(関係の応用)

http://www.sw.it.aoyama.ac.jp/2015/Math1/lecture10.html

© 2005-15 Martin J. Dürst Aoyama Gakuin University

- Summary, leftovers, and homework for last lecture
- Composition of relations
- Classification of relations: Reflexive, symmetric, antisymmetric, transitive
- Equivalence relations, equivalence classes, and partitions
- Partial and total orders
- This week's homework

Please use the video recordings!

Often they are already available on Monday.

Representations of relations: Matrix, table, graph; inverse relations

- Combinations, permutations, factorial, unit element
- Definition of a relation: Subset of Cartesian product, set of tuples
- Representations of relations: Denotation, connotation, matrix, table, graph
- Inverse relations

- For two 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

(However, scalar multiplication and addition in the matrix multiplication are replaced by logical conjunction and disjunction)

- 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 0: `P`: Set of (player, team) tuples (e.g. soccer or
volleyball; (Keisuke Honda, AC Milan)); `Q`: Set of (team, hometown)
tuples (e.g. (AC Milan, Milan)); `R` = `P`∘`Q`:
Set of (player, hometown) tuples (e.g. (Keisuke Honda, Milan)).

Example 1: `P`: Set of (parent, child) tuples;
`P`∘`P`: Set of (grandparent, grandchild) tuples

Example 2: `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 binary relation on `A` can be:

- Reflexive:
`∀`:`x`∈`A``xRx`; ∀`x`∈`A`: (`x`,`x`) ∈`R` - Symmetric: ∀
`x`,`y`∈`A`:`xRy`⇔`yRx`;

∀`x`,`y`∈`A`: (`x`,`y`) ∈`R`⇔ (`y`,`x`) ∈`R` - Antisymmetric: ∀
`x`,`y`∈`A`:`xRy`∧`yRx`⇒`x`=`y` - Transitive: ∀
`x`,`y`,`z`∈`A`:`xRy`∧`yRz`⇒`xR``z`

- Definition: In a reflexive relation
`R`, ∀`x`∈`A`: (`x`,`x`) ∈`R` - Examples: =, ≤, divisible, subset, knows (people knowing each other), ...
- How to check: In the matrix representation, check that all entries on the (main) diagonal are 1

- Definition: In a symmetric relation
`R`, ∀(`x`,`y`)∈`R`: (`y`,`x`) ∈`R` - Examples: =, sibling, spouse, friend??, ...
- How to check: The matrix representation is identical with its
transposition.

(The transposition of a matrix is its rotation or mirroring along the (main) diagonal.)

- Definition: A relation
`R`is antisymmetric if ∀`x`,`y`∈`A`:`xRy`∧`yRx`⇒`x`=`y`. - Examples: =, <, ≤, divisible, anchestor, descendant, ...
- How to check: In the matrix representation, check that for each entry 1 not on the (main) diagonal, the entry in opposite position (mirrored along the (main) diagonal) is 0. In other words, of the two opposite entries, at most one can be 1.
**Anti**symmetric relation is**not**the opposite of symmetric relation.- The opposite of symmetric relation is called
**a**symmetric relation.

- Definition: If and only if for all
`x`,`y`, and`z`,`xRy`∧`yRz`⇒`xR`z, then`R`is transitive.

(∀`x`,`y`,`z`:`xRy`∧`yRz`⇒`xR``z`) ⇔`R`is transitive - Examples: =, ≤, ≥, descendant, anchestor, divisible, ...
- How to check: Compose
`R`with itself. If the result is`R`(i.e. if`R`∘`R`=`R`), then`R`is transitive.

- The
*transitive closure*of a relation`R`is the result of repeatedly concatenating`R`with itself until the result does not change anymore

`R`∘`R``∘R∘...` - "Repetition until there is no change anymore" is a frequent concept in Information Technology
- In the programming language C, this is the general structure:

int change = 1; while (change) { change = 0;/* process data */if (/* data changed */) change = 1; }

Trying to calculate the transitive closure of a relation may not be possible.

The calculation may not converge to a fixpoint.

Relations on sets of size 2:

- 11 relations are transitive
- For 4 relations,
`R`∘`R`is transitive - 1 relation alternates between two states [
`R`= (0 1, 1 0) =`R`(odd(^{x}`x`)); (1, 0, 0, 1) =`R`(even(^{y}`y`))]

Relations on sets of size 3:

- 123 relations are transitive
- 252 relations are transitive from
`R`^{2} - 66 relations are transitive from
`R`^{3} - 18 relations are transitive from
`R`^{4} - 6 relations are transitive from
`R`^{5} - 33 relations alternate between two states
- 12 relations alternate between two states starting with
`R`^{2} - 2 relations alternate between 3 states

- An
`n`-ary relation is a function`f`from`n`arguments to a Boolean value (T/F)

`R`= {(`x`,`y`,`z`) |`f`(`x`,`y`,`z`)=T } - A function returns only one result for each input
- An
`n`-ary relation can be seen as a function`g`with`n`-1 arguments and a set as a return value

