Sets

(集合)

Discrete Mathematics I

6th lecture, October 23, 2020

https://www.sw.it.aoyama.ac.jp/2020/Math1/lecture6.html

Martin J. Dürst

Today's Schedule

Intermediate test
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

Intermediate Test

授業中の poll/quiz、ミニテスト:
- 他人との協力、情報交換は一切禁止
- プログラムなどの使用も一切禁止
- 資料、メモ、ウェブ検索は使用可
Prepare a new sheet of A4 paper, write your name and student number in the top right corner on both sides, divide the front and the back of the page in three parts, and label the parts with problem numbers. Prepare your smartphone for scanning.
Your personal test will arrive by email. Do not solve any other test.
Submit the test at https://moo.sw.it.aoyama.ac.jp/mod/assign/view.php?id=2629

Schedule for the Next Few Weeks

October 23 (today): Intermediate test, sets
October 30: No lectures (Aoyama Festival)
November 6: Predicate logic and quantifiers

Summary of Last Lecture

Logic circuits can be built from gates to implement Boolean functions.
The main gates are AND, OR, NOT, NAND, NOR, XOR (⊕).
All Boolean formulæ can be expressed using only NAND (⊼) or only NOR (⊽).
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 (↔).

Last Week's Homework: Problem 1

For each of the 16 Boolean functions of two Boolean variables A and B (same as problem 1 of last lecture), find the shortest formula using only NAND. You can use NAND with any number of arguments ≧1, but you cannot use T or F.

Hint: Start with simple formulæ using NAND and find out which functions they represent.

How to Solve Problem 1

Method 1: Use the solutions from problem one of lecture 4, replace ∨, ∧, and ¬ with their representation by NOR.
Problem: Solution may not be shortest one.
Method 2: Work backwards: Take the negation of the results of a Boolean function and try to create it from the conjunction of two Boolean functions for which you already have a solution.
Example: TFFT = ¬(FTTF) = ¬(FTTT ∧ TTTF) = NAND(NAND(NAND(A),NAND(B)),NAND(A, B))
Method 3: Try to use a program
- Use method 1 or 2
- Create all possible formulæ and evaluate them

A Program to Solve Problem 1

The program uses the program languge Ruby
Formulæ are represented by arrays (shown as [element1, element2,...])
Examples: NAND(A) ⇒ [A], NAND(A,B,C)⇒[A,B,C], NAND(NAND(A), B) ⇒ [[A],B]
Start with simple formulæ, create longer, more complicated formulæ:
- The simplest formulæ are single variables: A, B
- These form the set of expressions E₀ = {A, B}
- Create all formulæ with one or less NORs:
  All combinations of formulæ from E₀ without the empty combination, plus the formulæ in E₀
- E₁ = {A, B, NAND(A), NAND(B), NAND(A, B)}
- Continue in the same way for E₂, E₃, and so on
- The number of formulæ increases dramatically, so at each step, only keep one of the formulæ that produce the same result

Last Week's Homework: Problem 2

Draw logic circuits of the following three Boolean formulæ:

A ∨ ¬D ∧ C
NAND(G, XOR(H, K), H)
NOR(¬E, C) ∧ G

The Importance of Sets

Sets are one of the most fundamental concepts of Mathematics
Sets can be used to represent natural numbers
Sets are very important for Information Technology

Examples of Sets

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}

The Concept of a Set

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
  (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 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))

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
Reading of ∀x: for all x

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}}}}

Notations for Sets: Denotation

Denotation uses enumeration to describe a set
The elements are listed up separated by commas, and enclosed 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.

Notations for Sets: Connotation

Connotation is a formal description of the conditions for membership
Examples:
A = {n|n ∈ ℕ, n>0, n<5}
B = {{c, d}| c,d∈ℕ, c>3, c<10, d=3c-4}
Alternatives:
A = {n|n ∈ ℕ ∧ n>0 ∧ n<5}
B = {{c, d}| c∈ℕ∧ d∈ℕ ∧ c>3 ∧ c<9 ∧ d=3c-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}}

Connotation Details

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

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 context)
ℤ: Integers (whole numbers; German: Zahlen (numbers))
(-7, 13, -43, 99,...)
ℚ: Rational numbers (the Q comes from quotient)
(¼, ½, -23, ¾, -⁵/₁₁, ⁵⁶⁷/₈₉,...)
ℝ: Real numbers (0.37, π, e, sin(53°),...)
ℂ: Complex numbers (23.7, √-1, -i, 7+3i,...)

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
The universal set can also be the set of all possible elements (including all possible sets)

The Empty Set

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

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 = {d| d∈A∨d∈B}
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}

Operation on Sets: Intersection

(also: product)

The intersection of sets A and B is written A ∩ B.
The intersection of sets A and B is the set of elements that belong to both A and B:
A∩B = {d|d∈A∧d∈B}
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}

Neutral Element of an Operation

(also unit element, identity element, identity)

An element e is a neutral element for an operation △ if and only if
∀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 subtraction:
does not exist, but 0 is a rigth identity (satisfying only ∀x: x = x△e)
Neutral element of set union: {}
Neutral element of set intersection: U

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

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}; B = {2, 4, 6, 8, 10}
A^c = {5, 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.
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.)

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
Properties of U and {}: U∩A = A, U∪A = U, {}∩A = {}, {}∪A = A

Natural Numbers as Sets

It is possible to represent natural numbers as sets,
by using 0 ≙ {} and succ(a) = a∪{a}.
This leads to the following values:
- 0 ≙ {}
- 1 ≙ {{}} (= {} ∪ {{}})
- 2 ≙ {{}, {{}}} (= {{}} ∪ {{{}}})
- 3 ≙ {{}, {{}}, {{}, {{}}}}
- 4 ≙ {{}, {{}}, {{}, {{}}}, {{}, {{}}, {{}, {{}}}}} (= {{}, {{}}, {{}, {{}}}} ∪ {{{}, {{}}, {{}, {{}}}}})
- 5 ≙ …
The axioms of natural number arithmetic (Peano axioms) can then be derived from the axioms of set theory

Limits of Sets

Set theory seems to be able to deal with anything, but there are limits.
The following paradox shows the limits of set theory.
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: Powersets

Deadline: October 29, 2020 (Thursday), 22:00.

Format: A4 scanned PDF (more than one page is okay, but single file; NO cover page), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Moodle assignment

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 ⇒ |{S|S⊂A∧|S|=n}| = 3

Additional Homework: Proofs of Logic Formulæ

Deadline: Thursday November 5, 2020, 22:00.

Format: A4 scanned PDF (more than one page is okay, but single file; NO cover page), easily readable handwriting (NO printouts), name (kanji and kana) and student number at the top right

Where to submit: Moodle assignment

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

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

Problem 2: Prove transitivity of implication by formula manipulation:

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.

Glossary

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