Theory of Relations (1)

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Theory of Relations (1)Course of Mathematics

Pusan National University

Yoshhiro Mizoguchi

Institute of Mathematics for IndustryKyushu University, [email protected]

September 29-30, 2011

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

Table of Contents

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1 Relational CalculusBasic NotationsMatchingsDedekind Formula

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2 Cardinality of relationsBasic conceptsProperties

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3 Product and CoproductCoproduct relationsProduct relations

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4 Matching TheoremHall’s Marriage Theorem

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5 Report


Introduction

There are many network structures (relations between certainobjects) considered in applications of mathematics in other sciences.

We use many calculations of numbers and equations of numbers inmathematical analysis in application areas.

We seldom do calculations in mathematical analysis of networkstructures or equations of structures.

A sufficiently developed theory of relations has been existing for along while.

In this lecture, we review several elementary mathematical conceptsfrom the viewpoint of a theory of relations.

Managing the calculations of relations, we reexamine properties ofnetwork structures.

It is also intended to construct a theory of relations with computerverifiable 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 andaxiomaization 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 computermathematics. Bull. Inform. Cybernet. 23 (1988), pp67–78.Theory of Programs (program verification)Y.Kawahara and Y.Mizoguchi, Categorical assertion semantics intoposes, 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 databasedependencies. Bull. Inform. Cybernet. 32 (2000), 91–104.Formal Concept Analysis (model of data)T.Ishida, K.Honda, Y.Kawahara, Formal concepts in Dedekindcategories. 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 Cartesianproduct 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 isa 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 inclusionrelation, union, intersection and difference of them are available asusual and denoted by v, t, u and −, respectively.

(5) The identity relation idA : A ⇁ A is a relation withidA = {(a, a) ∈ A × A|a ∈ A}.

(6) The empty relation φ ⊆ A × B is denoted by 0AB. The entire setA × B is called the universal relation and denoted by ∇AB.

(7) The one point set {∗} is denoted by I. We note that ∇II = idI.


Union, Intersection, Complement

α : A ⇁ B, β : A ⇁ B : relations.

α t β = {(a, b) | (a, b) ∈ α ∨ (a, b) ∈ β}α u 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.

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


Distributive Law (t and u)

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Proposition

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Let α : A ⇁ B, β : A ⇁ B and βλ : A ⇁ B (λ ∈ Λ) be relations. Then wehave

α u (tλ∈Λβλ) = tλ∈Λ(α u βλ)α t (uλ∈Λβλ) = uλ∈Λ(α t βλ)α = α, (α u β) = α t β, (α t β) = α u β.0AB = ∇AB, ∇AB = 0AB.


I-Category

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Proposition (I-Category)

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Let α, α′ : A ⇁ B, β, β′ : B ⇁ C and γ : C ⇁ D be relations. Then

(1) (αβ)γ = α(βγ),(2) idAα = αidB = α,

(3) (α])] = α, (αβ)] = β]α],(4) If α v α′ and β v β′ then αβ v α′β′ and α] v (α′)].


Distributive Law (Composition and (t, u))

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Proposition

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

α(tλ∈Λβλ) = tλ∈Λ(αβλ)(tλ∈Λβλ)γ = tλ∈Λ(βλγ)α(uλ∈Λβλ) v uλ∈Λ(αβλ)(uλ∈Λβλ)γ v uλ∈Λ(βλγ)


Empty & Universal relation

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Proposition

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For a relation α : A ⇁ B,

0X Aα = 0XB, α0BY = 0AY .

If B , φ, then∇AB∇BC = ∇AC

Note: If B = φ, then ∇AB∇BC = 0AC.

If α : A ⇁ B is not empty, then

∇AAα∇BB = ∇AB.


Inverse relation

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Proposition

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

(tλ∈Λαλ)] = tλ∈Λα]λ, (uλ∈Λαλ)] = uλ∈Λα]λ.

(α)] = (α]), (α − β)] = α] − β].0]

AB= 0BA, ∇]

AB= ∇BA.

id]A= idA.

∇AB = ∇]IA∇IB.


