Theory of Relations (1)

Theory of Relations (1)
Course of Mathematics
Pusan National University .

Yoshhiro Mizoguchi
.
Institute of Mathematics for Industry
Kyushu University, JAPAN
ym@imi.kyushu-u.ac.jp

September 29-30, 2011

Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 1 / 35

Table of Contents

1 Relational Calculus
Basic Notations
Matchings
Dedekind Formula

2 Cardinality of relations
Basic concepts
Properties

3 Product and Coproduct
Coproduct relations
Product relations

4 Matching Theorem
Hall’s Marriage Theorem

5 Report


Introduction

There are many network structures (relations between certain
objects) considered in applications of mathematics in other sciences.
We use many calculations of numbers and equations of numbers in
mathematical analysis in application areas.
We seldom do calculations in mathematical analysis of network
structures or equations of structures.
A sufﬁciently developed theory of relations has been existing for a
long while.
In this lecture, we review several elementary mathematical concepts
from the viewpoint of a theory of relations.
Managing the calculations of relations, we reexamine properties of
network structures.
It is also intended to construct a theory of relations with computer
veriﬁable proofs.


Historical Background

The modern story of an algebra of logic is started by G. Boole (1847).
Complement, Converse (Inverse) and Composition of relations.
(De Morgen(1864))
To create an algebra out of logic. (C. S. Peirce(1870))
Axiomatization and Representability (A. Tarski(1941),
R.Lyndon(1950))
Relations in categories. (S. MacLane(1961), D. Puppe(1962),
Y. Kawahara(1973))
Fuzzy relations and its axiomatization and representability.
(L. A. Zadeh(1965), Y. Kawahara(1999))
† R. D. Maddux, The origin of relation algebras in the development and
axiomaization of the calculus of relations, Studia Logica 50(1991),
421–455.
† G. Schmidt, Relational Mathematics, Cambridge University Press,
2010, 582pages.

Applications to Computer Science

Theory of Automata (model of computing)
Y.Kawahara, Applications of relational calculus to computer
mathematics. Bull. Inform. Cybernet. 23 (1988), pp67–78.
Theory of Programs (program veriﬁcation)
Y.Kawahara and Y.Mizoguchi, Categorical assertion semantics in
toposes, Advances in Software Science and Technology, Vol.4(1992),
137–150.
Graph Rewriting System (model of computation)
Y.Mizoguchi and Y.Kawahara, Relational graph rewritings. Theoret.
Comput. Sci. 141 (1995), 311–328.
Relational Databases (model of data)
H.Okuma and Y.Kawahara, Relational aspects of relational database
dependencies. Bull. Inform. Cybernet. 32 (2000), 91–104.
Formal Concept Analysis (model of data)
T.Ishida, K.Honda, Y.Kawahara, Formal concepts in Dedekind
categories. Lecture Notes in Comput. Sci., Vol.4988(2008) 221–233.

Basic Notations

(1) A relation α of a set A into another set B is a subset of the Cartesian
product A × B and denoted by α : A B.
(2) The inverse relation α : B A of α is a relation such that
(b, a) ∈ α if and only if (a, b) ∈ α.
(3) The composite αβ : A C of α : A B followed by β : B C is
a relation such that (a, c) ∈ αβ if and only if there exists b ∈ B with
(a, b) ∈ α and (b, c) ∈ β.
(4) As a relation of a set A into a set B is a subset of A × B, the inclusion
relation, union, intersection and difference of them are available as
usual and denoted by , , and −, respectively.
(5) The identity relation id A : A A is a relation with
id A = {(a, a) ∈ A × A|a ∈ A}.
(6) The empty relation φ ⊆ A × B is denoted by 0 AB . The entire set
A × B is called the universal relation and denoted by ∇ AB .
(7) The one point set {∗} is denoted by I. We note that ∇ II = id I .

Union, Intersection, Complement

α: A B, β : A B : relations.

α β = {(a, b) | (a, b) ∈ α ∨ (a, b) ∈ β}
α β = {(a, b) | (a, b) ∈ α ∧ (a, b) ∈ β}
α = {(a, b) | (a, b)
¯ α}
α − β = {(a, b) | (a, b) ∈ α ∧ (a, b) β}

{αλ : A B | λ ∈ Λ}, {βλ : A B | λ ∈ Λ} : classes of relations.

