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Theory of Relations (1)                               Course of Mathematics                              Pusan National Un...
Table of Contents1   Relational Calculus      Basic Notations      Matchings      Dedekind Formula2   Cardinality of relat...
Introduction  There are many network structures (relations between certain  objects) considered in applications of mathema...
Historical Background  The modern story of an algebra of logic is started by G. Boole (1847).  Complement, Converse (Inver...
Applications to Computer Science  Theory of Automata (model of computing)  Y.Kawahara, Applications of relational calculus...
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...
Union, Intersection, Complementα: A         B, β : A            B : relations.                        α       β = {(a, b) ...
Distributive Law (                 and   )PropositionLet α : A  B, β : A                    B and βλ : A               B (...
I-CategoryProposition (I-Category)Let α, α : A       B , β, β : B C and γ : C D be relations. Then (1) (αβ)γ = α(βγ), (2) ...
Distributive Law (Composition and ( ,                         ))PropositionLet α : A  B, β : B                 C , βλ : A ...
Empty & Universal relationProposition    For a relation α : A             B,                                   0 X A α = 0...
Inverse relationPropositionLet α : A  B, β : A                    B, αλ : A              B (λ ∈ Λ) be relations.    (    λ...
Equivalence and Ordering (1)For a relation θ : A                    A, we define the following laws:                       ...
Equivalence and Ordering (2)  A relation θ : A  A is an equivalence relation on A, if θ satisfies  reflexsive, symmetric and...
Functions and Mappings                                          DefinitionDefinitionLet α : A     B be a relation. (1) α is ...
MatchingsDefinitionA relation f : X                 Y is matching, if f f                idY and f f       id X .Definition ...
Functions and Mappings                                            PropositionsProposition(1) If f : A → B and g : B → C ar...
RationalityProposition     Let q : X    Y be a surjection and f : X → Z a function. If qq    ff     then there exists an u...
Dedekind Formula                                             ConceptsProposition (∗)Let α : A  B, β :               B   C ...
Dedekind Formula I                                            PropertiesLemma(1) Let α : A                B be a relation....
Dedekind Formula II                                        PropertiesProposition (epi-mono factorization)Let f : A → B be ...
Cardinality of relations                                          DefinitionDefinition (Cartinarity)The cardinality |α| of o...
Cardinality of relations I                                                PropositionProposition (∗)Let X, Y and Z be finit...
Cardinality of relations II                                          PropositionPropositionLet X, Y and Z be finite sets, f...
Coproduct relation                                            DefinitionDefinition (Coproduct)Let X and Y be sets. The copro...
Coproduct relation I                                      PropositionsPropositionLet α : X     Z and β : Y   Z be relation...
Coproduct relation II                                       PropositionsProposition (Coproduct)Let X and Y be sets i : X →...
Coproduct relation III                                            PropositionsPropositionLet X0 + Y0 , X1 + Y1 and X2 + Y2...
Product relation                                              DefinitionDefinition (Product)Let X and Y be sets. The product...
Product relation I                                     PropositionsLet α : V          X and β : V    Y be relations. We de...
Product relation II                                            PropositionsPropositionLet f : V → X and g : V → Y be funct...
Product relation III                                      PropositionsProposition (Product)Let X and Y be sets p : Z → X a...
Product relation IV                                      PropositionsProposition (∗)Let X0 × Y0 , X1 × Y1 and X2 × Y2 be p...
Matching TheoremLet X and Y be finite sets.PropositionLet f : X  Y and α : X                Y be relations. If f is a match...
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...
