Giáo trình bài tập dsp midterm hki 2010 2011

Huynh Tuong Nguyen, Vinh Tan Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of RelationsPower of Relations Definition Let R be a

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Chapter 5

Relations

Discrete Structures for Computing on 22 March 2012

Huynh Tuong Nguyen, Tran Huong Lan, Tran Vinh Tan

Faculty of Computer Science and Engineering

University of Technology - VNUHCM

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of RelationsIntroduction

Function?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of RelationsIntroduction

Function?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relation

Definition

Let A and B be sets Abinary relation(quan hệ hai ngôi ) from a

set A to a set B is a set

R ⊆ A × B

• Notations:

(a, b) ∈ R ←→ aRb

• n-ary relations?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relation

Definition

Let A and B be sets Abinary relation(quan hệ hai ngôi ) from a

set A to a set B is a set

R ⊆ A × B

• Notations:

(a, b) ∈ R ←→ aRb

• n-ary relations?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relation

Definition

Let A and B be sets Abinary relation(quan hệ hai ngôi ) from a

set A to a set B is a set

R ⊆ A × B

• Notations:

(a, b) ∈ R ←→ aRb

• n-ary relations?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relation

Definition

Let A and B be sets Abinary relation(quan hệ hai ngôi ) from a

set A to a set B is a set

R ⊆ A × B

• Notations:

(a, b) ∈ R ←→ aRb

• n-ary relations?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Example

Example

Let A = {a, b, c} be the set of students, B = {l, c, s, g} be the set

of the available optional courses We can have relation R that

consists of pairs (a, b), where a is a student enrolled in course b

R = {(a, l), (a, s), (a, g), (b, c),

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Example

Example

Let A = {a, b, c} be the set of students, B = {l, c, s, g} be the set

of the available optional courses We can have relation R that

consists of pairs (a, b), where a is a student enrolled in course b

R = {(a, l), (a, s), (a, g), (b, c),

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Example

Example

Let A = {a, b, c} be the set of students, B = {l, c, s, g} be the set

of the available optional courses We can have relation R that

consists of pairs (a, b), where a is a student enrolled in course b

R = {(a, l), (a, s), (a, g), (b, c),

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relations on a Set

Definition

Arelation on the set A is a relation from A to A

Example

Let A be the set {1, 2, 3, 4} Which ordered pairs are in the

relation R = {(a, b) | a divides b} (a là ước số của b)?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relations on a Set

Definition

Arelation on the set A is a relation from A to A

Example

Let A be the set {1, 2, 3, 4} Which ordered pairs are in the

relation R = {(a, b) | a divides b} (a là ước số của b)?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

———————————————————–

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of RelationsRelations can have special properties

Reflexive xRx, ∀x ∈ A(phản xạ)

Symmetric xRy →yRx, ∀x, y ∈ A(đối xứng )

Antisymmetric (xRy∧yRx) →x=y, ∀x, y ∈ A(phản đối xứng )

Transitive (xRy∧yRz) →xRz, ∀x, y, z ∈ A(bắc cầu)

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relations can have special properties

Reflexive xRx, ∀x ∈ A

(phản xạ)

Symmetric xRy →yRx, ∀x, y ∈ A(đối xứng )

Antisymmetric (xRy∧yRx) →x=y, ∀x, y ∈ A(phản đối xứng )

Transitive (xRy∧yRz) →xRz, ∀x, y, z ∈ A(bắc cầu)

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relations can have special properties

Transitive (xRy∧yRz) →xRz, ∀x, y, z ∈ A(bắc cầu)

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relations can have special properties

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Relations can have special properties

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

———————————————————–

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Combining Relations

Because relations from A to B are subsetsof A × B, two

relations from A to B can be combined in any way two sets can

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Combining Relations

Because relations from A to B are subsetsof A × B, two

relations from A to B can be combined in any way two sets can

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Combining Relations

Because relations from A to B are subsetsof A × B, two

relations from A to B can be combined in any way two sets can

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Combining Relations

Because relations from A to B are subsetsof A × B, two

relations from A to B can be combined in any way two sets can

Let A and B be the set of all students and the set of all courses at

school, respectively SupposeR1= {(a, b) | a has taken the course

b}andR2= {(a, b) | a requires course b to graduate} What are

the relations R1∪ R2, R1∩ R2, R1− R2, R2− R1?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of RelationsComposition of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of RelationsPower of Relations

Definition

Let R be a relation on the set A Thepowers(lũy thừa)

Rn, n = 1, 2, 3, are defined recursively by

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Power of Relations

Definition

Let R be a relation on the set A Thepowers(lũy thừa)

Rn, n = 1, 2, 3, are defined recursively by

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Power of Relations

Definition

Let R be a relation on the set A Thepowers(lũy thừa)

Rn, n = 1, 2, 3, are defined recursively by

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Power of Relations

Definition

Let R be a relation on the set A Thepowers(lũy thừa)

Rn, n = 1, 2, 3, are defined recursively by

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Power of Relations

Definition

Let R be a relation on the set A Thepowers(lũy thừa)

Rn, n = 1, 2, 3, are defined recursively by

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Power of Relations

Definition

Let R be a relation on the set A Thepowers(lũy thừa)

Rn, n = 1, 2, 3, are defined recursively by

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Power of Relations

Definition

Let R be a relation on the set A Thepowers(lũy thừa)

Rn, n = 1, 2, 3, are defined recursively by

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Representing Relations Using Matrices

Definition

Suppose R is a relation from A = {a1, a2, , am} to

B = {b1, b2, , bn}, R can be represented by thematrix

R is relation from A = {1, 2, 3} to B = {1, 2} Let

R = {(2, 1), (3, 1), (3, 2)}, the matrix for R is

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Representing Relations Using Matrices

Definition

Suppose R is a relation from A = {a1, a2, , am} to

B = {b1, b2, , bn}, R can be represented by thematrix

R is relation from A = {1, 2, 3} to B = {1, 2} Let

R = {(2, 1), (3, 1), (3, 2)}, the matrix for R is

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Representing Relations Using Matrices

Definition

Suppose R is a relation on A = {a1, a2, , am}, R can be

represented by thematrix MR= [mij], where

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Representing Relations Using Matrices

Definition

Suppose R is a relation on A = {a1, a2, , am}, R can be

represented by thematrix MR= [mij], where

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Representing Relations Using Matrices

Definition

Suppose R is a relation on A = {a1, a2, , am}, R can be

represented by thematrix MR= [mij], where

Can we determine whether the relation has certain properties

(reflexive, symmetric, antisymmetric, ) by investigating its

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Representing Relations Using Matrices

Examples

Determine if the relations represented by the following matrices

have special properties?

Huynh Tuong Nguyen, Vinh Tan

Contents Properties of Relations Combining Relations Representing Relations Closures of Relations Types of Relations

Representing Relations Using Digraphs

Definition

Suppose R is a relation in A = {a1, a2, , am}, R can be

represented by thedigraph(đồ thị có hướng ) G = (V, E), where

V = A(ai, aj) ∈ E if (ai, aj) ∈ R