(集合)

http://www.sw.it.aoyama.ac.jp/2018/Math1/lecture6.html

© 2005-18 Martin J. Dürst Aoyama Gakuin University

- Schedule for the next few weeks
- Leftovers/summary/homework of last lecture
- Sets:
- Set membership and notations
- Operations on sets
- Subsets, powersets, the empty set
- Cardinality of sets
- Laws for sets
- Limits of set theory

- October 26 (today): Sets
- November 2:
**No lectures**(Aoyama Festival) - November 9: Predicate Logic, Quantifiers
- November 16: Applications of Predicate Logic
- November 23:
**No lectures**(Labor Thanksgiving Day) - November 30: Relations

- All Boolean formulæ can be expressed using only NAND (⊼) or only NOR (⊽).
- Logic circuits can be built from gates to implement Boolean functions.
- The main gates are AND, OR, NOT, NAND, NOR, XOR (⊕).
- There are many different ways to axiomatize Boolean logic (learn at least one set of axioms).
- Logical operations important for symbolic logic are implication (→) and equivalence (↔).

Sorry, it was removed! :)

Sorry, it was removed! :)

- Sets are one of the most fundamental concepts of Mathematics
- Sets can be used to represent natural numbers, similar to Peano
Arithmetic

E.g.: 0 ≙ {}, 1 ≙ {{}}, 2 ≙ {{{}}} - Sets are very important for Information Technology

The set of integers from 1 to 5: {1, 2, 3, 4, 5}

The set of prefectures in the Kanto area: {Kanagawa, Saitama, Chiba, Gunma, Tochigi, Ibaraki}

The set of campuses of Aoyama Gakuin University: {Sagamihara, Aoyama}

- An
*unordered*collection of objects

(i.e. {Sagamihara, Aoyama} = {Aoyama, Sagamihara}) - Conditions:
- It must be clear whether an object belongs to a set or not
- It must be clear whether two objects are the same or not

(one and*the same object*can belong to a set only*once*)

- Sets are usually denoted with upper-case letters (e.g.
`A`,`B`,`C`)

- The objects belonging to a set are called its
*elements* - Usually, lower-case letters are used to denote elements
- If an element
`b`belongs to a set`C`, we write`b`∈`C`(or`C`∋`b`)

(read:`b`is an element of set`C`;`b`is a member of`C`; element`b`belongs to set`C`;`C`contains element`b`) - If an element does not belong to a set, we write
`b`∉`C`or`C`∌`b`

(`b`∉`C`⇔`¬b`∈`C`; ∈, ∋, ∉, and ∌ are predicates written in the form of operators.)

- Elements can be anything: instances, categories, types, concepts,...

Examples:- Set of categories/types: {dog, cat, cow, horse, sheep, goat}
- Set of instances: {Garfield, Tom, Crookshanks, コロ、Sunny}

- There is no need for the elements in a set to be uniform

Example: {cow, happyness, Garfield, Mt. Fuji} - A set is also an object. Therefore, it can become an element of another
set:

Example: {1, {1,2}, {{1}, {1, {1,2}}}}

- Denotation (enumeration):

List up the elements separated by commas and enclose them in braces ({})

Examples: {a, b, c}, {1, 2, 3, 4}, {1, 3, 5, 7,...}

Reading for {a, b, c}: The set (with elements/members) a, b, and c. - Connotation (description of membership conditions):

Define the condition for elements

Examples:`A`= {`n`|`n`∈ ℕ,`n`>0,`n`<5},`B`= {{`c`,`d`}|`c`,`d`∈ℕ,`c`>3,`c`<10,`d`=3`c`-4}

Alternative:`A`= {`n`|`n`∈ ℕ ∧`n`>0 ∧`n`<5},`B`= {{`c`,`d`}|`c`∈ℕ∧`d`∈ℕ ∧`c`>3 ∧`c`<9 ∧`d`=3`c`-4}

Reading for {`n`|`n`∈ ℕ,`n`>0,`n`<5}: The set of all`n`, where`n`is a(n element of the) natural number(s),`n`is greater than 0, and`n`is smaller than 5

