Sets

(集合)

Discrete Mathematics I

8th lecture, Nov. 17, 2017

http://www.sw.it.aoyama.ac.jp/2017/Math1/lecture8.html

Martin J. Dürst

Today's Schedule

Minitest
Summary and homework for last lecture, leftovers
Sets:
- Set membership and notations
- Operations on sets
- Subsets, powersets, the empty set
- Cardinality of sets
- Laws for sets
- Limits of set theory

Minitest: Preparation

(ミニテストの注意点)

Follow instructions by Professor and TA
(先生と TA の指導に従うこと)
Do not use the middle chair
(机の外側だけ使用)
Use only pencils and erasers during the test
(試験中は鉛筆と消しゴムだけ使用可)
Put everything else (including pen case) into your bag, and put your bag below your chair
(ペンケースなども含め荷物は全て鞄の中にまとめ、椅子の下に置く)

Minitest: Latecommers

ミニテストの注意点 (遅刻者)

Follow instructions by Professor and TA
(先生と TA の指導に従うこと)
Wait in a single line on the right side of the lecture hall
(講堂の右端に一列に並んで待つ)
While waiting, take out pencil and eraser from your bag
(並んでいる間に鉛筆と消しゴムを荷物から出す)
When directed to do so, put your bag on the podium and sit down where directed
(指示あるとき、荷物を前の台に置いて、指示される席に着席)

Minitest: Collection

ミニテストの注意点 (終了時)

Follow instructions by Professor and TA
(先生と TA の指導に従うこと)
Do not move around or talk before collection of all examinations is completed
(試験用紙の回収が完全に終了するまでに一切動かない、音を出さない)

Summary of Last Lecture

Important points for quantifiers:

What is the universal set?
(Example: ∀i ∈ℕ⁺: i>0)
Notation (colons, parentheses)
Definition of used predicates
Free vs. bound variables

The Concept of a Set

An unordered collection of objects
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)

Elements and Membership

The objects belonging to a set are called its elements
Usually, lower-case letters are used to denote elements
If an element a belongs to a set B, we write a ∈ B (or B ∋ a)
(a is an element of set B; a is a member of B; element a belongs to set B; B contains element a)
If an element does not belong to a set, we write a ∉ B or B ∌ a
(a∉B = a∈B; ¬∈, ∋, ∉, and ∌ are predicates written in the form of operators.)

Notation for Sets

Denotation (enumeration):
We list up the elements separated by commas and enclose them in braces ({})
Examples: {a, b, c}, {1, 2, 3, 4}, {1, 3, 5, 7,...}
Connotation (description of membership conditions):
We define the condition for elements
Examples: A = {n|n ∈ ℤ, n>0, n<5}, B = {{c, d}| c,d∈ℕ, c>3, c<10, d=3c-4}
Alternative: A = {n|n ∈ ℤ ∧ n>0 ∧ n<5}, B = {{c, d}| c∈ℕ∧ d∈ℕ ∧ c>3 ∧ c<10 ∧ d=3c-4}

Convert A to denotation:

A = {1, 2, 3, 4} B={{4, 8}, {5, 11}, {23, 9}, {6, 14}, {17, 7}, {8, 20}}

Frequently used Sets of Numbers

ℕ: (set of) natural numbers (5, 12, 47,...)
ℕ₀: ℕ including 0; ℕ⁺: positive ℕ, not including 0
(ℕ may denote ℕ₀ or ℕ⁺ depending on notation)
ℤ: Integers (whole numbers, German: Zahlen (numbers))
(-7, 13, -43, 99,...)
ℚ: Rational numbers (the Q comes from quotient)
(¼, ½, ¾, -⁵/₁₁, ⁵⁶⁷/₈₉,...)
ℝ: Real numbers (0.37, π, e, sin(53°),...)
ℂ: Complex numbers (23.7, -i, 7+3i,...)

Equality of Sets

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

Element Uniformity

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}}}}

Operation on Sets: Union

(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, 5}; B = {2, 4, 6, 8, 10}; C = {1, 5, 6, 8, 9}
  A ∪ B = {1, 2, 3, 4, 5, 6, 8, 10}
  
  A ∪ C = {1, 2, 3, 4, 5, 6, 8, 9}
  
  B ∪ C = {1, 2, 4, 5, 6, 8, 9, 10}

Operation on Sets: Intersection

(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, 5}; B = {2, 4, 6, 8, 10}; C = {1, 5, 6, 8, 9}
  A ∩ B = {2, 4}
  
  A ∩ C = {1, 5}
  
  B ∩ C = {6, 8}

Operation on Sets: Set Difference

(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, 5}; B = {2, 4, 6, 8, 10}; C = {1, 5, 6, 8, 9}
  A - B = {1, 3, 5}; B - A = {6, 8, 10}
  
  A - C = {2, 3, 4}; C - A = {6, 8, 9}
  
  B - C = {2, 4, 10}; C - B = {1, 5, 9}

Universal Set

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

Operation on Sets: Complement

(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, 5}; B = {2, 4, 6, 8, 10}
A^c = {6, 7, 8, 9, 10}
B^c = {1, 3, 5, 7, 9}

Venn Diagram

Subset

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.

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

The Empty Set

The empty set is the set that contains no (zero) elements
The empty set is written {} or ∅
The empty set is a subset of every set:
∀A: {} ⊂ A
(reason: ∀A: ∀x: x∈{}→x∈A)

Size of a Set

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

Power Set

(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({}) = {{}}

Size of Infinite Sets

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 א₀ (aleph zero)
|ℚ| is also א‎₀
|ℝ| > א‎₀; |ℝ| = א‎₁
In general: |S| = א_n ⇒ |P(S)| = א_n+1
It is unknown whether there is a cardinality between א‎₀ and א‎₁,... (Cantor's continuum hypothesis)

Laws for Sets

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

Limits of Sets

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={}):
1. A: The set of all sets that include themselves (A = {a|a ∈ U, a ∈ a})
2. 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.

This Week's Homework

Deadline: November 23, 2017 (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

Glossary

set: 集合
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