Relations

(関係)

Discrete Mathematics I

9th lecture, December 6, 2019

https://www.sw.it.aoyama.ac.jp/2019/Math1/lecture9.html

Martin J. Dürst

© 2005-19 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Minitest
• Leftovers, summary, and homework for last lecture
• Relations:
  
  • Tuples
  • Cross product
  • Relations
  • Representations of relations
• This week's homework

Minitest: Preparation

ミニテストの注意点 (準備)

• Follow instructions by Professor and TA
  (先生と TA の指導に従うこと)
• Do not use the middle chair or the seats behind the camera
  (机の外側だけ使用、カメラの後ろに座らないこと)
• Use only pencils and eraser(s) 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)
• Sit at the place designated by the Professor
  (指定された席に座ること)
• Take out pencils and eraser(s) from bag, leave everything else in bag and put bag below your chair (鞄から鉛筆と消しゴムだけ取り出し、残り全部を鞄の中で椅子の下に置く)

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)

When to Hold Makeup Class

• Before New Year vacations or after?
• Weekday or Saturday?
• 1st or 6th period?

About makeup classes: The material in the makeup class is part of the final exam. If you have another makeup class at the same time, please inform the teacher as soon as possible.

補講について: 補講の内容は期末試験の対象。補講が別の補講とぶつかる場合には事前に申し出ること。

Homework 1 from Last Lecture: Ternary Logic

For ternary (three-valued) logic, create defining truth tables for conjunction, disjunction, and negation. The three values are T, F, and ?, where ? stands for "unknown" (in more words: "maybe true, maybe false, we don't know").

Hint: What's the result of "?∨T"? ? can be T or F, but in both cases the result will be T, so ?∨T=T.

[Sorry, removed!]

Homework 2 from Last Lecture:
Proof of Formula for Combinations

Prove _nC_m = n!/(m! (n-m)!) for 0≦n, 0≦m≦n using _nC_m = _n-1C_m-1 + _n-1C_m
(Hint: Prove first for m=0 and m=n, then for 0<m<n)

[Sorry, removed!]

Summary of Last Lecture

• Predicate logic allows more general statements and inferences than propositional logic
• The universal quantifier (∀) and the existential quantifier (∃) correspond to operations such as sum (∑) and product (∏)
• Universal quantification over an empty set produces T because the neutral element of conjunction is T (vacuous truth)

• The number of subsets of size m of a set of size n is the number of combinations of size m that can be taken from n elements
• These numbers form Pascal's triangle

Relations

• Importance of relations in IT
• Definition of relation
• Representation of relations

Importance of Relations for IT

• Relational databases
• Relations and graphs
• Relations and logical operations

Tuples

• Sets are not ordered. Tuples are ordered.
• An ordered pair is a tuple with two elements.
• The ordered pair of a and b is written (a, b).
• {a, b} = {b, a}. (a, b) ≠ (b, a).
• An n-tuple is an ordered sequence of n elements.
• Tuples with a fixed number of elements are called
  triple (3), quadruple (4), quintuple (5), sextuple (6), septuple (7), octuple (8), nonuple (9),...
• Example: Quintuple of (lecture, teacher, room, time, student)
  (Discrete Mathematics I, Martin J. Dürst, E-202, Friday-2, Hanako Aoyama)

Cartesian Product

• The Cartesian product (set) of two sets A and B is the set of all ordered pairs of elements from A and B.
• The Cartesian product of A and B is written A × B.
• A × B = {(x, y) | x ∈ A, y ∈ B}
  Example: A = {2, 3}, B = {5, 6}, A × B = {(2, 5), (2, 6), (3, 5), (3, 6)}
• Size of A × B: |A × B| = |A|·|B|
• Instead of A × A, one often writes A².
• The Cartesian product is also defined for more than two sets.
  Example 1: Cartesian product of A, B, C, D:
  A × B × C × D = {(x, y, z, v) | x ∈ A∧ y ∈ B ∧ z ∈ C ∧ v ∈ D}
  Example 2: Cartesian product of lectures, teachers, rooms, and students at Aoyama Gakuin University
  (totally about 3000×1000×200×20000 ≅ 10¹³ quadruples)

Definition of Relation

• A relation R between two sets A and B is defined as a subset of the Cartesian product A × B.
• Example: A = {1, 2, 3, 4, 5, 6, 7, 8}, B = {3, 4, 5}; R is the relation "is divisible by" (also called divisibility)
  R = {(3, 3), (6, 3), (4, 4), (8, 4), (5, 5)}
• (x, y) ∈ R can be written as x R y.
  Examples: x>y,...
• A relation between two sets is called a binary relation.
  There are also ternary relations, and so on.
• A binary relation between A and A is called a binary relation on A.
  Example: A = {1, 2, 3, 4}, a>b: {(2,1), (3,1), (4,1), (4,2), (4,3), (3,2)}
• Example: The relation including all quadruples of (lecture l, teacher t, room r, student s)
  where student s takes lecture l with teacher t in room r

Representation of Relations

• A relation is a set. We can therefore use set representations:
  • Denotation
    Example: R = {(3, 3), (6, 3), (4, 4), (8, 4), (5, 5)}
  • Connotation
    Example: R = {(x, y)| x ∈ A, y ∈ B, x mod y = 0}
• Matrix representation
• Table representation
• Graph representation

Matrix Representation

A relation between sets A and B is represented as a matrix where:

• Each row of the matrix corresponds to an element of A
• Each column of the matrix corresponds to an element of B
• If the row and column elements are related,
  the entry is 1 (true), otherwise 0 (false)

Matrix representation is suited for binary relations. For ternary,... relations, we need a tensor.