Express `A` and `B` using denotation:

`A` = {1, 2, 3, 4}

`B`={{4, 8}, {5, 11}, {23, 9}, {6, 14}, {17, 7},
{8, 20}}

Elements of the notation, in order from left to right (example:
{`n`|`n` ∈ ℕ, `n`>0, `n`<5})

- {: Opening brace
`n`: Variable or expression using variable(s)- |: Separator
`n`∈ ℕ,`n`>0,`n`<5: Conditions, connected by commas (for ∧) or logical operators; set-related conditions usually come first- }: Closing brace

- ℕ: (set of) natural
**n**umbers (5, 12, 47,...)

ℕ_{0}: ℕ including 0; ℕ^{+}: positive ℕ, not including 0

(ℕ may denote ℕ_{0}or ℕ^{+}depending on context) - ℤ: Integers (whole numbers; German:
**Z**ahlen (numbers))

(-7, 13, -43, 99,...) - ℚ: Rational numbers (the Q comes from
**q**uotient)

(¼, ½, -23, ¾, -⁵/₁₁, ⁵⁶⁷/₈₉,...) - ℝ:
**R**eal numbers (0.37, π, e, sin(53°),...) - ℂ:
**C**omplex numbers (23.7, √-1, -`i`, 7+3`i`,...)

- An element can belong to a set only once.
- The order of elements in a set is irrelevant.
- Example: {1, 2} = {2, 1} = {2, 1, 2},...
- More formally:

`A`=`B`⇔ ∀`x`:`x`∈`A`↔`x`∈`B` - Reading of ∀
`x`: for all`x`

- The
*empty set*is the set that contains no (zero) elements - The empty set is written {} or ∅
- When working with sets, always check for the empty set

(also *unit element*, *identity element*,
*identity*)

For an operation △, the neutral element `e` satisfies

∀`x`: `e`△`x` = `x` =
`x`△`e`

- Neutral element of addition: 0
- Neutral element of multiplication: 1
- Neutral element of conjunction (∧):
`true`

- Neutral element of disjunction (∨):
`false` - Neutral element of substraction:

does not exist, but 0 is a rigth identity (satisfying only ∀`x`:`x`=`x`△`e`)

(also: *sum*)

- The union of two sets
`A`and`B`is written`A`∪`B.` - The union of sets
`A`and`B`is the set of elements that belong to`A`**or**`B`(or both):

`A`∪`B`= {`e`|`e`∈`A`∨`e`∈`B`} - Neutral element of set union: {}
- Examples:
`A`= {1, 2, 3, 4};`B`= {2, 4, 6, 8, 10};`C`= {3, 4, 5, 6}`A`∪`B`= {1, 2, 3, 4, 6, 8}`A`∪`C`= {1, 2, 3, 4, 5, 6}`B`∪`C`= {2, 3, 4, 5, 6, 8, 10}

(also: *product*)

- The intersection of two sets
`A`and`B`is written`A`∩`B`. - The intersection of sets
`A`and`B`is the set of elements that belong to`A`**and**`B`:

`A`∩`B`= {`e`|`e`∈`A`∧`e`∈`B`} - Neutral element of set intersection:
`U` - Examples:
`A`= {1, 2, 3, 4};`B`= {2, 4, 6, 8, 10};`C`= {3, 4, 5, 6}`A`∩`B`= {2, 4}`A`∩`C`= {3, 4}`B`∩`C`= {4, 6}

- For logic, arithmetic, and other fields of mathematics, it is often
convenient to limit the objects used to be uniform.

Examples: Integers, students taking this lecture,... - Often, there is only one main kind of objects of interest
- In such cases, the set of all such objects is called the
*universal set*

- The universal set is often written
`U` - The universal set can also be the set of all possible elements

Example:`A`=`B`⇔ ∀`x`:`x`∈`A`↔`x`∈`B`

(result is called *difference set*)

- The difference set of
`A`and`B`is written`A`-`B`(or`A`∖`B`). - The difference set of sets
`A`and`B`is the set of elements that belong to`A`**but not**to`B`.

