Chapter 3 Sets and Functions Discrete Structures for Computer Science (CO1007)

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Chapter 4
Sets and Functions

Contents
Sets

Discrete Structures for Computer Science (CO1007) on Ngày
7 tháng 10 năm 2016

Set Operation
Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

Nguyen An Khuong, Huynh Tuong Nguyen
Faculty of Computer Science and Engineering
University of Technology, VNU-HCM
4.1


Contents

Sets and Functions
Nguyen An Khuong,

Huynh Tuong Nguyen

1 Sets
Contents
Sets

2 Set Operation

Set Operation
Functions

3 Functions

One-to-one and Onto
Functions
Sequences and
Summation
Recursion

4 One-to-one and Onto Functions
5 Sequences and Summation
6 Recursion

4.2


Set Definition

Sets and Functions
Nguyen An Khuong,

Huynh Tuong Nguyen

• Set is a fundamental discrete structure on which all discrete

structures are built
• Sets are used to group objects, which often have the same

properties

Contents
Sets
Set Operation

Example
• Set of all the students who are currently taking Discrete

Mathematics 1 course.
• Set of all the subjects that K2011 students have to take in

Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

the first semester.
• Set of natural numbers N
Definition

A set is an unordered collection of objects.
The objects in a set are called the elements (phần tử ) of the set.
A set is said to contain (chứa) its elements.
4.3


Notations

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Definition
• a ∈ A: a is an element of the set A
• a∈
/ A: a is not an element of the set A

Contents
Sets
Set Operation
Functions
One-to-one and Onto
Functions

Definition (Set Description)
• The set V of all vowels in English alphabet, V = {a, e, i, o, u}

Sequences and
Summation
Recursion

• Set of all real numbers greater than 1???

{x | x ∈ R, x > 1}
{x | x > 1}
{x : x > 1}

4.4


Equal Sets

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Contents

Definition

Sets

Two sets are equal iff they have the same elements.

Set Operation
Functions

• (A = B) ↔ ∀x(x ∈ A ↔ x ∈ B)

One-to-one and Onto

Functions
Sequences and
Summation

Example

Recursion

• {1, 3, 5} = {3, 5, 1}
• {1, 3, 5} = {1, 3, 3, 3, 5, 5, 5, 5}

4.5


Venn Diagram

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Contents

• John Venn in 1881
• Universal set (tập vũ trụ) is

represented by a rectangle
• Circles and other

geometrical figures are used
to represent sets

Sets
Set Operation
Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

• Points are used to represent

particular elements in set

4.6


Special Sets

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Contents
Sets
Set Operation

• Empty set (tập rỗng ) has no elements, denoted by ∅, or {}
• A set with one element is called a singleton set

Functions
One-to-one and Onto
Functions

• What is {∅}?

Sequences and
Summation

• Answer: singleton

Recursion

4.7


Subset

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Definition

The set A is called a subset (tập con) of B iff every element of A
is also an element of B, denoted by A ⊆ B.

Contents
Sets
Set Operation

Functions

If A = B, we write A ⊂ B and say A is a proper subset (tập con
thực sự) of B.

One-to-one and Onto
Functions
Sequences and
Summation
Recursion

• ∀x(x ∈ A → x ∈ B)
• For every set S,

(i) ∅ ⊆ S, (ii) S ⊆ S.

4.8


Cardinality

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Definition

If S has exactly n distinct elements where n is non-negative
integers, S is finite set (tập hữu hạn), and n is cardinality (bản
số ) of S, denoted by |S|.

Contents
Sets
Set Operation

Example
• A is the set of odd positive integers less than 10. |A| = 5.
• S is the letters in Vietnamese alphabet, |S| = 29.
• Null set |∅| = 0.

Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

Definition

A set that is infinite if it is not finite.
Example
• Set of positive integers is infinite
4.9


Power Set

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Definition

Given a set S, the power set (tập lũy thừa) of S is the set of all
subsets of the set S, denoted by P (S).