A matrix with only 1 or 0 as entries is called a logical matrix (also binary matrix, relation matrix, or Boolean matrix)

Table Representation

A relation between several sets is represented in a table as follows:

• Use a column for each set of the relation
  (i.e. two columns for a binary relation, three columns for a ternary relation)
• Use a row for each element of the relation (each tuple)

Table representation is suited for relations of any arity.

Table representation is suited for sparse relations
(relations with very few entries).

Table representation is used in relational databases.

Graph Representation

A relation between sets A and B is represented as a graph as follows:

• The elements of A and B are represented as vertices.
• A relation from an element of A to an element of B is represented as a directed edge between the corresponding vertices.
• If the vertices of A and B are well separated (e.g. A on the left, B on the right), then there may be no need to indicate direction.
• For a binary relation on A, the vertices are often drawn only once.

Graph representation is only suited for binary relations.

Inverse Relation

• The inverse relation of a binary relation R is written R^-1.
• The inverse relation is the relation with the order of the pairs reversed.
• xRy ⇔ yR^-1x; R^-1 = {(y, x) | (x, y) ∈ R}
• Example: R = {(3, 3), (6, 3), (4, 4), (8, 4), (5, 5)}
  R^-1 = {(3, 3), (3, 6), (4, 4), (4, 8), (5, 5)}
• (R^-1)^-1 = R

Composition of Relations

• For two binary relations P (from A to B) and Q (from B to C), we can define the composition R of P and Q
• We write the composition R of P and Q as R = P∘Q
  
• R = {(x, z) | (x, y) ∈ P ∧ (y, z) ∈ Q}
• The composition of two relations corresponds to the matrix multiplication of their matrix representations
  (in the matrix multiplication, scalar multiplication is replaced by conjunction, and addition is replaced by disjunction, or F and T are represented by 0 and 1, and results >1 are changed to 1)
  
• Attention: Depending on the field of mathematics, sometimes Q∘P is also used
  • P∘Q is derived from matrix multiplication
  • Q∘P is derived from function composition
    [the composition of functions p() and q() is q(p())]
  • In this lecture, we use P∘Q

Examples of Composition of Relations

Example 1: P: Set of (player, team) tuples (e.g. soccer or volleyball; (Shinji Kagawa, Borussia Dortmund)); Q: Set of (team, hometown) tuples (e.g. (Borussia Dortmund, Dortmund)); R = P∘Q: Set of (player, hometown) tuples (e.g. (Shinji Kagawa, Dortmund)).

Example 2: P: Set of (parent, child) tuples (e.g. (Ieyasu, Hidetada), (Hidetada, Iemitsu)); P∘P: Set of (grandparent, grandchild) tuples (e.g. (Ieyasu, Iemitsu))

Example 3: T: Trips made by riding on a single train ((Fuchinobe, Nagatsuta) ∈ T) → trips made by changing trains once (i.e. two train rides): (Fuchinobe, Shibuya) ∈ T∘T

Summary

• A relation is a mathematical concept important in many fields of information technology
• A relation is a subset of the cartesian product of a number of sets
• Relations can be represented as sets (using denotation or connotation), as a matrix, as a graph, or as a table
• Binary relations can be inverted and composed

This Week's Homework

Deadline: December 12, 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)

Describe three relations from the real world that can be expressed as mathematical relations:

1. A binary relation on a single set.
2. A binary relation between two different sets.
3. A relation between more than two sets.

For each relation, describe the set(s) used (including approximate size), the conditions for a tuple to be a member of the relation, the approximate size of the Cartesian product, and the approximate size of the relation, and give three examples of tuples belonging to the relation.

Example (for a binary relation between two different sets): Teachers (size ~1000) and lecture halls (size ~200) at AGU: The relation is true if a teacher t teaches in lecture hall l. Size of Cartesian product: ~200,000; size of relation: ~2000; Example elements: (Martin Dürst, E-202), (Martin Dürst, E-203).

Hint: If you do not understand the concept of relation very well yet, consult additional references (books, the Web)

There will be a deduction if different students submit the same relations.

Glossary

relational database

関係データベース

tuple

タプル

ordered pair

順序対

n-tuple

n 項組、n 字組

triple

三項組、三字組

quadruple

四項組、四字組

quintuple

五項組、五字組

sextuple

六項組、六字組

septuple

七項組、七字組

octuple

八項組、八字組

nonuple

九項組、九字組

Cartesian product (set)

直積 (集合)

definition

定義

divisible

割り切りが可能

binary relation

2項関係

ternary relation

3項関係

(binary) relation on A

A の中の関係、A の上の関係、A における関係

representation

表現

matrix

行列

binary (logical) matrix

論理行列

row

行

column

列、欄

correspond to

と対応する

tensor

テンソル

arity

アリティ

sparse

スパース、まばら (な)

vertex (plural: vertices)

頂点、節

edge

辺

directed

有向 (の)

opposite

反対

inverse relation

逆関係

composition

合成

matrix multiplication

行列の掛け算、(通常の) 行列の積