`A`-`B`= {`e`|`e`∈`A`∧`e`∉`B`}

- Examples:
`A`= {1, 2, 3, 4};`B`= {2, 4, 6, 8, 10};`C`= {3, 4, 5, 6}`A`-`B`= {1, 3};`B`-`A`= {6, 8}`A`-`C`= {1, 2};`C`-`A`= {5, 6}`B`-`C``= {2, 8, 10};``C`-`B``= {3, 5}`

(also: *complementary set*)

- The complement of
`A`is written`A`^{c}. - The complement of set
`A`is the set of all elements that do not belong to`A`(but belong to the universal set`U`).

`A`^{c}= {`e`|`e`∈`U`∧`e`∉`A`} =`U`-`A`. - Examples:

`U`= {1,...,10};`A`= {1, 2, 3, 4};`B`= {2, 4, 6, 8, 10}

`A`^{c}= {5, 6, 7, 8, 9, 10}

`B`^{c}= {1, 3, 5, 7, 9}

- A
*subset*of a set`A`is a set of some (zero or more) of the elements of`A` - We write
`B`⊂`A`(`B`is a subset of`A`) or`A`⊃`B`(`A`is a*superset*of`B`) `B`⊂`A`⇔ ∀`x`:`x`∈`B`→`x`∈`A`- ∀
`A`:`A`⊂`A`(any set is a subset of itself) - If
`B`⊂`A`and`B`≠`A`, then`B`is a*proper*subset of`A``.` - The empty set is a subset of every set (∀
`A`: {} ⊂`A`)

(reason: ∀`A`: ∀`x`:`x`∈{}→`x`∈`A`)

(Notation: Sometimes, ⊊ is used to denote proper subsets. Some authors use ⊂ for proper subsets, and ⊆ for subsets in general.)

- A finite set is a set with a finite number of elements.
- The number of elements in a set
`A`is written |`A`|. - Examples:
- |{dog, cat, cow, horse, sheep, goat}| = 6
- |{}| = 0
- |{
`n`|`n`≤20, prime(`n`)}| = 8 - |{1, {1,2}, {{1}, {1, {1,2}}}}| = 3

(also: *powerset*)

- The
*power set*of`A`is denoted`P`(`A`). - The power set of a set
`A`is the set of all subsets of`A`:

`P`(`A`) = {`B`|`B`⊂`A`} - Examples:
`P`({1, 2}) = {{}, {1}, {2}, {1, 2}}`P`({dog, cow, sheep}`) = {{}, {dog}, {cow}, {sheep}, {dog, cow}, {dog, sheep}, {cow, sheep}, {dog, cow, sheep}}``P`({Mt. Fuji}) = {{}, {Mt. Fuji}}`P`({}) = {{}}

- All infinite subsets of ℕ and ℤ have the same
*cardinality*

Example: |{1, 2, 3,...}| = |{1, 3, 5,...}|

Proof: 1↔1, 2↔3, 3↔5,... - This cardinality is denoted by א
_{0}(aleph zero)

- |ℚ| is also א
_{0} - |ℝ| > א
_{0}; |ℝ| = א_{1} - In general: |
`S`| = א_{n}⇒ |`P`(`S`)| = א_{n+1} - It is unknown whether there is a cardinality between
א
_{0}and א_{1},... (Cantor's continuum hypothesis)

- Idempotent laws:
`A`∩`A`=`A`;`A`∪`A`=`A` - Commutative laws:
`A`∩`B`=`B`∩`A`;`A`∪`B`=`B`∪`A` - Associative laws: (
`A`∩`B`) ∩`C`=`A`∩ (`B`∩`C`); (`A`∪`B`) ∪`C`=`A`∪ (`B`∪`C`) - Distributive laws: (
`A`∪`B`) ∩`C`= (`A`∩`C`) ∪(`B`∩`C`);

(`A`∩`B`) ∪`C`= (`A`∪`C`) ∩ (`B`∪`C`) - Absorption laws:
`A`∩ (`A`∪`B`) =`A`;`A`∪ (`A`∩`B`) =`A` - Involution law:
`A`= (`A`^{c})^{c} - Law of the excluded middle:
`A`∪`A`^{c}=`U` - Law of (non)contradiction:
`A`∩`A`^{c}= {} - De Morgan's laws: (
`A`∩`B`)^{c}=`A`^{c}∪`B`^{c};