Contents
Sets
Set Operation
Functions

Example

What is the power set of {0, 1, 2}?
P ({0, 1, 2}) = {∅, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}}

One-to-one and Onto
Functions
Sequences and
Summation
Recursion

Example
• What is the power set of the empty set?
• What is the power set of the set {∅}

4.10


Sets and Functions

Power Set

Nguyen An Khuong,
Huynh Tuong Nguyen

Contents
Sets
Set Operation
Functions

Theorem
n

If a set has n elements, then its power set has 2 elements.

One-to-one and Onto
Functions
Sequences and
Summation

Prove using induction!

Recursion

4.11


Ordered n-tuples

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Definition

The ordered n-tuple (dãy sắp thứ tự) (a1 , a2 , . . . , an ) is the
ordered collection that has a1 as its first element, a2 as its second
element, . . ., and an as its nth element.

Contents
Sets
Set Operation
Functions
One-to-one and Onto
Functions

Definition

Two ordered n-tuples (a1 , a2 , . . . , an ) = (b1 , b2 , . . . , bn ) iff ai = bi ,
for i = 1, 2, . . . , n.

Sequences and
Summation
Recursion

Example

2-tuples, or ordered pairs (cặp), (a, b) and (c, d) are equal iff
a = c and b = d

4.12


Cartesian Product

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

• René Descartes (1596–1650)
Contents

Definition

Sets

Let A and B be sets. The Cartesian product (tích Đề-các) of A
and B, denoted by A × B, is the set of ordered pairs (a, b), where
a ∈ A and b ∈ B. Hence,
A × B = {(a, b) | a ∈ A ∧ b ∈ B}

Set Operation
Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

Example

Cartesian product of A = {1, 2} and B = {a, b, c}. Then
A × B = {(1, a), (1, b), (1, c), (2, a), (2, b), (2, c)}
Show that A × B = B × A

4.13


Sets and Functions

Cartesian Product

Nguyen An Khuong,
Huynh Tuong Nguyen

Definition
Contents
Sets

A1 ×A2 ×· · ·×An = {(a1 , a2 , . . . , an ) | ai ∈ Ai for i = 1, 2, . . . , n}

Set Operation
Functions
One-to-one and Onto
Functions

Example

Sequences and

Summation

A = {0, 1}, B = {1, 2}, C = {0, 1, 2}. What is A × B × C?

Recursion

A×B×C

= {(0, 1, 0), (0, 1, 1), (0, 1, 2), (0, 2, 0), (0, 2, 1),
(0, 2, 2), (1, 1, 0), (1, 1, 1), (1, 1, 2), (1, 2, 0),
(1, 2, 1), (1, 2, 2)}

4.14


Sets and Functions

Union

Nguyen An Khuong,
Huynh Tuong Nguyen

Definition

The union (hợp) of A and B

Contents
Sets

A ∪ B = {x | x ∈ A ∨ x ∈ B}

Set Operation
Functions
One-to-one and Onto
Functions

A∪B

Sequences and
Summation
Recursion

A

B

• Example:
• {1,2,3} ∪ {2,4} = {1,2,3,4}
• {1,2,3} ∪ ∅ = {1,2,3}

4.15


Sets and Functions

Intersection

Nguyen An Khuong,
Huynh Tuong Nguyen

Definition

The intersection (giao) of A and B

Contents
Sets

A ∩ B = {x | x ∈ A ∧ x ∈ B}

Set Operation
Functions
One-to-one and Onto
Functions

A∩B

Sequences and
Summation
Recursion

A

B

Example:
• {1,2,3} ∩ {2,4} = {2}
• {1,2,3} ∩ N = {1,2,3}

4.16

Union/Intersection

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Contents
Sets
Set Operation
Functions

n

and Onto
Ai = A1 ∪ A2 ∪ ... ∪ An = {x | x ∈ A1 ∨ x ∈ A2 ∨ ... ∨ x ∈ An } One-to-one
Functions