Equivalence and Ordering (1)

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

idA v θ (Reflexive Law)θ] v θ (Symmetric Law)θθ v θ (Transitive Law)θ u θ] v idA (Antisymmetric Law)θ t θ] = ∇AA (Linear Law)


Equivalence and Ordering (2)

A relation θ : A ⇁ A is an equivalence relation on A, if θ satisfiesreflexsive, symmetric and transitive laws.

A relation θ : A ⇁ A is a partial ordering on A, if θ satisfiesreflexsive, transitive and antisymmetric laws.

A partial ordering θ : A ⇁ A is a total ordering if it satisfies thelinear law.


Functions and MappingsDefinition

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Definition

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Let α : A ⇁ B be a relation.

(1) α is total, if idA v αα].(2) α is univalent, if α]α v idB.

(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 = idB.

(4) A (total) function α : A ⇁ B is injection, if αα] = idA.

(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

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A relation f : X ⇁ Y is matching, if f ] f v idY and f f ] v idX.

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Let α : X ⇁ Y be a relation. A relation f : X ⇁ Y is matching of α, iff ] f v idY , f f ] v idX and f v α.


Functions and MappingsPropositions

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Proposition

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(1) If f : A → B and g : B → C are functions, then the compositionf g : A → C is a function.

(2) If f : A → B and g : A → B are functions and f v g, then f = g.

(3) If f : A → B is a function, then f f ] f = f .

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Proposition

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Let f : X → A, g : Y → B be functions and βλ : A ⇁ B (λ ∈ Λ)relations. Then

f (uλ∈Λβλ)g] = uλ∈Λ( fβλg])


Rationality

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Let q : X � Y be a surjection and f : X → Z a function. If qq] v f f ]then there exists an unique function g : Y → Z such that f = qg.

Let m : Y � X be an injection and f : Z → X a function. Ifm]m v f ] f then there exists an unique function g : Z → Y such thatf = gm.

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Theorem (Rationality)

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For a relation α : A ⇁ B, there exist functions f : R → A and g : R → Bsuch that α = f ]g and f f ] u gg] = idR hold.

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Corollary

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For a relation ρ : I ⇁ X, there exists an injection A � X such thatρ = ∇IAi.


Dedekind FormulaConcepts

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Proposition (∗)

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Let α : A ⇁ B, β : B ⇁ C and γ : A ⇁ C be relations.

(1) αβ u γ v α(β u α]γ),(2) αβ u γ v (α u γβ])β,(3) αβ u γ v (α u γβ])(β u α]γ).


Dedekind Formula IProperties

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Lemma

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(1) Let α : A ⇁ B be a relation. Then α v αα]α.

(2) Let α : A ⇁ A and β : A ⇁ A be relations. If α v idA and β v idA,then α] = α, αα = α and αβ = α u β.

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Proposition

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Let α : A ⇁ B, β : B ⇁ A be relations. If αβ = idA and βα = idB then αand β are both bijections and β = α].

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Proposition

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Let α : A → B, β : B ⇁ C and γ : B ⇁ C be relations. If α]α v idB andγ v β, then α(β − γ) = αβ − αγ.


Dedekind Formula IIProperties

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Proposition (epi-mono factorization)

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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).

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Corollary

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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 relationsDefinition

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Definition (Cartinarity)

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The cardinality |α| of of relation α : A ⇁ B is the cardinality of α as asubset 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 = 0XY ,

(2) |α t α′| = |α| + |α′| − |α u α′|,(3) α v α′ ⇒ |α| ≤ |α′|,(4) |α]| = |α|,(5) |idI| = 1.


Cardinality of relations IProposition

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Proposition (∗)

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Let X, Y and Z be finite sets, α : X ⇁ Y, β : Y ⇁ Z and γ : X ⇁ Zrelations. If α is univalent, i.e. α]α v idY , then

|β u α]γ| ≤ |αβ u γ| ∧ |α u γβ]| ≤ |αβ u γ|.

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Proposition

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Let X, Y and Z be finite sets, α : X ⇁ Y, β : Y ⇁ Z and γ : X ⇁ Zrelations.

(1) If α and β are univalent, then |αβ u γ| = |α u γβ]|.(2) If α is a matching, then |αβ u γ| = |β u 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 IIProposition

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Proposition

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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 v idX then |∇IXu| = |u|. Especially, |∇IX| = |idX| = |X|.(3) If f is an injection, then |β| = |β f |.(4) If f is an injection, then |∇IX| ≤ |∇IY |.


