(関係)

http://www.sw.it.aoyama.ac.jp/2015/Math1/lecture9.html

© 2005-15 Martin J. Dürst Aoyama Gakuin University

- 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, powerset
- Sets of numbers: Natural numbers (ℕ), integers (ℤ), rationals (ℚ), reals (ℝ), complex numbers (ℂ)
- Laws for sets (parallel to laws for Boolean operations)

(Pascal's triangle)

Start with a single `1`

in the first row, surrounded by zeroes
(`(0 ... 0) 1 (0 ... 0)`

).

Create row by row by adding the number above and to the left and the number
above and to the right.

1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1

- 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)_{n}`C`_{0}= 1 (the only subset of size 0 is {})_{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* - This is the same as the subsets of a given size
- The number of combinations is written
_{n}`C`_{m} - There are also
`permutations`(considering order), repeated permutations, and repeated combinations

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

(prove it as a homework)

Notation: `n`!

Definition: `n`! = 1 · 2 · ... (`n`-1) · `n` =
∏^{n}_{i=1} `i`

(∏ is called *product*)

Question:

1! = 1

0! = 1

(also *unit element*, *identity element*)

- Neutral element of addition: 0
- Neutral element of multiplication: 1
- Neutral element of set union: {}
- Neutral element of set intersection:
`U` - Neutral element of conjunction:
`true`

(this is the reason why ∀`x`∈{}:`R`(`x`) =`true`) - Neutral element of disjunction:
`false` - Neutral element of substraction:

Concrete example (sum):

int sum = 0; for (i=0; i<end; i++) sum += array[i];

In programming language C:

typeresult =単位元; for (i=0; i<end; i++) result = result演算子array[i];

In programming language Ruby:

array.inject(単位元) do |memo, next| memo演算子next end

- 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, 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`A`×`B`. - Example:
`A`= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, B = {3, 4}; R is the relation "is divisible by"

`R`= {(3, 3), (6, 3), (9, 3), (4, 4), (8, 4)} - (
`a`,`b`) ∈`R`can be written as`a``R``b`.

Examples:`a`>`b`,... - 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, 5},`a`>`b`: {(2,1), (3,1), (4,1), (4,2), (5,2), (4,3), (5,1), (3,2), (5,3), (5,4)}

- A relation is a set. We can therefore use set representations:
- Denotation

Example:`R`= {(3, 3), (6, 3), (9, 3), (4, 4), (8, 4)} - 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.

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
`R`is written`R`^{-1}. - The inverse relation of a binary relation is the relation with the order of the pairs reversed.
- Example:
`R`= {(3, 3), (6, 3), (9, 3), (4, 4), (8, 4)}

`R`^{-1}= {(3, 3), (3, 6), (3, 9), (4, 4), (4, 8)} `xR``y`⇔`y``R`^{-1}`x`- (
`R`^{-1})^{-1}=`R`

Deadline: November 26, 2015 (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`n`>0, 0<`m`<`n`-0 using_{n}`C`_{m}=_{n-1}`C`_{m-1}+_{n-1}`C`_{m} - Describe three relations from the real world that can be expressed as
mathematical relations, including examples of elements in the relation.

Example: Teachers and lecture halls: The relation is true if a teacher`t`teaches in lecture hall`l`. Example elements: (Martin Dürst, E-202), (Martin Dürst, E-203).

There will be a deduction if different students submit the same 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
- row
- 行
- column
- 列、欄
- correspond to
- と対応する
- 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