第二章：

章節彙總：

• 集合：
  • 集合語言，集合操作，集合
• 函式：
  • 函式種類，函式操作，可計算性
• 序列：
  • 序列種類，求和公式
• 總和：
  • 可數集
• 矩陣：
• 矩陣算術

Chapter Summary

• Sets
  • The Language of Sets, Set Operations, Set Identities
• Functions
  • Types of Functions, Operations on Functions,
  • Computability
• Sequences and Summations
  • Types of Sequences, Summation Formulae
• Set Cardinality
  • Countable Sets
• Matrices
  • Matrix Arithmetic

集合：

小節彙總：

• 集的定義
• 描述集
  • 名冊方法
  • Set-Builder表示法
• 數學中的一些重要集合
• 空集和通用集
• 子集和設定平等
• 集合的基數
• 元組
• 笛卡爾積

Section Summary

• Definition of sets
• Describing Sets
  • Roster Method
  • Set-Builder Notation
• Some Important Sets in Mathematics
• Empty Set and Universal Set
• Subsets and Set Equality
• Cardinality of Sets
• Tuples
• Cartesian Product

簡介

• 集合是離散數學中考慮的物件型別的基本構建塊之一。
  • 計數很重要。
  • 程式語言已設定操作。
• 集合論是數學的一個重要分支。
  • 許多不同的公理系統已被用於發展集合論。
  • 這裡我們不關心集合論的正式公理集。 相反，我們將使用所謂的樸素集理論（第118頁）。
• Sets are one of the basic building blocks for the types of objects considered in discrete mathematics.
  • Important for counting.
  • Programming languages have set operations.
• Set theory is an important branch of mathematics.
  • Many different systems of axioms have been used to develop set theory.
  • Here we are not concerned with a formal set of axioms for set theory. Instead, we will use what is called naïve set theory( page 118).

集合：

• 集合是無序的物件集合。
  • 本課程的學生
  • 這個房間的椅子
• 集合中的物件稱為元素或集合的成員。 據說一組包含其元素。
• 符號a∈A表示a是集合A的元素。
• 如果a不是A的成員，請寫一個∉A

set：

• A set is an unordered collection of objects.
  • the students in this class
  • the chairs in this room
• The objects in a set are called the elements, or members of the set. A set is said to contain its elements.
• The notation a ∈ A denotes that a is an element of the set A.
• If a is not a member of A, write a ∉ A

描述集合：名冊方法

• S= {a，b，c，d}
• 順序不重要
• S = {a，b，c，d} = {b，c，a，d}
• 每個不同的物件都是成員; 不止一次列出同一成員不會改變集合。
• S = {a，b，c，d} = {a，b，c，b，c，d}
• 當規律明瞭時，Elipses（...）可用於描述一個集合而不列出所有成員。
• S = {a，b，c，d，......，z}

Describing a Set: Roster Method

• S = {a,b,c,d}
• Order not important
• S = {a,b,c,d} = {b,c,a,d}
• Each distinct object is either a member or not; listing more than once does not change the set.
• S = {a,b,c,d} = {a,b,c,b,c,d}
• Elipses (...) may be used to describe a set without listing all of the members when the pattern is clear.
• S = {a,b,c,d, ......,z }

• 英語字母表中所有母音的集合：V = {a，e，i，o，u}
• 小於10的所有奇數正整數的集合：O = {1,3,5,7,9}
• 小於100的所有正整數的集合：S = {1,2,3，........，99}
• 小於0的所有整數的集合：S = {....，-3，-2，-1}
• Set of all vowels in the English alphabet: V = {a,e,i,o,u}
• Set of all odd positive integers less than 10: O = {1,3,5,7,9}
• Set of all positive integers less than 100: S = {1,2,3,........,99}
• Set of all integers less than 0: S = {...., -3,-2,-1}

一些重要的集合

• N =自然數= {0,1,2,3 ....}
• Z =整數= {...， - 3，-2，-1,0,1,2,3，...}
• Z + = 正整數= {1,2,3，.....}
• R =實數集
• R + =正實數集合
• C =複數集合。
• Q =有理數集

Some Important Sets

• N = natural numbers = {0,1,2,3....}
• Z = integers = {...,-3,-2,-1,0,1,2,3,...}
• Z+ = positive integers = {1,2,3,.....}
• R = set of real numbers
• R+ = set of positive real numbers
• C = set of complex numbers.
• Q = set of rational numbers

描述一個集合：集生成器表示法

• 指定所有成員必須滿足的屬性：
• S = {x | x是小於100的正整數}
• O = {x | x是小於10的奇數正整數} O = {x∈Z+ | x是奇數，x <10}
• 可以使用謂詞：S = {x | P（x）}
• 示例：S = {x | Prime（x）}
• 正有理數：
• Q + = {x∈R| x = p / q，對於某些正整數p，q}