λ∈Λ αλ = {(a, b) | ∃λ ∈ Λ, (a, b) ∈ αλ )}
λ∈Λ αλ = {(a, b) | ∀λ ∈ Λ, (a, b) ∈ αλ )}


Distributive Law ( and )

Proposition
Let α : A B, β : A B and βλ : A B (λ ∈ Λ) be relations. Then we
have
α ( λ∈Λ βλ ) = λ∈Λ (α βλ )
α ( λ∈Λ βλ ) = λ∈Λ (α βλ )
α = α, (α
¯ β) = α
¯ β, (α
¯ β) = α
¯ β.
¯
0 AB = ∇ AB , ∇ AB = 0 AB .
.


I-Category

Proposition (I-Category)
Let α, α : A B , β, β : B C and γ : C D be relations. Then
(1) (αβ)γ = α(βγ),
(2) id A α = αid B = α,
(3) (α ) = α, (αβ) = β α ,
(4) If α α and β β then αβ α β and α (α ) .

.


Distributive Law (Composition and ( , ))

Proposition
Let α : A B, β : B C , βλ : A B (λ ∈ Λ) and γ : C D be
relations. Then we have

α( λ∈Λ βλ ) = λ∈Λ (αβλ )
( λ∈Λ βλ )γ = λ∈Λ (βλ γ)
α( λ∈Λ βλ ) λ∈Λ (αβλ )
( λ∈Λ βλ )γ λ∈Λ (βλ γ)
.


Empty & Universal relation

Proposition
For a relation α : A B,

0 X A α = 0 XB , α0 BY = 0 AY .

If B φ, then
∇ AB ∇ BC = ∇ AC
Note: If B = φ, then ∇ AB ∇ BC = 0 AC .
If α : A B is not empty, then

∇ AA α∇ BB = ∇ AB . .


Inverse relation

Proposition
Let α : A B, β : A B, αλ : A B (λ ∈ Λ) be relations.
( λ∈Λ αλ ) = λ∈Λ α , ( λ∈Λ αλ ) = λ∈Λ α .
λ λ

(α) = (α ), (α − β) = α − β .
¯
0 = 0B A, ∇ = ∇ BA .
AB AB
id = id A .
A
∇ AB = ∇ ∇ IB .
IA
.


Equivalence and Ordering (1)

For a relation θ : A A, we deﬁne the following laws:

id A θ (Reﬂexive Law)
θ θ (Symmetric Law)
θθ θ (Transitive Law)
θ θ id A (Antisymmetric Law)
θ θ = ∇AA (Linear Law)


Equivalence and Ordering (2)

A relation θ : A A is an equivalence relation on A, if θ satisfies
reflexsive, symmetric and transitive laws.
A relation θ : A A is a partial ordering on A, if θ satisfies
reflexsive, transitive and antisymmetric laws.
A partial ordering θ : A A is a total ordering if it satisfies the
linear law.


Functions and Mappings
Deﬁnition

Deﬁnition
Let α : A B be a relation.
(1) α is total, if id A αα .
(2) α is univalent, if α α id B .
(3) A univalent relation is also called as a partial function.
(4) α is (total) function, if α is total and univalent.
(3) A (total) function α : A B is surjection, if α α = id B .
(4) A (total) function α : A B is injection, if αα = id A .
(5) A (total) function is bijection, if it is surjection and injection.
Note. We use letters f , g, h, · · · for (total) functions. For a function,
surjection and injection, we use an arrow symbol →, and . .


Matchings

Deﬁnition
A relation f : X Y is matching, if f f idY and f f id X .

Deﬁnition .
Let α : X Y be a relation. A relation f : X Y is matching of α, if
f f idY , f f id X and f α. .

.


Functions and Mappings
Propositions

Proposition
(1) If f : A → B and g : B → C are functions, then the composition
f g : A → C is a function.
(2) If f : A → B and g : A → B are functions and f g, then f = g.
(3) If f : A → B is a function, then f f f = f .
.
Proposition
Let f : X → A, g : Y → B be functions and βλ : A B (λ ∈ Λ)
relations. Then
f( λ∈Λ βλ ) g = λ∈Λ ( f βλ g )
.


Rationality

Proposition
Let q : X Y be a surjection and f : X → Z a function. If qq ff
then there exists an unique function g : Y → Z such that f = q g.
Let m : Y X be an injection and f : Z → X a function. If
.
m m f f then there exists an unique function g : Z → Y such that
f = gm.

