# Relations

(関係)

## Discrete Mathematics I

### 9th lecture, Nov. 24, 2017

http://www.sw.it.aoyama.ac.jp/2017/Math1/lecture9.html

# Today's Schedule

• Leftovers, summary, and homework for last lecture
• Pascal's triangle and combinations
• Factorial and neutral element
• Relations:
• Tuples
• Cross products
• Relations
• Representations of relations
• This week's homework

# Summary of Last Lecture

• Sets are a central concept of Mathematics
• Representation of sets: Denotation, connotation, Venn diagram
• Member (bA), subset (BA), powerset (P(A)), universal set (U)
• Set operations: Union, intersection, difference, complement
• Sets of numbers: Natural numbers (ℕ), integers (ℤ), rationals (ℚ), reals (ℝ), complex numbers (ℂ)
• Laws for sets (parallel to laws for Boolean operations)

# Subsets and Pascal's Triangle

• For a set A with |A| = n, we can write
|{B|BA∧|B|=m}| as nCm
• nCn = 1 (the only subset of size n is A itself, {B|BA∧|B|=|A|=n} = {{A}})
• nC0 = 1 (the only subset of size 0 is {}, {B|BA∧|B|=1} = {{}})
• nCm = n-1Cm-1 + n-1Cm (n>0, 0<m<n)

# Subsets and Combinations

• Combinatorics is very important for Information Technology
• Combinatorics deals with counting the number of different things under various conditions or restrictions
• The word combinations refers to the choices of a given size from a set without repetitions and without considering order
• Combinations of a certain size selected from a set are the same as the subsets of a given size
• The number of combinations is written nCm
• There are also permutations (considering order), repeated permutations (allowing an element to be selected more than once), and repeated combinations

# Direct Formula for Combinations

• nCm = n!/(m! (n-m)!)

(prove it as a homework)

# 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: Quadruple of (lecture, teacher, room, student)
(Discrete Mathematics I, Martin J. Dürst, E202, 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) | xA, yB}
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) | xAy ∈ B ∧ z ∈ C ∧ v ∈ D}
Example 2: Cartesian product of lectures, teachers, rooms, and students

# 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 at Aoyama Gakuin University

# 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)| xA, 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 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.
• xRyyR-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

# This Week's Homework

Deadline: November 30, 2017 (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. Prove nCm = n!/(m! (n-m)!) for 0≦n, 0≦mn using nCm = n-1Cm-1 + n-1Cm
(Hint: Prove first for m=0 and m=n, then for 0<m<n)
2. 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 relation.

# Glossary

Pascal's triangle
パスカルの三角形
combinatorics

combination

permutation

repeated combination

repeated permutation

factorial

product (∏)

neutral element

relational database

tuple
タプル
ordered pair

n-tuple
n 項組、n 字組
triple

quintuple

sextuple

septuple

octuple

nonuple

Cartesian product (set)

definition

divisible

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

matrix

binary (logical) matrix

row
column

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)

edge
directed

opposite

inverse relation