Describing a Set: Set-Builder Notation

• Specify the property or properties that all members must satisfy:
• S = {x | x is a positive integer less than 100}
• O = {x | x is an odd positive integer less than 10} O = {x ∈ Z+ | x is odd and x < 10}
• A predicate may be used: S={x|P(x)}
• Example: S = {x | Prime(x)}
• Positive rational numbers:
• Q+ = {x ∈ R | x = p/q, for some positive integers p,q}

間隔符號

• [a,b] = {x | a ≤ x ≤ b}
• [a,b)={x|a≤x<b}
• (a,b]={x|a<x≤b}
• (a,b)={x|a<x<b}
• 閉區間[a，b]
• 開區間（a，b）

Interval Notation

• [a,b] = {x | a ≤ x ≤ b}
• [a,b)={x|a≤x<b}
• (a,b]={x|a<x≤b}
• (a,b)={x|a<x<b}
• closed interval [a,b]
• open interval (a,b)

包含其他集的集

• 示例：集合{N，Z，Q，R}是一個包含四個元素的集合，每個元素都是一個集合。 這個集合的四個元素是

N，自然數的集合; Z，整數集; Q，有理數的集合; 和R，一組實數。

Sets containing other sets

• Example: The set {N,Z,Q,R} is a set containing four elements, each of which is a set. The four elements

of this set are N, the set of natural numbers; Z, the set of integers; Q, the set of rational numbers; and R, the set of real numbers.

電腦科學中的資料型別

• 電腦科學中資料型別或型別的概念建立在集合的概念之上。
• 特別是，資料型別或型別是集合的名稱，以及可以對該集合中的物件執行的一組操作。
• 例如，boolean是集合{0,1}的名稱以及該集合的一個或多個元素上的運算子，例如AND，OR和NOT。
• 你還有另一個例子嗎？

Datatype in Computer Science

• The concept of a datatype, or type, in computer science is built upon the concept of a set.
• In particular, a datatype or type is the name of a set, together with a set of operations that can be performed on objects from that set.
• For example, boolean is the name of the set {0, 1} together with operators on one or more elements of this set, such as AND, OR, and NOT.
• Do you have another example?

全集和空集

• 全集U是包含當前正在考慮的所有內容的集合。
  • 有時暗示
  • 有時明確說明。
  • 內容取決於具體情況。
• 空集（空集）是沒有元素的集合。 符號化∅，但也使用{}。

Universal Set and Empty Set

•  The universal set U is the set containing everything currently under consideration.
  • Sometimes implicit
  • Sometimes explicitly stated.
  • Contents depend on the context.
• The empty set(null set)is the set with no elements. Symbolized ∅, but {} also used.

要記住一些事情

• 集合可以是集合的元素。

{{1,2,3}，a，{b，c}}

{N，Z，Q，R}

• 空集與包含空集的集不同。

∅≠{∅}

Some things to remember

• Sets can be elements of sets.

{{1,2,3},a, {b,c}}

{N,Z,Q,R}

• The empty set is different from a set containing the empty set.

∅ ≠{ ∅ }

集合相等

• 定義：當且僅當它們具有相同的元素時，兩個集合是相等的。
  • 因此，如果A和B是集合，當且僅當A和B是相同的集合，則A和B相等，

我們認為A = B if A 和 B是等集 。

{1,3,5} = {3,5,1} {1,5,5,5,3,3,1} = {1,3,5}

Set Equality

Definition: Two sets are equal if and only if they have the same elements.

• Therefore if A and B are sets, then A and B are equal if and only if Any x(x∈A<->x∈B）
• We write A = B if A and B are equal sets.

{1,3,5} = {3, 5, 1}

{1,5,5,5,3,3,1} = {1,3,5}

維恩圖

John Venn（1834-1923）英國劍橋

• 可以用英語數學家約翰·維恩（John Venn）命名的圖表，以圖表的形式使用文字圖，他們在1881年介紹了它們的用法。

• 在維恩圖中，包含所有考慮物件的通用集U由矩形表示。

Venn Diagrams

John Venn (1834-1923) Cambridge, UK

• Sets can be represented graphically using Venn diagrams, named after the English mathematician John Venn, who introduced their use in 1881.

• In Venn diagrams the universal set U, which contains all the objects under consideration, is represented by a rectangle.

• Inside this rectangle, circles or other geometrical figures are used to represent sets. Sometimes points are used to represent the particular elements of the set.

• Venn diagrams are often used to indicate the relationships between sets.

