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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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
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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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