Applications of Relations

(関係の応用)

Discrete Mathematics I

10th lecture, Dec. 1, 2017

http://www.sw.it.aoyama.ac.jp/2017/Math1/lecture10.html

Martin J. Dürst

AGU

© 2005-17 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

 

Video

Please use the video recordings!

Often they are already available on Monday afternoon.

 

Summary of Last Lecture

 

Composition of Relations

 

Examples of Composition of Relations

Example 0: P: Set of (player, team) tuples (e.g. soccer or volleyball; (Sinji Kagawa, Borussia Dortmund)); Q: Set of (team, hometown) tuples (e.g. (Borussia Dortmund, Dortmund)); R = PQ: Set of (player, hometown) tuples (e.g. (Shinji Kagawa, Dortmund)).

Example 1: P: Set of (parent, child) tuples; PP: 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) ∈ TT

 

Properties of Relations

A binary relation on A can be:

  1. Reflexive: xA:xRx; ∀xA: (x, x) ∈ R
  2. Symmetric: ∀x, yA: xRyyRx;
    x, yA: (x, y) ∈ R ⇔ (y, x) ∈ R
  3. Antisymmetric: ∀x, yA: xRyyRxx=y
  4. Transitive: ∀x, y, zA: xRyyRzxRz

 

Reflexive Relation

 

Symmetric Relation

 

Antisymmetric Relation

 

Transitive Relation

 

Transitive Closure

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

 

Cautions about Transitive Closure

Calculating the transitive closure of a relation may not be possible.

The calculation may not converge to a fixpoint.

Relations on sets of size 2:

 

Relations and Functions

 

Relations and Predicates

 

Equivalence Relation

 

Partial Order

 

Examples of Order Relations

Some examples need a careful definition:

 

Hasse Diagram

An order relation can be represented by a Hasse diagram.

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

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

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

 

Equivalence Relations and Order Relations in Matrix Representation

 

Total Order

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

if either bc or cb, then

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

(∀b, cA: b≥c ∨cb ⇔ ≥ is a total order on A)

The Hasse diagram of a total order is a single line, without branches

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

 

This Week's Homework

Deadline: December 7, 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)

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. One combination needs 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.

Additional homework: Bring some small scissors to the next lecture.

 

Glossary

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)