Sets

(集合)

Discrete Mathematics I

6th lecture, November 8, 2019

https://www.sw.it.aoyama.ac.jp/2019/Math1/lecture6.html

Martin J. Dürst

© 2005-19 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• 第二回 情報テクノロジー学科同窓会 (2019年11月30日)
• 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
• Return of homework due October 19

第二回 情報テクノロジー学科同窓会

• 時間: 2019年11月30日 (土), 18:00-21:00
• 場所: アイビーホール青山・表参道 (青山キャンパス隣接)
• 詳細情報は学科の LMS からメールで配信予定

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 NOR. You can use NOR with any number of arguments ≧1, but you cannot use T or F.

Hint: Start with simple formulæ using NOR and find out which functions they represent.

How to Solve Problem 1

[Sorry, removed!]

A Program to Solve Problem 1

• The program uses the program languge Ruby
• Formulæ are represented by arrays (shown as [element1, element2,...])
  Examples: NOR(A) ⇒ [A], NOR(A,B,C)⇒[A,B,C], NOR(NOR(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, NOR(A), NOR(B), NOR(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:

1. A ∨ ¬D ∧ C
2. NAND(G, XOR(H, K), H)
3. 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, similar to Peano Arithmetic
  E.g.: 0 ≙ {}, 1 ≙ {{}}, 2 ≙ {{{}}}
• 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
    (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 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.)

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 (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=3c-4}
  Alternative: 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,...)

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

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

Neutral Element of an Operation

(also unit element, identity element, identity)

An element e is a neutral element for an operation △, 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)

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}; 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 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}

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}

    A - C = {1, 2}; C - A = {5, 6}

    B - C = {2, 8, 10}; C - B = {3, 5}

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
  

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

1. Idempotent laws: A ∩ A = A; A ∪ A = A
2. Commutative laws: A ∩ B = B ∩ A; A ∪ B = B ∪ A
3. Associative laws: (A∩B) ∩ C = A ∩ (B∩C); (A∪B) ∪ C = A ∪ (B∪C)
4. Distributive laws: (A∪B) ∩ C = (A∩C) ∪(B∩C);
   (A∩B) ∪ C = (A∪C) ∩ (B∪C)
5. Absorption laws: A ∩ (A∪B) = A; A ∪ (A∩B) = A
6. Involution law: A = (A^c)^c
7. Law of the excluded middle: A ∪ A^c = U
8. Law of (non)contradiction: A ∩ A^c = {}
9. 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 14, 2019 (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)

1. Create a set with four elements. If you use the same elements as other students, there will be a deduction.
2. Create the powerset of the set you created in problem 1.
3. 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	?
   ...	?

4. Express the relationship between the size of a set A and the size of its powerset P(A) as a formula.
5. Explain the reason behind the formula in problem 4.
6. 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

About Returns of Tests and Homeworks

• Today, the graded homeworks due October 19 will be returned
• This is not part of the lecture itself (i.e. after 12:30)
• If you have some other commitment after 12:30, you can come to my office in the afternoon (after 16:00) to pick up your homework
• Homeworks including names in kana/Latin letters will be distributed first, then those without
• Homeworks with higher points will be distributed before those with lower points
• When your name is called, immediately and very clearly raise your hand, and come to the front
• When taking your homework, make sure it is really yours
• NEVER take the homework of somebody else (a friend,...)
• Carefully analize your mistakes and work on fixing them and avoiding them in the future
• Feel free to ask questions

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