Coproduct relationDefinition

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Definition (Coproduct)

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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.

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Proposition (∗)

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Functions i and j are both injections and the following equations holds:

ii] = idX, j j] = idY , i j] = 0XY , i]i t j] j = idX+Y


Coproduct relation IPropositions

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Proposition

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Let α : X ⇁ Z and β : Y ⇁ Z be relations. Then there exists a uniquerelation γ : X + Y ⇁ Z which satisfies

iγ = α, and jγ = β.

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

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Proposition

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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 IIPropositions

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Proposition (Coproduct)

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Let X and Y be sets i′ : X → Z and j′ : Y → Z relations which satisfiesfollowing conditions:

i′i′] = idX, j′ j′] = idY , i′ j′] = 0XY , and i′]i′ t j′] j′ = idZ .

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


Coproduct relation IIIPropositions

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Proposition

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Let X0 + Y0, X1 + Y1 and X2 + Y2 be coproducts, and ik, jk inclusions forXk + Yk (k = 0, 1, 2).For relations αk : Xk−1 ⇁ Xk and βk : Yk−1 ⇁ Yk (k = 1, 2),

(α1 + β1)(α2 + β2) = ((α1α2) + (β1β2)),

where αk + βk = (αkik)⊥(βk jk) (k = 1, 2).


Product relationDefinition

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Definition (Product)

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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) = xand q(x, y) = y for x ∈ X and y ∈ Y.

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

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

p]p = idX, q]q = idY , p]q = ∇XY , and pp] u qq] = idX×Y


Product relation IPropositions

Let α : V ⇁ X and β : V ⇁ Y be relations. We define a relationα>β : V ⇁ X × Y by

α>β = αp] u βq].

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Proposition

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(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 IIPropositions

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Proposition

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

hp = f, and hq = g.

Especially, h = ( f>g).


Product relation IIIPropositions

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Proposition (Product)

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Let X and Y be sets p′ : Z → X and q′ : Z → Y functions which satisfiesfollowing conditions:

p′]p′ = idX, q′]q′ = idY , p′]q′ = ∇XY , and p′p′] u q′q′] = idX×Y

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


Product relation IVPropositions

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Proposition (∗)

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Let X0 × Y0, X1 × Y1 and X2 × Y2 be products, and pk, qk projections forXk × Yk (k = 0, 1, 2). Let αk : Xk ⇁ Xk+1 and βk : Yk−1 ⇁ Yk (k = 0, 1)be relations and αk × βk is defined by (pkαk)>(qkβk) (k = 0, 1). Then, wehave

((p0α0)>(q0β0))((p2α]

1)>(q2β

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(α0 × β0)(α1 × β1) = ((α0α1) × (β0β1)).


Matching Theorem

Let X and Y be finite sets.

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Proposition

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Let f : X ⇁ Y and α : X ⇁ Y be relations. If f is a matching in α then wehave

| f | ≤ |∇IX| − (|ρ| − |ρα|)

for any relation ρ : I ⇁ X.

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Definition (Marriage condition)

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

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Theorem (Hall 1935)

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Let α : X ⇁ Y be a relation where |X| , 0. There exists a total matchingf : X ⇁ Y in α if and only if α satisfies the marriage condition.


Exercises

(1) Let α : X ⇁ A, β1, β2 : A ⇁ B and γ : Y ⇁ B be relations. If α andγ are univalent (i.e. α]α v idA, γ]γ v idB), then

α(β1 u β2)γ] = (αβ1γ]) u (αβ2γ

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cf. f , g : function⇒ f (uλ∈Λβλ)g] = uλ∈Λ( fβλg])(2) Let α : A ⇁ B, β : B ⇁ C and γ : A ⇁ C be relations.

(αβ u γ) @ (α u γβ])(β u α]γ)

(3) Let α : A ⇁ B be a relation.The equation α = αα]α holds if and only if there exist injections mand 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 andsurjection e such that f = em

(4) A relation θ : A ⇁ A is an equivalence relation if and only if thereexists a surjection p : A � X such that θ = pp].


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