Properties of Relations

(関係の性質)

Discrete Mathematics I

10th lecture, December 2, 2022

https://www.sw.it.aoyama.ac.jp/2022/Math1/lecture10.html

Martin J. Dürst

© 2005-22 Martin J. Dürst Aoyama Gakuin University

Today's Schedule

• Leftovers, summary, and homework for last lecture
• Classification of relations: Reflexive, symmetric, antisymmetric, transitive
• Equivalence relations, equivalence classes, and partitions
• Partial and total orders
• This week's homework

Leftovers from Last Lecture

Summary of Last Lecture

• Definition of a relation: Subset of Cartesian product, set of tuples
• Representations of relations: Denotation, connotation, matrix, table, graph
• Operations on relations: Inverse relation, composition of relations

Last Week's Homework
Examples of Relations

Comment: As always, homework can be submitted in Japanese

Describe three relations from the real world that can be expressed as mathematical relations.
For each relation, describe the sets used (including their size), the conditions for a tuple to be a member of the relation, the size of the Cartesian product, and the size of the relation, and give three examples of tuples belonging to the relation.

1. A binary relation on a single set.

   Example solution: Relation on set of train stations (size ~1'000 for Kanto region), tuples of stations that can be reached from each other without changing trains, size of Cartesian product: ~1'000'000, size of relation ~50'000.
   Examples of tuples: (Fuchinobe, Nagatsuta), (Nagatsuta, Shibuya), (Hashimoto, Shinjuku); counterexample: (Fuchinobe, Shibuya)

2. A binary relation between two different sets.

   Example solution: Relation between Sumo wrestler and stable (heya); size of set of wrestlers: ~600, size of set of stables: ~40; Cartesian product is ~25·10³, size of relation is ~600
   Examples of tuples: (Terunofuji, Isegahama), (Abi, Shikoroyama), (Tobizaru, Oitekaze)

3. A relation between more than two sets.

   Example solution: Baseball: Relation between a set of teams (~100), a set of players (~10000), a set of positions (9), and a set of numbers in batting order (9), size of Cartesian product is ~81'000'000, size of relation is ~900
   Example (outdated): (Miami Marlins, Ichiro Suzuki, right field, 1st),...

Properties of Relations

A binary relation on a set A can be:

1. Reflexive
2. Symmetric
3. Antisymmetric
4. Transitive

Reflexive Relation

• Definition: In a reflexive relation R on set A, ∀x∈A: (x, x) ∈ R
• Examples: =, ≤, ≥, divisible (for ℕ⁺), subset, knows (people know themselves), ...
• How to check: In the matrix representation, check that all entries on the (main) diagonal are 1

Symmetric Relation

• Definition: In a symmetric relation R, ∀x, y: (x, y)∈R → (y, x)∈R; R = R^-1
• Examples: =, sibling (brother/sister), spouse (husband/wife), ... (, friend??)
• How to check: The matrix representation is identical with its transposition, i.e. R = R^-1 (for each symmetric pair (mirroring accross main diagonal), the values are the same)
  

Antisymmetric Relation

• Definition: A relation R on set A is antisymmetric if ∀x, y ∈A: xRy ∧ yRx ⇒ x=y.
  Alternative: ∀x, y ∈A, x≠y: (x, y)∈R → (y, x)∉R
• Examples: =, <, ≤, >, ≥, divisible, parent, child, anchestor, descendant, 'senpai', 'kohai',...
• How to check (two methods):
  1. In the matrix representation, check that for each entry 1 not on the (main) diagonal, the entry in the opposite position (mirrored along the (main) diagonal) is 0.
     In other words, of two opposite entries, at most one can be 1.

  2. ?	0	0	0
     ?	?	0	1
     ?	?	?	0
     ?	?	?	?

     Reorder rows and columns (in the same way!).
     If there is a reordering so that all entries above (or all below) the diagonal are 0,
     then the relation is antisymmetric.

Row/Column Reordering

• Ordering of rows/column in matrix representation is arbitrary
• Relations on a set use same order for rows/columns
• Some property checks only work for specific row/column orderings
• When you write a matrix, make sure the row/column order is clear by
  
  • Using the 'obvious' order
  • Labeling the rows/columns

Antisymmetric and Asymmetric Relations

• Antisymmetric relation is not the opposite of symmetric relation.
• The opposite of symmetric relation is called asymmetric relation.
• An asymmetric relation is a relation that is not symmetric.
• It is possible for a relation to be both symmetric and antisymmetric:
  All entries in the matrix representation except for the main diagonal are 0 (shown on right).

