情報数学 I

第九回 (2012年12月 7日)

A の中の関係 R に対して次の性質がありうる:

関係の性質 (反射的、対称的、推移的、反対称的) の有無の組合せを一つ一つ調べ、できるだけ小さい具体例 (例えば {a, b, c} の中の関係とか {a, b, c, d} の中の関係として) を作りなさい。

反射的	対称的	推移的	反対称的	可能性・例など
T	T	T	T	{(a,a), (b,b), (c,c)} (半順序関係かつ同値関係)
T	T	T	F	直積集合など; 同値関係; {(a,a), (b,b), (c,c), (a,b), (b,a)}
T	T	F	T	不可能
T	T	F	F	{(a,a), (b,b), (c,c), (a,b), (b,c), (b,a), (c,b)}
T	F	T	T	半順序関係; {(a,a), (b,b), (c,c), (a,b)}
T	F	T	F	{(a,a), (a,b), (b,a), (b,b), (c,c), (c,d), (d,d)}*
T	F	F	T	{(a,a), (b,b), (c,c), (a,b), (b,c)}
T	F	F	F	{(a,a), (b,b), (c,c), (a,b), (b,c), (b,a)}
F	T	T	T	{(a,a), (c,c)}
F	T	T	F	{(a,a), (a,c), (c,a), (c,c)}
F	T	F	T	不可能
F	T	F	F	{(a,b), (b,c), (b,a), (c,b)}
F	F	T	T	{(a,a), (a,c)}
F	F	T	F	{(a,a), (a,b), (b,a), (b,b), (c,c), (c,d)}*
F	F	F	T	{(a,b), (b,c)}
F	F	F	F	{(a,b), (b,c), (b,a)}

例は * の場合 {a, b, c, d} の中の関係、それ以外は {a, b, c} の中の関係

If a relation is reflexive, antisymmetric, and transitive, then it is called a partial order relation
This is also often just called a 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 an 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

Some examples need a careful definition:

The relation on a set of tasks, where some tasks need be done after (or at the same time as) others
The relation "stronger (or equal) than" 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 indicated reflexivity
Rearange the vertices of the graph so that all arrows point upwards (or updownwards)
Remove the arrows that can be reconstructed using transitive closure
Remove the arrowheads

Example 1: The relation "needs to be done after (or at the same time as)" for the above tasks 1. to 4.

Example 2: The relation "divisible by" on the set {10, 5, 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 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 don't 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 ~~and above [or below]~~ are 1.

ある集合 A の全ての元 a と b に対して

a≥b 又はb≥a が成り立つ場合、

≥ を全順序関係 (total order relation)

(あるいは線形順序関係 (linear order relation) という

例: 整数、実数など; 日付や時間; 辞書での単語の順番

(algebraic system)

(group)

演算は二項演算「∘」一つ。公理:

例: (Z; +), (R-{0}; ×), 対称群 (symmetric group)

n 次対称群は n 個の要素の置き換えすべてを
含まれる
対称群の要素は 1..n の順列で表す
例: (2, 4, 1, 3) は (猫、犬、馬、牛) を
(犬、牛、猫、馬) に置き換える
対称群の二次演算は置き換えの連結
例: (2, 4, 1, 3) ∘ (1, 4, 2, 3) = (2, 3, 4, 1)
単位元は「何もしない」置き換え
例: (1, 2, 3, 4)
逆元: (2, 4, 1, 3) の逆元は (3, 1, 4, 2)
結合律は成立するが、交換律は成り立たない:
例: (2, 4, 1, 3) ∘ (1, 4, 2, 3) ≠
(1, 4, 2, 3) ∘ (2, 4, 1, 3)