(`A`∪`B`)^{c}=`A`^{c}∩`B`^{c}

- Set theory seems to be able to deal with anything, but there are limits.
- We can divide the set of all sets
`U`into two sets (`A`∪`B`=`U`,`A`∩`B`={}):`A`: The set of all sets that include themselves (`A`= {`a`|`a`∈`U`,`a`∈`a`})`B`: The set of all sets that do*not*include themselves (`B`= {`b`|`b`∈`U`,`b`∉`b`})

`B`is a set and so`B`∈`U`. But does`B`belong to`A`or to`B`?- Let's assume
`B`∈`A`:`B`∈`A`→`B`∉`B`→`B`∈`B`: contradiction - Let's assume
`B`∈`B`:`B`∈`B`→`B`∉`B`→`B`∈`A`: contradiction - There is no solution, so this is a paradox
- Concrete example: A library catalog of all library catalogs that do not list themselves.

Deadline: **November 1**, 2018 (Thursday), 19:00.

Format: **A4 single page** (using both sides is okay; NO cover
page; an additional page is okay if really necessary, but staple the pages
together at the top left corner), easily readable **handwriting**
(NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Box in front of room O-529 (building O, 5th floor)

Problem 1: Prove/check the following laws using truth tables:

- Reductio ad absurdum (A→¬
`A`= ¬A) - Contraposition
- The associative law for disjunction
- One of De Morgan's laws

Problem 2: Prove transitivity of implication (((`A`→`B`)
∧ (`B`→`C`)) ⇒ (`A`→`C`)) by
formula manipulation. For each step, indicate which law you used.

Hint: Show that ((`A`→`B`) ∧
(`B`→`C`)) → (`A`→`C`) is a tautology
by simplifying it to T.

Deadline: **November 8**, 2018 (Thursday), 19:00.

Format: **A4 single page** (using both sides is okay; NO cover
page), easily readable **handwriting** (NO printouts), name (kanji
and kana) and student number at the top right

Where to submit: Box in front of room O-529 (building O, 5th floor)

- Create a set with four elements. If you use the same elements as other students, there will be a deduction.
- Create the powerset of the set you created in problem 1.
- For sets
`A`of size zero to six, create a table of the sizes of the powersets (|`P`(`A`)|). Example:

| `A`|| `P`(`A`)|0 ? 1 ? ... ? - Express the relationship between the size of a set
`A`and the size of its powerset`P`(`A`) as a formula. - Explain the reason behind the formula in problem 4.
- Create a table that shows, for sets
`A`of size zero to five, and for each`n`(size of sets in`P`(`A`)), the number of such sets.

Example: |`A`|=3,`n`=2 ⇒ |{`B`|`B`⊂`A`∧|`B`|=`n`}| = 3

- set
- 集合
- prefecture
- 県
- element
- 元・要素
- denotation
- 外延的記法
- brace (curly bracket)
- 波括弧
- connotation
- 内包的記法
- natural number
- 自然数
- integer
- 整数
- rational number
- 有理数
- real number
- 実数
- complex number
- 複素数
- equality
- 同一性
- uniformity
- 一貫性
- instance
- 個体
- universal set
- 全体集合・普遍集合
- (set) union
- 和集合
- (set) intersection
- 積集合
- difference set/set difference
- 差集合
- complement, complementary set
- 補集合
- Venn diagram
- ベン図
- subset
- 部分集合
- superset
- 上位集合
- proper subset
- 真 (しん) の部分集合
- empty set
- 空 (くう) 集合
- size of a set
- 集合の大きさ
- finite
- 有限
- finite set
- 有限集合
- power set
- べき (冪) 集合
- infinite set
- 無限集合
- cardinality, cardinal number
- 濃数
- aleph zero
- アレフ・ゼロ
- continuum hypothesis
- 連続体仮説
- involution law
- 対合律
- paradox
- パラドックス
- library catalog
- 図書目録
- deduction (of points)
- 減点