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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


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