子集

定義：集合A是B的子集，當且僅當A的每個元素也是B的元素時。

• 符號A⊆B用於表示A是集合B的子集。

• 當且僅當a為真時，A⊆B成立。

        1.因為對於每一組S，a∈∅總是假的，對於任何 S，∅⊆S。

        2.因為αS→a∈S，S⊆S，forevery setS。

        ⊆：子集

Subsets

Definition: The set A is a subset of B, if and only if every element of A is also an element of B.

• The notation A ⊆ B is used to indicate that A is a subset of the set B.

• A ⊆ B holds if and only if   is true.

            1. Because a ∈ ∅ is always false, ∅ ⊆ S , for every set S.

            2. Becausea∈S →a∈S, S⊆S, for every set S.

            ⊆ : subset of

表明一個集合是否是另一個集合的子集

• 表明A是一個B的子集：為表明A⊆B，表明如果x屬於A，那麼x也屬於B.

• 表明A不是B的子集：為了表明A不是B的子集，A⊈B，找一個屬於元素 x∈A與x∉B。（這樣的x是聲稱x∈Aimpliesx∈B的反例。）

例子：

1.你學校所有電腦科學專業的學生都是你學校所有學生的一部分。

2.平方小於100的整數集不是非負整數集的子集。

Showing a Set is or is not a Subset of Another Set

• Showing that A is a Subset of B: To show that A⊆B, show that if x belongs to A, then x also belongs to B.

• Showing that A is not a Subset of B: To show that A isnotasubsetofB,A⊈B, find an elementx∈Awith x ∉ B. (Such an x is a counterexample to the claim that x∈Aimpliesx∈B.)

Examples:

1. The set of all computer science majors at your school is a subset of all students at your school.

2. The set of integers with squares less than 100 is not a subset of the set of nonnegative integers.

再看看集合的平等性

• 回想一下，兩組A和B相等，用A = B表示，iff

• Any x（x∈A<->x∈B）

• 使用邏輯等價，我們得到A = B iff

• 這等價於

                            A⊆B和B⊆A

Another look at Equality of Sets

• Recall that two sets A and B are equal, denoted by A = B, iff

  • Any x（x∈A<->x∈B）

• Using logical equivalences we have that A = B iff

• This is equivalent to

                            A⊆B and B⊆A

真子集

定義：如果A⊆B，但A≠B，那麼wesayAisa B的真子集，用A⊂B表示。如果A⊂B，那麼

是真的。

Proper Subsets

Definition:IfA⊆B,butA ≠B,thenwesayAisa proper subset of B, denoted by A ⊂ B. If A ⊂ B, thenis true.

集合的基數

定義：如果S中恰好有n個不同的元素，其中n是非負整數，我們說S是有限的。 否則它是無限的。

定義：有限集A的基數，由| A |表示，是A的（不同）元素的數量。

例子：

1. |ø| = 0

2.設S是英文字母的字母。 然後| S | = 26

3. | {1,2,3} | = 3

4. | {ø} | = 1

5. 整數集是無限的。

Set Cardinality

Definition: If there are exactly n distinct elements in S where n is a nonnegative integer, we say that S is finite. Otherwise it is infinite.

Definition: The cardinality of a finite set A, denoted by |A|, is the number of (distinct) elements of A.

Examples:

1. |ø| = 0

2. Let S be the letters of the English alphabet. Then |S| = 26

3. |{1,2,3}| = 3

4. |{ø}| = 1

5. The set of integers is infinite.

冪集

定義：集合A的所有子集的集合，表示為P（A），稱為A的冪集。

示例：如果A = {a，b}則

P（A）= {ø，{a}，{b}，{a，b}}總共4個子集 - 2^2

• 如果一個集合有n個元素，那麼冪集的基數是2^n。

• 讓我們以另一種方式看同一個例子。 示例：如果A = {a，b}，則P（A）= {ø，{a}，{b}，{a，b}}

使用二進位制數字1表示元素包含在子集中; 使用二進位制數字0表示元素未包含在子集中。

00-子集中既不包含a也不包含b。ø

01 - b，而不是a，包含在子集中。{b}

10 - a，而不是b，包含在子集中。 {a}

11 - a或b都包含在子集中。{a，b}

• 如果一個集合有n個元素，那麼冪集的基數是2^n。

示例：購買錘子和螺絲刀的方式有多少種？

解：

A = {錘子，螺絲刀}然後

P（A）= {ø，

{一個錘子}，

{一把螺絲起子}，

{錘子，螺絲刀}}

示例：空集的冪集是多少？ 集合{∅}的冪集是多少？

解：

空集恰好有一個子集，即它本身。 所以，

P（∅）= {∅}。

集合{∅}恰好有兩個子集，即∅和

設定{∅}本身。 因此，P（{∅}）= {∅，{∅}}。

Power Sets

Definition: The set of all subsets of a set A, denoted P(A), is called the power set of A.

