Chapter 4 functions

Functions Functions Huynh Tuong Nguyen, Tran Vinh Tan Contents Functions One to one and Onto Functions Sequences and Summation Recursion 4 1 Chapter 4 Functions Discrete Structures for Computing on 13[.]

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.1

Chapter 4

Functions

Discrete Structures for Computing on 13 March 2012

Huynh Tuong Nguyen, Tran Vinh TanFaculty of Computer Science and Engineering

University of Technology - VNUHCM

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.2

Contents

1 Functions

2 One-to-one and Onto Functions

3 Sequences and Summation

4 Recursion

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.3

Introduction

• Each student is assigned a grade from set

{0, 0.1, 0.2, 0.3, , 9.9, 10.0} at the end of semester

• Function is extremely important in mathematics andcomputer science

• linear, polynomial, exponential, logarithmic,

• Don’t worry! For discrete mathematics, we need tounderstand functions at a basic set theoretic level

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.3

Introduction

• Each student is assigned a grade from set

{0, 0.1, 0.2, 0.3, , 9.9, 10.0} at the end of semester

• Function is extremely important in mathematics and

computer science

• linear, polynomial, exponential, logarithmic,

• Don’t worry! For discrete mathematics, we need tounderstand functions at a basic set theoretic level

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.3

Introduction

• Each student is assigned a grade from set

{0, 0.1, 0.2, 0.3, , 9.9, 10.0} at the end of semester

• Function is extremely important in mathematics and

computer science

• linear, polynomial, exponential, logarithmic,

• Don’t worry! For discrete mathematics, we need tounderstand functions at a basic set theoretic level

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.3

Introduction

• Each student is assigned a grade from set

{0, 0.1, 0.2, 0.3, , 9.9, 10.0} at the end of semester

• Function is extremely important in mathematics and

computer science

• linear, polynomial, exponential, logarithmic,

• Don’t worry! For discrete mathematics, we need to

understand functions at a basic set theoretic level

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.4

Function

Definition

Let A and B be nonempty sets A functionf from A to B is an

assignment ofexactly oneelement of B to each element of A

• f : A → B

• A:domain(miền xác định) of f

• B:codomain(miền giá trị) of f

• For each a ∈ A, if f (a) = b

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.6

Example

Example

What are domain, codomain, and range of the function that

assigns grades to students includes: student A: 5, B: 3.5, C: 9, D:

5.2, E: 4.9?

Example

Let f : Z → Z assign the the square of an integer to this integer

What is f (x)? Domain, codomain, range of f ?

• f (x) = x2

• Domain: set of all integers

• Codomain: Set of all integers

• Range of f : {0, 1, 4, 9, }

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.6

Example

Example

What are domain, codomain, and range of the function that

assigns grades to students includes: student A: 5, B: 3.5, C: 9, D:

5.2, E: 4.9?

Example

Let f : Z → Z assign the the square of an integer to this integer

What is f (x)? Domain, codomain, range of f ?

• f (x) = x2

• Domain: set of all integers

• Codomain: Set of all integers

• Range of f : {0, 1, 4, 9, }

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.6

Example

Example

What are domain, codomain, and range of the function that

assigns grades to students includes: student A: 5, B: 3.5, C: 9, D:

5.2, E: 4.9?

Example

Let f : Z → Z assign the the square of an integer to this integer

What is f (x)? Domain, codomain, range of f ?

• f (x) = x2

• Domain: set of all integers

• Codomain: Set of all integers

• Range of f : {0, 1, 4, 9, }

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.7

Add and multiply real-valued functions

Definition

Let f1and f2 be functions from A to R Then f1+ f2 and f1f2

are also functions from A to R defined by

(f1+ f2)(x) = f1(x) + f2(x)(f1f2)(x) = f1(x)f2(x)

Example

Let f1(x) = x2 and f2(x) = x − x2 What are the functions

f1+ f2 and f1f2?

(f1+ f2)(x) = f1(x) + f2(x) = x2+ x − x2= x(f1f2)(x) = f1(x)f2(x) = x2(x − x2) = x3− x4

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

4.7

Add and multiply real-valued functions

Definition

Let f1and f2 be functions from A to R Then f1+ f2 and f1f2

are also functions from A to R defined by

(f1+ f2)(x) = f1(x) + f2(x)(f1f2)(x) = f1(x)f2(x)

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

• Is f : Z → Z, f(x) = x2

one-to-one?

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

• Is f : Z → Z, f(x) = x2

one-to-one?

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

• Is f : Z → Z, f(x) = x2

onto?

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

4.11

One-to-one and onto (bijection)

Definition

f : A → B isbijective (one-to-one correspondence)(song ánh) if

and only if f isinjectiveandsurjective

• Let f be the function from{a, bc, d} to {1, 2, 3, 4} with

f (a) = 4, f (b) = 2,

f (c) = 1, f (d) = 3 Is f abijection?

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

4.12

Example

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

4.12

Example

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.12

Example

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.12

Example

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.12

Example

Huynh Tuong Nguyen, Tran Vinh Tan

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

A one-to-one correspondence is callinvertible(khả nghịch)

because we can define the inverse of this function

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

4.16

Function Composition

Definition

Given a pair of functions g : A → B and f : B → C Then the

composition(hợp thành) of f and g, denotedf ◦ g is defined by

f ◦ g : A → C

f ◦ g(a) = f (g(a))

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

4.17

Example

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

4.17

Example

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

4.17

Example

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

4.17

Example

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

(0, 0)(1, 1)(2, 4)(3, 9)

DefinitionLet f be a function from the set A to the set B Thegraphof thefunction f is the set of ordered pairs {(a, b) | a ∈ A and f (a) = b}

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

(0, 0)(1, 1)(2, 4)(3, 9)

DefinitionLet f be a function from the set A to the set B Thegraphof thefunction f is the set of ordered pairs {(a, b) | a ∈ A and f (a) = b}

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

(0, 0)(1, 1)(2, 4)(3, 9)

Definition

Let f be a function from the set A to the set B Thegraphof the

function f is the set of ordered pairs {(a, b) | a ∈ A and f (a) = b}

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

(3b) d−xe = −bxc(4a) bx + nc = bxc + n(4b) dx + ne = dxe + n

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

(3b) d−xe = −bxc(4a) bx + nc = bxc + n(4b) dx + ne = dxe + n

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

(2) x − 1 < bxc ≤ dxe < x + 1(3a) b−xc = −dxe

(3b) d−xe = −bxc(4a) bx + nc = bxc + n(4b) dx + ne = dxe + n

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

(3a) b−xc = −dxe(3b) d−xe = −bxc(4a) bx + nc = bxc + n(4b) dx + ne = dxe + n

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

(3b) d−xe = −bxc

(4a) bx + nc = bxc + n(4b) dx + ne = dxe + n

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

(3b) d−xe = −bxc(4a) bx + nc = bxc + n(4b) dx + ne = dxe + n

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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

Huynh Tuong Nguyen, Tran Vinh Tan

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