(関係)

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

© 2005-19 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

- 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。

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.

A |
B |
A ∨ B |
A ∧ B |
¬B |
---|---|---|---|---|

F | F | F | F | T |

F | ? | ? | F | ? |

F | T | T | F | F |

? | F | ? | F | |

? | ? | ? | ? | |

? | T | T | ? | |

T | F | T | F | |

T | ? | T | ? | |

T | T | T | T |

or (for ∨ and ∧):

A ∨ B |
F | ? | T |

F | F | ? | T |

? | ? | ? | T |

T | T | T | T |

A∧ B |
F | ? | T |

F | F | F | F |

? | F | ? | ? |

T | F | ? | T |

Proof of Formula for Combinations

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`)

Formula we want to prove:
_{n}`C`_{m} =
`n`!/(`m`!·(`n`-`m`)!)

Right edge of Pascal triangle:
_{n}`C`_{n} =
`n`! / (`n`!·(`n`-`n`)!) = `n`! /
`n`! = 1

Left edge of Pascal triangle:
_{n}`C`_{0} = `n`! /
(0!·(`n`-0)!) = `n`! / `n`! = 1

General case: We need to prove that
_{n-1}`C`_{m-1} +
_{n-1}`C`_{m} =
`n`!/(`m`! (`n`-`m`)!) for
`n`>0, 0<`m`<`n`

_{n-1}`C`_{m-1}
+ _{n-1}`C`_{m} =
(`n`-1)! / ((`m`-1)!·((`n`-1)-(`m`-1))!) +
(`n`-1)! / (`m`!·((`n`-1)-`m`)!) =

= `n`!·`m` /
(`n`·`m`!·(`n`-`m`)!) +
`n`!·(`n`-`m`) /
(`n`·`m`!·(`n`-`m`)!) =

= (`n`!·`m` + `n`!·(`n`-`m`))
/ (`n`·`m`!·(`n`-`m`)!) =
`n`!·(`m`+(`n`-`m`)) /
(`n`·`m`!·(`n`-`m`)!) =

= `n`!·`n` /
(`n`·`m`!·(`n`-`m`)!) = `n`! /
(`m`!·(`n`-`m`)!) Q.E.D.

- 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 (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`

- 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

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:

- 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 relations.

- 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