Theorem (Rationality)
For a relation α : A B, there exist functions f : R → A and g : R → B
such that α = f g and f f g g = id R hold.
.

Corollary
For a relation ρ : I X, there exists an injection A X such that
ρ = ∇ I A i.

Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) .
September 29-30, 2011 18 / 35

Dedekind Formula
Concepts

Proposition (∗)
Let α : A B, β : B C and γ : A C be relations.
(1) αβ γ α(β α γ),
(2) αβ γ (α γβ )β,
(3) αβ γ (α γβ )(β α γ).

.


Dedekind Formula I
Properties

Lemma
(1) Let α : A B be a relation. Then α αα α.
(2) Let α : A A and β : A A be relations. If α id A and β id A ,
then α = α, αα = α and αβ = α β. .

Proposition
Let α : A B, β : B A be relations. If αβ = id A and βα = id B then α
and β are both bijections and β = α . .

Proposition
Let α : A → B, β : B C and γ : B C be relations. If α α id B and
γ β, then α(β − γ) = αβ − αγ.
.

Dedekind Formula II
Properties

Proposition (epi-mono factorization)
Let f : A → B be a function. Then there exist a surjection e : A B
and an injection m : B B such that f = em.
.
We denote the set B deﬁned in above proposition as f (A).

Corollary
If f : A B be an injection, then there exist a bijection e : . A f (A)
and an injection m : f (A) B such that f = em.


Cardinality of relations
Definition

Definition (Cartinarity)
The cardinality |α| of of relation α : A B is the cardinality of α as a
subset of A × B.
.
In this lecture, we are going to consider only finite cardinality.
Let X, Y and Z be a finite sets. Then
.
(1) |α| = 0 ⇔ α = 0 XY ,
(2) |α α | = |α| + |α | − |α α |,
(3) α α ⇒ |α| ≤ |α |,
(4) |α | = |α|,
(5) |id I | = 1.


Cardinality of relations I
Proposition

Proposition (∗)
Let X, Y and Z be ﬁnite sets, α : X Y, β : Y Z and γ : X Z
relations. If α is univalent, i.e. α α idY , then
.
|β α γ| ≤ |αβ γ| ∧ |α γβ | ≤ |αβ γ|.

Proposition
Let X, Y and Z be ﬁnite sets, α : X Y, β : Y Z and γ : X Z
relations. .
(1) If α and β are univalent, then |αβ γ| = |α γβ |.
(2) If α is a matching, then |αβ γ| = |β α γ|.
(3) If α is a partial function and β is a total function, then |αβ| = |α|.
(4) If α is a matching, then |α αβ| = |αβ|.


Cardinality of relations II
Proposition

Proposition
Let X, Y and Z be ﬁnite sets, f : X Y and β : Z X relations.
(1) If f is a matching, then |∇ IX f | = | f |.
(2) If u id X then |∇ IX u| = |u|. Especially, |∇ IX | = |id X | = |X|.
(3) If f is an injection, then |β| = |β f |.
(4) If f is an injection, then |∇ IX | ≤ |∇ IY |.

.


Coproduct relation
Definition

Definition (Coproduct)
Let X and Y be sets. The coproduct X + Y of X and Y is a set

X + Y = (X × {0}) ∪ (Y × {1}).

Functions i : X → X + Y and j : Y → X + Y are defined by i(x) = (x, 0)
and j(y) = (y, 1) for x ∈ X and y ∈ Y . .

We call i and j inclusion functions for X + Y .

Proposition (∗)
Functions i and j are both injections and the following equations holds:

ii = id X , j j = idY , i j = 0 XY , i i j j = id X+Y
.


Coproduct relation I
Propositions

Proposition
Let α : X Z and β : Y Z be relations. Then there exists a unique
relation γ : X + Y Z which satisﬁes

iγ = α, and jγ = β. .

We denote the relation γ deﬁned in above proposition as α⊥β.

Proposition
Let δ : X + Y → Z be a relation. .
(1) (α⊥β)δ = (αδ)⊥(βδ).
(2) If α and β are univalent relations then α⊥β is also univalent.
(3) If α and β are total relations then α⊥β is also total.