`g`(`x`,`y`) = {`z`| (`x`,`y`,`z`) ∈`R`}

- A function with
`n`-1 arguments can be expressed as an`n`-ary relation

`f`(`x`,`y`) =`z`⇒`R`= {(`x`,`y`,`z`) |`f`(`x`,`y`) =`z`}

- Example of function: father (
`x`) =`y`(the father of`x`is`y`) - Example of predicate: father (
`y`,`x`) (`y`is the father of`x`) - Predicates express properties (mainly predicates with 1 argument) and relations (predicates with 2 or more arguments)
- Relations and predicates are very closely related concepts
- The difference is mostly in field of use (logic: predicates; databases,...: relations)

- Examples: People with the same birthday, month of birth, year of birth, zodiac sign; people from the same prefecture/country, cities in the same prefecture/country
- An
`equivalence relation`is a relation that is reflexive, symmetric, and transitive - An equivalece relation allows to define the set of all elements related
to a given element
`a` - Such sets are called
*equivalence classes*, and written [`a`]

([`a`] = {`x`|`x``R``a`}) `a``is a representative (element) of [``a`]- An equivalence relation creates a
*partition*of the original set`A` - A partition is a set of sets so that:
- The union of these sets is the original set
`A` - The intersection of any two distinct sets in the partition is {}

- The union of these sets is the original set
- The Cartesian product is also an equivalence relation

(where the partition consists of a single set, namely A itself)

- If a relation is reflexive, antisymmetric, and transitive, then it is
called a
*partial order relation* - This is also often just called an
*order relation* - The set on which the relation is defined is called a
*partially ordered set*or just an*ordered set* - The symbol ≤ is often used for order relations
- For any order relation ≤, the order relation ≥ and the relations > and < are also defined
- In any order relation, two elements
`x`and`y`can be in any of four mutually exclusive relationships:

`x`<`y`,`x`=`y`,`x`>`y`, or there is no relationship between`x`and`y`

- The
*divisible by*relation on the set of integers ≥1, or a subset thereof - The
*subset*relation on a set of sets

Some examples need a careful definition:

- The relation on a set of tasks, where some tasks need be done before
*or at the same time as*others - The relation "stronger than
*or as strong as*" in a Tennis tournament, defined by (the transitive closure of) the tournament results

An order relation can be represented by a Hasse diagram.

How to convert a directed graph of an order relation to a Hasse diagram:

- Remove arrows that indicate reflexivity
- Rearange the vertices of the graph so that all arrows point upwards (or downwards)
- Remove the arrows that can be reconstructed using transitive closure
- Remove the arrowheads

Example: The relation "divisible by" on the set {12, 6, 4, 3, 2, 1}

- The elements in a set
`A`are not ordered - Therefore, we can exchange (permute) the rows and the columns in the
matrix representation of a relation on
`A`if and only if we use the same permutation for both rows and columns. - A relation on a set
`A`is an equivalence relation if and only if we can permute the rows and columns so that we obtain the following:

- The areas of 1s form squares
- The centers of the squares are on the (main) diagonal of the matrix
- The squares do not overlap
- The entries on the (main) diagonal are all 1

- A relation on a set
`A`is an order relation if and only if we can permute the rows and columns so that we obtain the following:- All entries below the (main) diagonal [or above] are 0
- All entries on the (main) diagonal are 1

If for all elements `b` and `c` in a set `A`,

either `b`≥`c` or
`c``≥``b`, then

≥ is a `total order` (relation) or `linear order`
(relation)

Examples: ≥ for integers or reals; dates or time; order of words in a dictionary

Deadline: December 3, 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)

Investigate all combinations of the four properties of relations introduced in this lecture (reflexive, symmetric, antisymmetric, transitive). For each combination, give a minimal example or explain why such a combination is impossible.

Hint: There are 16 combinations. Two combinations are impossible. Two combinations need a set of four elements for a minimal example. Three combinations need a set of two elements for a minimal example. Two combinations need a set of one element for a minimal example. The other combinations need a set of three elements for a minimal example.

Hint: Use {`a`, `b`, `c`} for a set with three
elements.

Hint: Present the 16 combinations in a table similar to the tables used in the homework of lecture 4.

- composition
- 合成
- matrix multiplication
- 行列の掛け算、(通常の) 行列の積
- reflexive relation
- 反射的関係
- (main) diagonal
- (主) 対角線
- symmetric relation
- 対称的関係
- (matrix) transposition
- (行列) 転置
- sibling
- 兄弟 (姉妹も含む)
- antisymmetric relation
- 反対称的関係
- opposite
- 反対
- asymmetric relation
- 非対称的関係
- transitive relation
- 推移的関係
- descendant
- 子孫
- anchestor
- 先祖
- transitive closure
- 推移的閉包
- converge
- 収束
- fixpoint
- 不動点
- equivalence relation
- 同値関係
- equivalence class
- 同値類
- representative (element)
- 代表元
- partition
- 分割
- partial order
- 半順序
- partial order relation
- 半順序関係
- order relation
- 順序関係
- partially ordered set
- 半順序集合
- ordered set
- 順序集合
- mutually exclusive
- 相互排他的な
- Hasse diagram
- ハッセ図
- vertex (plural vertices)
- (グラフの) 節、頂点
- reconstruct
- 復元する
- square
- 正方形
- overlap
- 重なる、重複する
- total order (relation)
- 全順序 (関係)、線形順序 (関係)