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

AGU

© 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

 

Summary of Last Lecture

 

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

 

A Program to Solve Problem 1

 

Last Week's Homework: Problem 2

Draw logic circuits of the following three Boolean formulæ:

  1. A¬DC
  2. NAND(G, XOR(H, K), H)
  3. NOR(¬E, C) ∧ G

 

The Importance of Sets

 

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

 

Elements and Membership

 

Element Uniformity

 

Notations for Sets

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})

 

Frequently used Sets of Numbers

 

Equality of Sets

 

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: ex = x = xe

 

Operation on Sets: Union

(also: sum)

 

Operation on Sets: Intersection

(also: product)

 

Operation on Sets: Set Difference

(result is called difference set)

 

Universal Set

 

Operation on Sets: Complement

(also: complementary set)

 

Venn Diagram

 
 
 
 
 
 
 

Subset

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

 

Size of a Set

 

Power Set

(also: powerset)

 

Size of Infinite Sets

 

Laws for Sets

  1. Idempotent laws: AA = A; AA = A
  2. Commutative laws: AB = BA; A ∪ B = B ∪ A
  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

 

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|BA∧|B|=n}| = 3

 

About Returns of Tests and Homeworks

 

Glossary

set
集合
prefecture
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