# Sets

(集合)

## Discrete Mathematics I

### 8th lecture, Nov. 18, 2016

http://www.sw.it.aoyama.ac.jp/2016/Math1/lecture8.html

# Today's Schedule

• Minitest
• Summary and homework for 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

# 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 aB (or Ba)
(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 aB or Ba
(∈, ∋, ∉, 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}
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
0: ℕ including 0; ℕ+: positive ℕ
(ℕ may denote ℕ0 or ℕ+ depending on notation)
• ℤ: Integers (whole numbers, German: Zahlen (numbers))
• ℚ: Rational numbers (the Q comes from quotient)
• ℝ: Real numbers
• ℂ: Complex numbers

# 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: xAxB

# Element Uniformity

• Objects can be 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:
Examples: {1, {1,2}, {{1}, {1, {1,2}}}}

# Operation on Sets: Union

(also: sum)

• The union of two sets A and B is written AB.
• The union of sets A and B is the set of elements that belong to A or B (or both):
AB = {e|eAeB}
• Examples:
• A = {1, 2, 3, 4, 5}; B = {2, 4, 6, 8, 10}; C = {1, 5, 6, 8, 9}

AB = {1, 2, 3, 4, 5, 6, 8, 10}

AC = {1, 2, 3, 4, 5, 6, 8, 9}

BC = {1, 2, 4, 5, 6, 8, 9, 10}

# Operation on Sets: Intersection

(also: product)

• The intersection of two sets A and B is written AB.
• The intersection of sets A and B is the set of elements that belong to A and B:
AB = {e|eAeB}
• Examples:
• A = {1, 2, 3, 4, 5}; B = {2, 4, 6, 8, 10}; C = {1, 5, 6, 8, 9}

AB = {2, 4}

AC = {1, 5}

BC = {6, 8}

# Operation on Sets: Difference Set

(also: set difference)

• The difference set of A and B is written A - B (or AB).
• The difference set of sets A and B is the set of elements that belong to A but not to B.
A - B = {e|eAeB}
• 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

(complementary set)

• The complement of A is written Ac.
• The complement of set A is the set of all elements that do not belong to A (but belong to the universal set U).
Ac = {e|eUeA}; Ac = U-A.
• Examples:
U = {1,...,10}; A = {1, 2, 3, 4, 5}; B = {2, 4, 6, 8, 10}
Ac = {6, 7, 8, 9, 10}
Bc = {1, 3, 5, 7, 9}

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# Subset

• A subset of a set A is a set of some (zero or more) of the elements of A.
• We write BA (B is a subset of A) or AB (A is a superset of B)
• BA ⇔ ∀x: xBxA
• A: AA (any set is a subset of itself)
• If BA and BA, 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
(∀A: ∀x: x∈{}→xA)

# 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)}| =
• |{1, {1,2}, {{1}, {1, {1,2}}}}| =

# 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|BA}

• 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
• |S| = אn ⇒ |P(S)| = 2‎אn = אn+1
• It is unknown whether there is a cardinality between א‎0 and א‎1,... (Cantor's continuum hypothesis)

# Laws for Sets

1. Idempotent laws: AA = A; AA = A
2. Commutative laws: AB = BA; AB = BA
3. Associative laws: (AB) ∩ C = A ∩ (BC); (AB) ∪ C = A ∪ (BC)
4. Distributive laws: (AB) ∩ C = (AC) ∪(BC);
(AB) ∪ C = (AC) ∩ (BC)
5. Absorption laws: A ∩ (AB) = A; A ∪ (AB) = A
6. Involution law: A = (Ac)c
7. Law of the excluded middle: AAc = U
8. Law of (non)contradiction: AAc = {}
9. De Morgan's laws: (AB)c = AcBc;
(AB)c = AcBc

# 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 (AB=U, AB={}):
1. A: The set of all sets that include themselves (A = {a|aU, aa})
2. B: The set of all sets that do not include themselves (B = {b|bU, bb})
• B is a set and so BU. But does B belong to A or to B?
• Let's assume BA: BABBBB: contradiction
• Let's assume BB: BBBBBA: 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 24, 2016 (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|BA∧|B|=n}| = 3

# Glossary

set

element

denotation

brace (curly bracket)

connotation

natural number

integer

rational number

real number

complex number

equality

uniformity

instance

universal set

(set) union

(set) intersection

difference set/set difference

complement, complementary set

Venn diagram
ベン図
subset

superset

proper subset

empty set

size of a set

finite

finite set

power set
べき (冪) 集合
infinite set

cardinality, cardinal number

aleph zero
アレフ・ゼロ
continuum hypothesis

involution law