Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Chapter 4

Functions

Discrete Structures for Computing on 13 March 2012

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

Huynh Tuong Nguyen, Tran Huong Lan

Faculty of Computer Science and Engineering

University of Technology - VNUHCM

4.1

Contents

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

1 One-to-one and Onto Functions

Contents

One-to-one and Onto

Functions

Sequences and

Summation

2 Sequences and Summation

Recursion

3 Recursion

4.2

Introduction

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

• Each student is assigned a grade from set

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

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.3

Introduction

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

• 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

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.3

Introduction

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

• 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,...

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.3

Introduction

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

• 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,...

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

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

understand functions at a basic set theoretic level

4.3

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element of B to each element of A.

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element of B to each element of A.

• f :A→B

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element of B to each element of A.

• f :A→B

• A: domain (miền xác định) of f

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element of B to each element of A.

• f :A→B

• A: domain (miền xác định) of f

• B: codomain (miền giá trị) of f

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

• b is an image (ảnh) of a

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

• b is an image (ảnh) of a

• a is pre-image (nghịch ảnh) of f (a)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

• b is an image (ảnh) of a

• a is pre-image (nghịch ảnh) of f (a)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

• Range of f is the set of all images of elements of A

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

• b is an image (ảnh) of a

• a is pre-image (nghịch ảnh) of f (a)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

• Range of f is the set of all images of elements of A

• f maps (ánh xạ) A to B

4.4

Functions

Function

Huynh Tuong Nguyen,

Tran Huong Lan

Definition

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

Contents

One-to-one and Onto

Functions

• b is an image (ảnh) of a

• a is pre-image (nghịch ảnh) of f (a)

Sequences and

Summation

Recursion

• Range of f is the set of all images of elements of A

• f maps (ánh xạ) A to B

f

a

b = f (a)

A

B

f

4.4

Example

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.5

Functions

Example

Huynh Tuong Nguyen,

Tran Huong Lan

Example:

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.5

Functions

Example

Huynh Tuong Nguyen,

Tran Huong Lan

Example:

• y is an image of d

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.5

Functions

Example

Huynh Tuong Nguyen,

Tran Huong Lan

Example:

• y is an image of d

• c is a pre-image of z

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.5

Example

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

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?

Contents

One-to-one and Onto

Functions

Example

Sequences and

Summation

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

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

Recursion

4.6

Example

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

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?

Contents

One-to-one and Onto

Functions

Example

Sequences and

Summation

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

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

Recursion

• f (x) = x2

• Domain: set of all integers

• Codomain: Set of all integers

• Range of f : {0, 1, 4, 9, . . .}

4.6

Example

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

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?

Contents

One-to-one and Onto

Functions

Example

Sequences and

Summation

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

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

Recursion

• f (x) = x2

• Domain: set of all integers

• Codomain: Set of all integers

• Range of f : {0, 1, 4, 9, . . .}

4.6

Add and multiply real-valued functions

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Definition

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

are also functions from A to R defined by

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

(f1 f2 )(x) = f1 (x)f2 (x)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.7

Add and multiply real-valued functions

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Definition

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

are also functions from A to R defined by

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

(f1 f2 )(x) = f1 (x)f2 (x)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

Example

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

f1 + f2 and f1 f2 ?

(f1 + f2 )(x) = f1 (x) + f2 (x) = x2 + x − x2 = x

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

4.7

Huynh Tuong Nguyen,

Tran Huong Lan

Chapter 4

Functions

Discrete Structures for Computing on 13 March 2012

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

Huynh Tuong Nguyen, Tran Huong Lan

Faculty of Computer Science and Engineering

University of Technology - VNUHCM

4.1

Contents

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

1 One-to-one and Onto Functions

Contents

One-to-one and Onto

Functions

Sequences and

Summation

2 Sequences and Summation

Recursion

3 Recursion

4.2

Introduction

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

• Each student is assigned a grade from set

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

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.3

Introduction

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

• 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

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.3

Introduction

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

• 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,...

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.3

Introduction

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

• 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,...

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

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

understand functions at a basic set theoretic level

4.3

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element of B to each element of A.

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element of B to each element of A.

• f :A→B

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element of B to each element of A.

• f :A→B

• A: domain (miền xác định) of f

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element of B to each element of A.

• f :A→B

• A: domain (miền xác định) of f

• B: codomain (miền giá trị) of f

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

• b is an image (ảnh) of a

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

• b is an image (ảnh) of a

• a is pre-image (nghịch ảnh) of f (a)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

• b is an image (ảnh) of a

• a is pre-image (nghịch ảnh) of f (a)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

• Range of f is the set of all images of elements of A

4.4

Function

Definition

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

• b is an image (ảnh) of a

• a is pre-image (nghịch ảnh) of f (a)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

• Range of f is the set of all images of elements of A

• f maps (ánh xạ) A to B

4.4

Functions

Function

Huynh Tuong Nguyen,

Tran Huong Lan

Definition

Let A and B be nonempty sets. A function f from A to B is an

assignment of exactly one element 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

Contents

One-to-one and Onto

Functions

• b is an image (ảnh) of a

• a is pre-image (nghịch ảnh) of f (a)

Sequences and

Summation

Recursion

• Range of f is the set of all images of elements of A

• f maps (ánh xạ) A to B

f

a

b = f (a)

A

B

f

4.4

Example

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.5

Functions

Example

Huynh Tuong Nguyen,

Tran Huong Lan

Example:

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.5

Functions

Example

Huynh Tuong Nguyen,

Tran Huong Lan

Example:

• y is an image of d

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.5

Functions

Example

Huynh Tuong Nguyen,

Tran Huong Lan

Example:

• y is an image of d

• c is a pre-image of z

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.5

Example

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

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?

Contents

One-to-one and Onto

Functions

Example

Sequences and

Summation

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

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

Recursion

4.6

Example

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

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?

Contents

One-to-one and Onto

Functions

Example

Sequences and

Summation

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

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

Recursion

• f (x) = x2

• Domain: set of all integers

• Codomain: Set of all integers

• Range of f : {0, 1, 4, 9, . . .}

4.6

Example

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

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?

Contents

One-to-one and Onto

Functions

Example

Sequences and

Summation

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

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

Recursion

• f (x) = x2

• Domain: set of all integers

• Codomain: Set of all integers

• Range of f : {0, 1, 4, 9, . . .}

4.6

Add and multiply real-valued functions

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Definition

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

are also functions from A to R defined by

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

(f1 f2 )(x) = f1 (x)f2 (x)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

4.7

Add and multiply real-valued functions

Functions

Huynh Tuong Nguyen,

Tran Huong Lan

Definition

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

are also functions from A to R defined by

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

(f1 f2 )(x) = f1 (x)f2 (x)

Contents

One-to-one and Onto

Functions

Sequences and

Summation

Recursion

Example

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

f1 + f2 and f1 f2 ?

(f1 + f2 )(x) = f1 (x) + f2 (x) = x2 + x − x2 = x

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

4.7

## Giáo trình bài tập kts2 ch4 hazards

## Giáo trình bài tập nhom 06

## Giáo trình bài tập 6 ontap kngt chg1 5 hdxbao

## Giáo trình bài tập bg06 c6 inclination rock distribution 2 bt3

## Giáo trình bài tập clctt6 b

## Giáo trình bài tập clctt5 b

## Giáo trình bài tập 5 vật rắn

## Giáo trình bài tập clctt3 b

## Giáo trình bài tập luyben cap2

## Giáo trình bài tập bdnldc baigiang2

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