Relations

(関係)

Discrete Mathematics I

9th lecture, November 25, 2022

https://www.sw.it.aoyama.ac.jp/2022/Math1/lecture9.html

Martin J. Dürst

© 2005-22 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Summary and homework from last lecture
• Importance of relations for IT
• Definitions: Tuples, cross products, and relations
• Representations of relations
• Operations on relations: Inversion, Composition
• Relations, functions, and predicates
• This week's homework

Homework 1: 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.

[solution remonved]

Homework 2:
Examples for Laws for Quantifiers

Give one example for law 4 and two examples (one for each direction) for law 11 of the laws for quantifiers. Do not use students in your example. There will be a deduction for examples that are too similar to those submitted by other students.

Summary of Last Lecture

• Predicate logic allows more general statements and inferences than propositional logic
• Formulæ including quantifiers can be rewritten using the laws for quantifiers
• 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)

Examples of Relations

• Relation between students, teachers, lectures, and rooms:
  Hanako Aoyama, Martin J. Dürst, Discrete Mathematics I, E-202
  Taro Sagamihara, Martin J. Dürst, Discrete Mathematics I, E-202
  ...
• Relation of train stations that can be reached without changing trains:
  Fuchinobe, Machida
  Machida, Shinjuku
  Yokohama, Narita
  ...

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 (student, teacher, lecture, room, time)
  (Hanako Aoyama, Martin J. Dürst, Discrete Mathematics I, E-202, Friday-2)

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.
• Example: The quaternary 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
• 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)}  

Representation of Relations

(Example: A = {1, 2, 3, 4, 5, 6, 7, 8}, B = {3, 4, 5}; R is the divisibility relation)

• Set representations (because every relation is a set):
  • 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 (shown as dots).
• A relation from an element of A to an element of B is represented as a directed edge between the corresponding vertices (shown as arrows).
• 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  

Representations of Inverse Relations

• Graph representation: Invert the direction of the edges in the graph
• Table representation: Exchange the two columns
• Matrix representation: Transpose the matrix (exchange vertical and horizontal direction)

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}

Examples of Composition of Relations

Example 1: P: Set of (student, lecture) tuples (e.g.: (Hanako Aoyama, Discrete Mathematics I)); Q: Set of (lecture, teacher) tuples (e.g.: (Discrete Mathematics I, Martin Dürst)); R = P∘Q: Set of (student, teacher) tuples (e.g.: (Hanako Aoyama, Martin Dürst)).

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

Composition of Relation: Additional Comments

• 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; in other words, when F and T are represented by 0 and 1, any 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

Relations and Functions

• An n-ary relation is equivalent to a function f from n arguments to a Boolean value (T/F)
  R = {(x, y, z) | f(x, y, z)=T }
• A function returns only one result for each input
• An n-ary relation can be seen as a function g with n-1 arguments and a set as a return value
  g(x, y) = {z | (x, y, z) ∈ R}
  
• A function with n-1 arguments can be expressed as an n-ary relation
  f(x, y) = z ⇒ R = {(x, y, z) | f(x, y) = z}

Relations and Predicates

• Example of function: parent (x) = y (the parent of x is y)
• Example of predicate: parent (y, x) (y is the parent of x)
• Predicates express properties (mainly predicates with 1 argument)
  and relations (predicates with 2 or more arguments)
• Relations and predicates are very closely related concepts
• The difference is mostly in field of use:
  predicates: logic
  relations: structure, databases
• There are also 1-tuples (singletons), and even a 0-tuple,
  but relations (subsets) defined with them are not very interesting

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 1, 2022 (Thursday), 18:40.

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 (not just made-up toy examples) 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) and the conditions for a tuple to be a member of the relation. Give the approximate sizes of the Cartesian product and the relation, and give three examples of tuples belonging to the relation. Make sure you describe the sets, and do not enumerate them. E.g. "prefectures of Japan", not {Tokyo, Kanagawa, Chiba, Saitama}

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)

Important: There will be a deduction if different students submit the same or very similar 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