(関係)

http://www.sw.it.aoyama.ac.jp/2018/Math1/lecture9.html

© 2005-18 Martin J. Dürst Aoyama Gakuin University

- Minitest
- Leftovers, summary, and homework for last lecture
- Relations:

- Tuples
- Cross product
- Relations
- Representations of relations

- This week's homework

Sorry, it was removed! :)

Sorry, it was removed! :)

Proof of Formula for Combinations

Sorry, it was removed! :)

- 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

- 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: Quintuple of (lecture, teacher, room, time, student)

(Discrete Mathematics I, Martin J. Dürst, E-202, Friday-2, 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 at Aoyama Gakuin University

(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`

- 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 only 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`

- 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`

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;
`P`∘`P`: Set of (grandparent, grandchild) tuples

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`

Deadline: December 6, 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)

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.

- 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