(関係)

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

© 2005-17 Martin J. Dürst Aoyama Gakuin University

- 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

- Sets are a central concept of Mathematics
- Representation of sets: Denotation, connotation, Venn diagram
- Member (
`b`∈`A`), subset (`B`⊂`A`), 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)

- For a set
`A`with |`A`| =`n`, we can write

|{`B`|`B`⊂`A`∧|`B`|=`m`}| as_{n}`C`_{m} _{n}`C`_{n}= 1 (the only subset of size`n`is`A`itself, {`B`|`B`⊂`A`∧|`B`|=|`A`|=`n`} = {{`A`}})_{n}`C`_{0}= 1 (the only subset of size 0 is {}, {`B`|`B`⊂`A`∧|`B`|=1} = {{}})_{n}`C`_{m}=_{n-1}`C`_{m-1}+_{n-1}`C`_{m}(`n`>0, 0<`m`<`n`)

- 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
**c**ombinations is written_{n}`C`_{m} - There are also
`permutations`(considering order),*repeated permutations*(allowing an element to be selected more than once), and*repeated combinations*

_{n}`C`_{m}=`n`!/(`m`! (`n`-`m`)!)

(prove it as a homework)

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

- Relational databases
- Relations and graphs
- Relations and logical operations

- 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)

- 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`A`^{2}. - 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

(totally about 3000×1000×200×20000 ≅ 10^{13}quadruples)

- 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

- 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}

- Denotation
- Matrix representation
- Table representation
- Graph 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*)

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*.

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.

- 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.
`xR``y`⇔`y``R`^{-1}`x`; 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`

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)

- Prove
_{n}`C`_{m}=`n`!/(`m`! (`n`-`m`)!) for 0≦`n`, 0≦`m`≦`n`using_{n}`C`_{m}=_{n-1}`C`_{m-1}+_{n-1}`C`_{m}

(Hint: Prove first for`m`=0 and`m`=`n`, then for 0<`m`<`n`) - Describe three relations from the real world that can be expressed as
mathematical relations:
- A binary relation
**on**a single set. - A binary relation between two different sets.
- 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.

- A binary relation

- Pascal's triangle
- パスカルの三角形
- combinatorics
- 組合せ論
- combination
- 組合せ
- permutation
- 順列
- repeated combination
- 重複組合せ
- repeated permutation
- 重複順列
- factorial
- 階乗
- product (∏)
- 総乗、総積
- neutral element
- 単位元
- 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