Transitive Relation

• Definition: If and only if for all x, y, and z, xRy ∧ yRz ⇒ xRz, then R is transitive.
  (∀x, y, z: xRy ∧ yRz ⇒ xRz) ⇔ R is transitive
• Examples: =, <, >, ≤, ≥, descendant, anchestor, divisible, ...
• How to check: Compose R with itself. If the result is in R (i.e. if R∘R ⊂ R), then R is transitive.

Exercise: Which of the following relations are transitive?

Properties of Relations (Summary)

A binary relation on A can be:

1. Reflexive: ∀x∈A: xRx; ∀x∈A: (x, x) ∈ R
2. Symmetric: ∀x, y ∈A: xRy ⇔ yRx;
   ∀x, y ∈A: (x, y) ∈ R ⇔ (y, x) ∈ R
3. Antisymmetric: ∀x, y ∈A: xRy ∧ yRx ⇒ x=y
4. Transitive: ∀x, y, z ∈A: xRy ∧ yRz ⇒ xRz

Idempotent Relations

• A relation for which R∘R = R is called idempotent
• All idempotent relations are transitive
• Relations that are reflexive and transitive are idempotent
• There are some idempotent relations that are not reflexive

Exercise: Which of the following relations are idempotent/reflexive/transitive?

Transitive Closure

• The transitive closure of a relation R is the result of repeatedly composing 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 pattern:

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 result of calculating a transitive closure, if it exists, is called a fixpoint.

Of all 16 relations on sets of size 2:

• 11 relations are idempotent (R∘R = R)
• 4 relations reach transitive closure at R∘R
• 1 relation alternates between two states
  [R = (0 1, 1 0) = R²ⁿ⁺¹; (1 0, 0 1) = R²ⁿ)]

Equivalence Relation

• If a relation is reflexive, symmetric, and transitive, then it is called an equivalence relation
• Examples:
  
  • People with the same birthday
  • People with the same month of birth
  • People with the same year of birth
  • People with the same zodiac sign
  • People from the same prefecture/country
  • Cities in the same prefecture/country
  • Natural numbers or integers with the same reminder when divided by a specific number
  • ...
• An equivalence relation allows to define the set of all elements related to a given element a
• The sets of equivalent elements (e.g. all people from the same prefecture,...) are called equivalence classes.

Equivalence Class

• The equivalence class of an element a is written [a]
  ([a] = {x|xRa})
• a is a representative (element) of [a]
• An equivalence relation creates a partition of the original set A
• The Cartesian product is also an equivalence relation
  (where the partition contains a single set, namely A itself)

Partition

• 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 {}
    (∀a, b: XOR([a]=[b], [a]∩[b]={}))
• Partition P of A: ∀b∈A: ∃C∈P: b∈C∧¬∃D∈(P-{C}): b∈D
  (every element b of A is an element of exactly one element C of P)

Equivalence Relations in Matrix Representation

A relation on a set A is an equivalence relation if and only if after suitable row/column reordering:

• The areas of 1s form squares
• The centers of the squares are on the (main) diagonal of the matrix (symmetry)
• The squares do not overlap
• The entries on the (main) diagonal are all 1 (reflexivity)
• If these conditions are met, transitivity is guaranteed

Partial Order

• 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:
  1. x < y
  2. x = y
  3. x > y
  4. There is no relationship between x and y

Examples of Order Relations

• The divisible by relation on the set of integers ≥1, or a subset thereof
• The subset relation on a set of sets (if degenerate cases are excluded)

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

Order Relations in Matrix Representation

A relation on a set A is an order relation if only if after suitable row/column reordering:

• All entries above (or below) the (main) diagonal are 0 (antisymmetry)
• All entries on the (main) diagonal are 1 (reflexivity)
• The relation is transitive (separate check needed)

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}

Total Order

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

if either b≥c or c≥b, then

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

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

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

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

Summary

• Binary relations on a set can be reflexive, symmetric, antisymmetric, or transitive
• Transitive closure is an operation often used in Information Technology
• Equivalence relations define a partition into equivalence classes
• (Partial) order relations can be represented with Hasse diagrams

This Week's Homework

(no need to submit)

Investigate all combinations of the four properties of relations introduced in this lecture (reflexive, symmetric, antisymmetric, transitive). For each combination, try to find an example relation (actual real-world data or small example expressed as a matrix).

Hint: Two combinations are impossible. Can you explain why?

Glossary

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)