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Sets

Huynh Tuong Nguyen,
Tran Huong Lan

Chapter 3
Sets
Discrete Structures for Computing on 21 March 2011

Huynh Tuong Nguyen, Tran Huong Lan
Faculty of Computer Science and Engineering
University of Technology - VNUHCM
3.1


Contents

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

3.2


Set Definition

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

• 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
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

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.
3.3


Notations

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

Definition
• a ∈ A: a is an element of the set A
• a∈
/ A: a is not an element of the set A
Definition (Set Description)
• The set V of all vowels in English alphabet, V = {a, e, i, o, u}
• Set of all real numbers greater than 1???

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

3.4


Equal Sets

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

Definition

Two sets are equal iff they have the same elements.
• (A = B) ↔ ∀x(x ∈ A ↔ x ∈ B)
Example
• {1, 3, 5} = {3, 5, 1}
• {1, 3, 5} = {1, 3, 3, 3, 5, 5, 5, 5}

3.5


Venn Diagram

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

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

represented by a rectangle
• Circles and other

geometrical figures are used
to represent sets
• Points are used to represent

particular elements in set

3.6


Special Sets

Sets

Huynh Tuong Nguyen,

Tran Huong Lan

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

3.7


Subset

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

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

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

(i) ∅ ⊆ S, (ii) S ⊆ S.

3.8

Cardinality

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

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|.
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.
Definition

A set that is infinite if it is not finite.
Example
• Set of positive integers is infinite
3.9


Power Set

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

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

3.10


Power Set

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

Theorem

If a set has n elements, then its power set has 2n elements.
Prove using induction!

3.11


Ordered n-tuples

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

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.
Definition

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

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

3.12


Cartesian Product

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

• René Descartes (1596–1650)
Definition

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}

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

3.13


Sets

Cartesian Product

Huynh Tuong Nguyen,
Tran Huong Lan

Definition

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

Example

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

3.14


Sets

Union

Huynh Tuong Nguyen,
Tran Huong Lan

Definition

The union (hợp) of A and B
A ∪ B = {x | x ∈ A ∨ x ∈ B}
A∪B

A

B

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

3.15


Sets

Intersection

Huynh Tuong Nguyen,
Tran Huong Lan

Definition

The intersection (giao) of A and B
A ∩ B = {x | x ∈ A ∧ x ∈ B}
A∩B

A

B

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

3.16

Union/Intersection

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

n

Ai = A1 ∪ A2 ∪ ... ∪ An = {x | x ∈ A1 ∨ x ∈ A2 ∨ ... ∨ x ∈ An }
i=1
n

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

3.17


Sets

Difference

Huynh Tuong Nguyen,
Tran Huong Lan

Definition

The difference (hiệu) of A and B
A − B = {x | x ∈ A ∧ x ∈
/ B}
A−B

A

B

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

3.18


Sets

Complement

Huynh Tuong Nguyen,
Tran Huong Lan

Definition

The complement (phần bù) of A
A = {x | x ∈A}
/

Example:

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

3.19


Sets

Set Identities

Huynh Tuong Nguyen,
Tran Huong Lan

A∪∅
A∩U

=
=

A
A

Identity laws
Luật đồng nhất

A∪U
A∩∅

=
=

U
∅

Domination laws
Luật nuốt

A∪A
A∩A

=
=

A
A

Idempotent laws
Luật lũy đẳng

¯
(A)

=

A

Complementation law
Luật bù

3.20

Sets

Set Identities

Huynh Tuong Nguyen,
Tran Huong Lan

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

A ∪ (B ∩ C)
A ∩ (B ∪ C)

=
=

(A ∪ B) ∩ (A ∪ C)
(A ∩ B) ∪ (A ∩ C)

Distributive laws
Luật phân phối

A∪B
A∩B

=
=

A∩B
A∪B

Commutative laws
Luật giao hoán

De Morgan’s laws
Luật De Morgan

3.21

Method of Proofs of Set Equations

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

To prove A = B, we could use
• Venn diagrams
• Prove that A ⊆ B and B ⊆ A
• Use membership table
• Use set builder notation and logical equivalences

3.22


Example (1)

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

Example

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

3.23

Example (2)

Sets

Huynh Tuong Nguyen,
Tran Huong Lan

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
(2) Show that A ∪ B ⊆ A ∩ B

3.24


Sets

Example (3)

Huynh Tuong Nguyen,
Tran Huong Lan

Prove: A ∩ B = A ∪ B
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

3.25