∈
A
”
to say that
x
is an element of
A
or
x
belongs to
A
, and write “
x /
∈
A
” to say
that
x
is not an element of
A
or
x
doesn’t belong to
A
. We usually denote sets by
uppercase letters such as
A, B, C, . . . , X, Y, Z
, and denote the elements of a set by
lowercase letters such as
a, b, c, . . . , x, y, z
, etc.
There are two ways to express a set. One way is to list all elements of the set;
the other way is to point out the attributes of the elements of the set. For example,
let
A
be the set of integers whose absolute values are less than or equal to 3. The
set
A
can be described in two ways:
A
=
{
3
,

2
,

1
,
0
,
1
,
2
,
3
}
or
A
=
{
a

a
is an integer,

a
 ≤
3
}
.
A set
X
whose elements satisfying Property
P
is denoted by
X
=
{
x

x
satisfies
P
}
or
X
=
{
x
:
x
satisfies
P
}
.
In this note, most of time we use the first notation
X
=
{
x

x
satisfies
P
}
, and
occasionally use the second notation when there is confusion to use the symbol “

”.
There are two important things to be noticed about the concept of sets. The
first one is that any set, when it is considered as an object, can not be an element of
itself, but can be an element of another set. The second one is that for a particular
object, it is possible to decide in principle whether or not the object is an element
of a given set.