Sets

(集合)

Discrete Mathematics I

6th lecture, October 26, 2018

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

Martin J. Dürst

© 2005-18 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• 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

Schedule for the Next Few Weeks

• 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

Leftovers of Last Lecture

Summary of Last Lecture

• 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 (↔).

Last Week's Homework: Problem 1

Sorry, it was removed! :)

Last Week's Homework: Problem 2

Sorry, it was removed! :)

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)

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)

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}

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
  Example: A=B ⇔ ∀x: x∈A↔x∈B

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}

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 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:

1. Reductio ad absurdum (A→¬A = ¬A)
2. Contraposition
3. The associative law for disjunction
4. 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.

Additional Homework

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)

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
   

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