i=1

Sequences and
Summation

n

Recursion

Ai = A1 ∩ A2 ∩ ... ∩ An = {x | x ∈ A1 ∧ x ∈ A2 ∧ ... ∧ x ∈ An }
i=1

4.17


Sets and Functions

Difference

Nguyen An Khuong,
Huynh Tuong Nguyen

Definition

The difference (hiệu) of A and B

Contents
Sets

A − B = {x | x ∈ A ∧ x ∈
/ B}

Set Operation
Functions
One-to-one and Onto
Functions

A−B

Sequences and
Summation
Recursion

A

B

Example:
• {1,2,3} - {2,4} = {1,3}
• {1,2,3} - N = ∅

4.18


Sets and Functions

Complement

Nguyen An Khuong,
Huynh Tuong Nguyen

Definition

The complement (phần bù) of A

Contents

A = {x | x ∈A}
/

Sets
Set Operation

Functions
One-to-one and Onto
Functions

Example:
• A = {1,2,3} then A = ???
• Note that A - B = A ∩ B

Sequences and
Summation
Recursion

4.19


Sets and Functions

Set Identities

Nguyen An Khuong,
Huynh Tuong Nguyen

Contents

A∪∅
A∩U

=
=

A
A

Identity laws
Luật đồng nhất

A∪U
A∩∅

=
=

U
∅

Domination laws
Luật nuốt

One-to-one and Onto
Functions

A∪A
A∩A

=
=

A
A

Idempotent laws
Luật lũy đẳng

Recursion

¯
(A)

=

A

Complementation law
Luật bù

Sets
Set Operation
Functions

Sequences and
Summation

4.20


Sets and Functions

Set Identities

Nguyen An Khuong,

Huynh Tuong Nguyen

Contents

A∪B
A∩B

=
=

B∪A
B∩A

A ∪ (B ∪ C)
A ∩ (B ∩ C)

=
=

(A ∪ B) ∪ C
(A ∩ B) ∩ C

Associative laws
Luật kết hợp

One-to-one and Onto
Functions

A ∪ (B ∩ C)
A ∩ (B ∪ C)

=
=

(A ∪ B) ∩ (A ∪ C)
(A ∩ B) ∪ (A ∩ C)

Distributive laws
Luật phân phối

Recursion

A∪B
A∩B

=
=

A∩B
A∪B

Commutative laws
Luật giao hoán

Sets
Set Operation
Functions

Sequences and
Summation

De Morgan’s laws
Luật De Morgan

4.21


Method of Proofs of Set Equations

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Contents
Sets

To prove A = B, we could use
• Venn diagrams
• Prove that A ⊆ B and B ⊆ A
• Use membership table

Set Operation
Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

• Use set builder notation and logical equivalences

4.22


Example (1)

Sets and Functions
Nguyen An Khuong,
Huynh Tuong Nguyen

Contents
Sets
Set Operation
Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

Example

Verify the distributive rule P ∪ (Q ∩ R) = (P ∪ Q) ∩ (P ∪ R)

4.23


Example (2)

Sets and Functions

Nguyen An Khuong,
Huynh Tuong Nguyen

Example

Prove: A ∩ B = A ∪ B
(1) Show that A ∩ B ⊆ A ∪ B
Suppose that x ∈ A ∩ B
By the definition of complement, x ∈
/ A∩B
So, x ∈
/ A or x ∈
/B
¯
Hence, x ∈ A¯ or x ∈ B
We conclude, x ∈ A ∪ B
Or, A ∩ B ⊆ A ∪ B

Contents
Sets
Set Operation
Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

(2) Show that A ∪ B ⊆ A ∩ B

4.24


Sets and Functions

Example (3)

Nguyen An Khuong,
Huynh Tuong Nguyen

Contents
Sets

Prove: A ∩ B = A ∪ B

Set Operation

A

B

A∩B

A∩B

¯
A¯ ∪ B

1
1

0
0

1
0
1
0

1
0
0
0

0
1
1
1

0
1
1
1

Functions
One-to-one and Onto
Functions
Sequences and
Summation
Recursion

4.25