Chapter 3 Sets and Functions Discrete Structures for Computer Science (CO1007)

Nguyen An Khuong, Huynh Tuong NguyenContents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion 4.1 Chapter 4 Sets and Functions Discrete Struct

Nguyen An Khuong, Huynh Tuong Nguyen

Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.1

Chapter 4

Sets and Functions

Discrete Structures for Computer Science (CO1007) on Ngày

Nguyen An Khuong, Huynh Tuong Nguyen

Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Contents

1 Sets

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.3

Set Definition

• Set is afundamentaldiscrete structure on which all discrete

structures are built

• Sets are used to group objects, which often have thesame

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

Asetis an unordered collection of objects

The objects in a set are called theelements(phần tử ) of the set

A set is said tocontain(chứa) its elements

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Notations

Definition

• a ∈ A: a is an element of the set A

• a /∈ A: a isnotan 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}

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Venn Diagram

• John Venn in 1881

• Universal set(tập vũ trụ) is

represented by a rectangle

• Circlesand other

geometrical figuresare used

to represent sets

• Pointsare used to represent

particular elements in set

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.7

Special Sets

• Empty set(tập rỗng ) has no elements, denoted by ∅, or {}

• A set with one element is called asingleton set

• What is {∅}?

• Answer: singleton

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Subset

Definition

The set A is called asubset(tập con) of B iff every element of A

is also an element of B, denoted by A ⊆ B

If A 6= B, we write A ⊂ B and say A is aproper subset(tập con

thực sự) of B

• ∀x(x ∈ A → x ∈ B)

• For every set S,

(i)∅ ⊆ S,(ii)S ⊆ S

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4.9

Cardinality

Definition

If S has exactly n distinct elements where n is non-negative

integers, S isfinite set(tập hữu hạn), and n iscardinality(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

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Power Set

Definition

Given a set S, thepower set(tập lũy thừa) of S is the set of all

subsets of the set S, denoted byP (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 {∅}

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.11

Power Set

Theorem

If a set has n elements, then its power set has 2n elements

Prove using induction!

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Ordered n-tuples

Definition

Theordered 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

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.13

Cartesian Product

• René Descartes (1596–1650)

Definition

Let A and B be sets TheCartesian product(tích Đề-các) of A

and B, denoted by A × B, is the set of ordered pairs (a, b), where

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Nguyen An Khuong, Huynh Tuong Nguyen

Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Nguyen An Khuong, Huynh Tuong Nguyen

Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Nguyen An Khuong, Huynh Tuong Nguyen

Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Method of Proofs of Set Equations

To prove A = B, we could use

• Venn diagrams

• Prove that A ⊆ B and B ⊆ A

• Usemembership table

• Use set builder notation and logical equivalences

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.27

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

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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

• For each a ∈ A, if f (a) = b

• b is animage(ảnh) of a

• a ispre-image(nghịch ảnh/tạo ảnh) of f (a)

• f maps (ánh xạ) A to B

f

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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, }

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.31

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)

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation 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 Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

• Is f : Z → Z, f(x) = x2

onto?

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.35

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, b, c, 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 Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Example

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Contents Sets Set Operation 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 Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Function Composition

Definition

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

f ◦ g : A → C

f ◦ g(a) = f (g(a))

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.41Example

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

Definition

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Contents Sets Set Operation 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 Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Definition (Fibonacci Sequence)

Initial condition: f0= 0 and f1= 1

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Exercise (1)

Initial deposit: $10,000

Interest: 11%/year,compoundedannually (lãi suất kép)

After 30 years, how much do you have in your account?

Let Pn be the amount in the account after n years The sequence

{Pn} satisfies the recurrence relation

Pn= Pn−1+ 0.11Pn−1= (1.11)Pn−1

The initial condition is P0= 10, 000

Step 1 Solve the recurrence relation (iteration technique)

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In this sequence, integer 1 appears once, the integer 2 appears

twice, the integer 3 appears three times, and so on Therefore

integer n appears n times in the sequence

We can prove that (try it!)

62

X

i=1

i = 1953

so the next 63 numbers (until 2016) is 63

Therefore, 2012th number in the sequence is 63

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

=Pn+1 k=1ark

=Pn k=0ark+ (arn+1− a)

= Sn+ (arn+1− a)

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.49

Recursion

Definition (Recurrence Relation)

An equation thatrecursively definesa sequence

Definition (Recursion (đệ quy))

The act of defining an object (usually a function) in terms of that

object itself

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Recursive Algorithms

Definition

An algorithm is calledrecursiveif it solves a problem by reducing

it to an instance of the same problem with smaller input

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

else if n = 1 then return 1

else return fibonacci (n-1) + fibonacci (n-2)

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Tower of Hanoi

There is a tower in Hanoi that has three pegs mounted on a board

together with64gold disks of different sizes

Initially, these disks are placed on the first peg in order of size,

with the largest on the borrom

The rules:

1 Move one at a time from one peg to another

2 A disk is never placed on top of a smaller disk

Goals: all the disks on the third peg in order of size

The myth says thatthe world will endwhen they finish the

puzzle

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Tower of Hanoi – 1 Disc

1

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

Tower of Hanoi – 1 Disc

1

OK

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4.57

Tower of Hanoi – 2 Discs

2

1

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Tower of Hanoi – 2 Discs

Moved disc from peg 1 to peg 2

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Tower of Hanoi – 2 Discs

21

Moved disc from peg 2 to peg 3

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Contents Sets Set Operation Functions One-to-one and Onto Functions Sequences and Summation Recursion

4.61

Tower of Hanoi – 2 Discs

21

OK