Example: If A = {a,b} then

P(A) = {ø, {a},{b},{a,b}} 4 subsets in total - 22

• If a set has n elements, then the cardinality of the power set is 2n.

• Let’ s look at the same example in another way. Example: If A = {a,b} then P(A) = {ø, {a},{b},{a,b}}

Use binary digit 1 to indicate an element is contained in a subset; use binary digit 0 to indicate an element is not contained in a subset.

00 – neither a nor b is contained in a subset. ø 01 – b, not a, is contained in a subset. {b}

10 – a, not b, is contained in a subset. {a}

11 – both a or b is contained in a subset. {a,b}

• If a set has n elements, then the cardinality of the power set is 2n.

Example: How many different ways are there to purchase a hammer and a screw driver?

Solution:

A = {a hammer, a screw driver} then P(A) = {ø,

{a hammer},

{a screw driver},

{a hammer, a screw driver}}

Example: What is the power set of the empty set? What is the power set of the set {∅}?

Solution:

The empty set has exactly one subset, namely, itself. Consequently,

P(∅) = {∅}.

The set {∅} has exactly two subsets, namely, ∅ and the

set {∅} itself. Therefore, P({∅}) = {∅, {∅}}.

有序的n元組

• 有序的n元組（a1，a2，.....，an）是有序集合，其中a1作為其第一個元素，a2作為其第二個元素，依此類推，直到作為其最後一個元素。

• 當且僅當它們的相應元素相等時，兩個n元組是相等的。

• 2元組稱為有序對。

• 當且僅當a = c且b = d時，有序對（a，b）和（c，d）相等。

Ordered n-Tuples

• The ordered n-tuple (a1,a2,.....,an) is the ordered collection that has a1 as its first element and a2 as its second element and so on until an as its last element.

• Two n-tuples are equal if and only if their corresponding elements are equal.

• 2-tuples are called ordered pairs.

• The ordered pairs (a,b) and (c,d) are equal if and only if a = c and b = d.

笛卡爾積

定義：由A×B表示的兩組A和B的笛卡爾乘積是有序對（a，b）的集合，其中a∈A和b∈B。

例：

A = {a，b} B = {1,2,3}

A×B = {（a，1），（a，2），（a，3），（b，1），（b，2），（b，3）}

• 定義：笛卡爾乘積A×B的子集R稱為從集合A到集合B的關係。關係將在第9章中詳細介紹。

• 定義：集合A1，A2，......，An的笛卡爾積，由A1×A2×......×An表示，是有序n元組（a1，a2，...）的集合。 .....，a）ai屬於Ai

對於i = 1，... n。

示例：什麼是A×B×C，其中A = {0,1}，B = {1,2}且C = {0,1,2}

解：A×B×C = {（0,1,0）,（0,1,1）,（0,1,2）,（0,2,0）,（0,2,1）,（ 0,2,2）,（1,1,0）,（1,1,1）,（1,1,2）,（1,2,0）,（1,2,1）,（1,2,2）}

Cartesian Product

Definition: The Cartesian Product of two sets A and B, denoted by A × B is the set of ordered pairs (a, b) where a∈A and b∈B.

Example:

A = {a,b} B = {1,2,3}

A × B = {(a,1),(a,2),(a,3), (b,1),(b,2),(b,3)}

• Definition: A subset R of the Cartesian product A × B is called a relation from the set A to the set B. (Relations will be covered in depth in Chapter 9. )

• Definition: The cartesian products of the sets A1,A2,......,An, denoted by A1 × A2 × ...... × An , is the set of ordered n-tuples (a1,a2,......,an) where ai belongs to Ai

for i = 1, ... n.

Example: What is A × B × C where A = {0,1}, B = {1,2} and C = {0,1,2}

Solution: A × B × C = {(0,1,0), (0,1,1), (0,1,2),(0,2,0), (0,2,1), (0,2,2),(1,1,0), (1,1,1), (1,1,2), (1,2,0), (1,2,1), (1,2,2)}

量詞的真實集合

• 給定謂詞P和域D，我們將P的真值集定義為D中P（x）為真的元素集。 P（x）的真值集用表示

• 例子：P（x）的真值集，其中域是整數，P（x）是“| x | = 1“是集合{-1,1}

Truth Sets of Quantifiers

• Given a predicate P and a domain D, we define the truth set of P to be the set of elements in D for which P(x) is true. The truth set of P(x) is denoted by

• Example: The truth set of P(x) where the domain is the integers and P(x) is “|x| = 1” is the set {-1,1}