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  1. 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, 2011Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 1 / 35
  2. 2. Table of Contents1 Relational Calculus Basic Notations Matchings Dedekind Formula2 Cardinality of relations Basic concepts Properties3 Product and Coproduct Coproduct relations Product relations4 Matching Theorem Hall’s Marriage Theorem5 Report Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 2 / 35
  3. 3. 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.Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 3 / 35
  4. 4. 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.Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 4 / 35
  5. 5. 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.Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 5 / 35
  6. 6. 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 . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 6 / 35
  7. 7. Union, Intersection, Complementα: 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 )} Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 7 / 35
  8. 8. Distributive Law ( and )PropositionLet α : A B, β : A B and βλ : A B (λ ∈ Λ) be relations. Then wehave 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 . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 8 / 35
  9. 9. I-CategoryProposition (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 ) . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 9 / 35
  10. 10. Distributive Law (Composition and ( , ))PropositionLet α : A B, β : B C , βλ : A B (λ ∈ Λ) and γ : C D berelations. 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) . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 10 / 35
  11. 11. Empty & Universal relationProposition 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 . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 11 / 35
  12. 12. Inverse relationPropositionLet 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 . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 12 / 35
  13. 13. 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) Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 13 / 35
  14. 14. 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.Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 14 / 35
  15. 15. Functions and Mappings DefinitionDefinitionLet 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 . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 15 / 35
  16. 16. MatchingsDefinitionA 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, iff f idY , f f id X and f α. . . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 16 / 35
  17. 17. Functions and Mappings PropositionsProposition(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 . .PropositionLet 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 ) . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 17 / 35
  18. 18. RationalityProposition 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 → Bsuch that α = f g and f f g g = id R hold. .CorollaryFor a relation ρ : I X, there exists an injection A X such thatρ = ∇ I A i. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) . September 29-30, 2011 18 / 35
  19. 19. Dedekind Formula ConceptsProposition (∗)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). . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 19 / 35
  20. 20. Dedekind Formula I PropertiesLemma(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. .PropositionLet α : A B, β : B A be relations. If αβ = id A and βα = id B then αand β are both bijections and β = α . .PropositionLet α : A → B, β : B C and γ : B C be relations. If α α id B andγ β, then α(β − γ) = αβ − αγ. . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 20 / 35
  21. 21. Dedekind Formula II PropertiesProposition (epi-mono factorization)Let f : A → B be a function. Then there exist a surjection e : A Band an injection m : B B such that f = em. .We denote the set B defined in above proposition as f (A).CorollaryIf 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. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 21 / 35
  22. 22. Cardinality of relations DefinitionDefinition (Cartinarity)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 = 0 XY ,(2) |α α | = |α| + |α | − |α α |,(3) α α ⇒ |α| ≤ |α |,(4) |α | = |α|,(5) |id I | = 1. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 22 / 35
  23. 23. Cardinality of relations I PropositionProposition (∗)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 idY , then . |β α γ| ≤ |αβ γ| ∧ |α γβ | ≤ |αβ γ|.PropositionLet 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 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|. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 23 / 35
  24. 24. Cardinality of relations II PropositionPropositionLet 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 |. . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 24 / 35
  25. 25. Coproduct relation DefinitionDefinition (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 . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 25 / 35
  26. 26. Coproduct relation I PropositionsPropositionLet 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.PropositionLet 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. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 26 / 35
  27. 27. Coproduct relation II PropositionsProposition (Coproduct)Let X and Y be sets i : X → Z and j : Y → Z relations which satisfiesfollowing 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 bijectionα : Z → X + Y such that i α = i and j α = j. . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 27 / 35
  28. 28. Coproduct relation III PropositionsPropositionLet X0 + Y0 , X1 + Y1 and X2 + Y2 be coproducts, and i k , j k inclusions forX 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). . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 28 / 35
  29. 29. Product relation DefinitionDefinition (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) = xand q(x, y) = y for x ∈ X and y ∈ Y . .We call p and q projection functions for X × Y .PropositionFunctions 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 . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 29 / 35
  30. 30. Product relation I PropositionsLet α : V X and β : V Y be relations. We define a relationα 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. Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 30 / 35
  31. 31. Product relation II PropositionsPropositionLet 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). . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 31 / 35
  32. 32. Product relation III PropositionsProposition (Product)Let X and Y be sets p : Z → X and q : Z → Y functions which satisfiesfollowing conditions: p p = id X , q q = idY , p q = ∇ XY , and p p q q = id X×YThen Z is the product of X and Y . That is there is a bijectionα : X × Y → Z such that αp = p and αq = q. . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 32 / 35
  33. 33. Product relation IV PropositionsProposition (∗)Let X0 × Y0 , X1 × Y1 and X2 × Y2 be products, and pk , q k projections forX 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, wehave ((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 )). . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 33 / 35
  34. 34. Matching TheoremLet X and Y be finite sets.PropositionLet 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.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 matchingf :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
  35. 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 . Y. Mizoguchi (Kyushu Univ.) Theory of Relations (1) September 29-30, 2011 35 / 35
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