Coproduct relation II
Propositions

Proposition (Coproduct)
Let X and Y be sets i : X → Z and j : Y → Z relations which satisﬁes
following conditions:

i i = id X , j j = idY , i j = 0 XY , and i i j j = id Z .

Then Z is the coproduct of X and Y . That is there is a bijection
α : Z → X + Y such that i α = i and j α = j.

.


Coproduct relation III
Propositions

Proposition
Let X0 + Y0 , X1 + Y1 and X2 + Y2 be coproducts, and i k , j k inclusions for
X k + Y k ( k = 0, 1, 2).
For relations α k : X k−1 X k and β k : Y k−1 Y k ( k = 1, 2),

(α1 + β1 )(α2 + β2 ) = ((α1 α2 ) + (β1 β2 )),

where α k + β k = (α k i k )⊥(β k j k ) ( k = 1, 2).

.


Product relation
Definition

Definition (Product)
Let X and Y be sets. The product X × Y of X and Y is a set

X × Y = {(x, y) | x ∈ X ∧ y ∈ Y}.

Functions p : X × Y → X and q : X × Y → Y are defined by p(x, y) = x
and q(x, y) = y for x ∈ X and y ∈ Y . .

We call p and q projection functions for X × Y .

Proposition
Functions p and q are both surjections and the following equations holds:

p p = id X , q q = idY , p q = ∇ XY , and pp qq = id X×Y
.


Product relation I
Propositions

Let α : V X and β : V Y be relations. We deﬁne a relation
α β:V X × Y by
α β = αp βq .

Proposition
(1) If α and β are univalent relations then α β is also univalent.
(2) If α and β are total relations then α β is also total.
(3) If α and β are functions then α β is also a function.


Product relation II
Propositions

Proposition
Let f : V → X and g : V → Y be functions. Then there exists a unique
relation h : V → X × Y which satisﬁes

hp = f, and hq = g.

Especially, h = ( f g).

.


Product relation III
Propositions

Proposition (Product)
Let X and Y be sets p : Z → X and q : Z → Y functions which satisﬁes
following conditions:

p p = id X , q q = idY , p q = ∇ XY , and p p q q = id X×Y

Then Z is the product of X and Y . That is there is a bijection
α : X × Y → Z such that αp = p and αq = q.

.


Product relation IV
Propositions

Proposition (∗)
Let X0 × Y0 , X1 × Y1 and X2 × Y2 be products, and pk , q k projections for
X k × Y k ( k = 0, 1, 2). Let α k : X k X k+1 and β k : Y k−1 Y k ( k = 0, 1)
be relations and α k × β k is deﬁned by (pk α k ) (q k β k ) ( k = 0, 1). Then, we
have
((p0 α0 ) (q0 β0 ))(( p2 α ) (q2 β )) = ( p0 α0 α1 p ) (q0 β0 β1 q ), and
1 1 2 2
(α0 × β0 )(α1 × β1 ) = ((α0 α1 ) × (β0 β1 )).

.


Matching Theorem

Let X and Y be finite sets.
Proposition
Let f : X Y and α : X Y be relations. If f is a matching in α then we
have
| f | ≤ |∇ IX | − (|ρ| − |ρα|)
.
for any relation ρ : I X.

Definition (Marriage condition)
A relation α : X Y satisfies the marriage condition if and only if
|ρ| ≤ |ρα| for any ρ : I X.
.

Theorem (Hall 1935)
Let α : X Y be a relation where |X| 0. There exists a total matching
f :X Y in α if and only if α satisfies the marriage condition.

Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) .
September 29-30, 2011 34 / 35

Exercises

(1) Let α : X A , β1 , β 2 : A B and γ : Y B be relations. If α and
γ are univalent (i.e. α α id A , γ γ id B ), then

α(β1 β2 )γ = (αβ1 γ ) (αβ2 γ )

cf. f , g : function ⇒ f ( λ∈Λ βλ ) g
= λ∈Λ ( f βλ g )
(2) Let α : A B, β : B C and γ : A C be relations.

(αβ γ) (α γβ )(β α γ)

(3) Let α : A B be a relation.
The equation α = αα α holds if and only if there exist injections m
and n and surjections p and q such that α = m pq n.
cf. If f is a function then f = f f f and there exist a injection m and
surjection e such that f = em
(4) A relation θ : A A is an equivalence relation if and only if there
exists a surjection p : A X such that θ = pp .

Theory of Relations (1)

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