Introduction Models Mathematical aspects Orthopairs Davide Ciucci Dipartimento di Informatica, Sistemistica e Comunicazione Università di Milano Bicocca May 2009/Milano Davide Ciucci (DISCo) Orthopairs RST 1 / 33
Introduction Models Mathematical aspects Outline Introduction 1 Models 2 Mathematical aspects 3 On three values and orthopairs Operations on orthopairs Operations on rough sets Davide Ciucci (DISCo) Orthopairs RST 2 / 33
Introduction Models Mathematical aspects Outline Introduction 1 Models 2 Mathematical aspects 3 On three values and orthopairs Operations on orthopairs Operations on rough sets Davide Ciucci (DISCo) Orthopairs RST 3 / 33
Introduction Models Mathematical aspects Orthopairs: definition Definition Let X a given universe, A , B ⊆ X form an orthopair iff they are disjoint or orthogonal ( A , B ) with A ∩ B = ∅ Any orthopair ( A , B ) is equal to a subset pair ( A , C ) such that A ⊆ C through the mapping C := B c The collection of all orthopairs on a universe X is denoted as O ( X ) Davide Ciucci (DISCo) Orthopairs RST 4 / 33
Introduction Models Mathematical aspects Orthopairs: definition Definition Let X a given universe, A , B ⊆ X form an orthopair iff they are disjoint or orthogonal ( A , B ) with A ∩ B = ∅ Any orthopair ( A , B ) is equal to a subset pair ( A , C ) such that A ⊆ C through the mapping C := B c The collection of all orthopairs on a universe X is denoted as O ( X ) Davide Ciucci (DISCo) Orthopairs RST 4 / 33
Introduction Models Mathematical aspects Orthopairs: definition Definition Let X a given universe, A , B ⊆ X form an orthopair iff they are disjoint or orthogonal ( A , B ) with A ∩ B = ∅ Any orthopair ( A , B ) is equal to a subset pair ( A , C ) such that A ⊆ C through the mapping C := B c The collection of all orthopairs on a universe X is denoted as O ( X ) Davide Ciucci (DISCo) Orthopairs RST 4 / 33
Introduction Models Mathematical aspects Historical notes Orthopairs studied in Narinyani, 1980: Sub-definite sets elements belonging/not belonging to a set A Cattaneo,Nistico, 1989: classical preclusivity propositions an example of BZ–lattices induced by a preclusivity relation Coker, 1996: Intuitionistic sets a weakening to the classical case of Atanassov intuitionistic fuzzy sets Subset pairs studied in Gentilhomme, 1968: Flou sets Klaua, 1968: Partial sets Iwinski, 1987 Yao, Li, 1997: Interval sets Davide Ciucci (DISCo) Orthopairs RST 5 / 33
Introduction Models Mathematical aspects Historical notes Orthopairs studied in Narinyani, 1980: Sub-definite sets elements belonging/not belonging to a set A Cattaneo,Nistico, 1989: classical preclusivity propositions an example of BZ–lattices induced by a preclusivity relation Coker, 1996: Intuitionistic sets a weakening to the classical case of Atanassov intuitionistic fuzzy sets Subset pairs studied in Gentilhomme, 1968: Flou sets Klaua, 1968: Partial sets Iwinski, 1987 Yao, Li, 1997: Interval sets Davide Ciucci (DISCo) Orthopairs RST 5 / 33
Introduction Models Mathematical aspects Historical notes Orthopairs studied in Narinyani, 1980: Sub-definite sets elements belonging/not belonging to a set A Cattaneo,Nistico, 1989: classical preclusivity propositions an example of BZ–lattices induced by a preclusivity relation Coker, 1996: Intuitionistic sets a weakening to the classical case of Atanassov intuitionistic fuzzy sets Subset pairs studied in Gentilhomme, 1968: Flou sets Klaua, 1968: Partial sets Iwinski, 1987 Yao, Li, 1997: Interval sets Davide Ciucci (DISCo) Orthopairs RST 5 / 33
Introduction Models Mathematical aspects Historical notes Orthopairs studied in Narinyani, 1980: Sub-definite sets elements belonging/not belonging to a set A Cattaneo,Nistico, 1989: classical preclusivity propositions an example of BZ–lattices induced by a preclusivity relation Coker, 1996: Intuitionistic sets a weakening to the classical case of Atanassov intuitionistic fuzzy sets Subset pairs studied in Gentilhomme, 1968: Flou sets Klaua, 1968: Partial sets Iwinski, 1987 Yao, Li, 1997: Interval sets Davide Ciucci (DISCo) Orthopairs RST 5 / 33
Introduction Models Mathematical aspects Models Several model which generate orthopairs (subset pairs) Boolean rough sets Pawlak, VPRS, decision theoretic, ... Twofold sets Shadowed sets A IM ( OF THIS PRESENTATION ) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets Davide Ciucci (DISCo) Orthopairs RST 6 / 33
Introduction Models Mathematical aspects Models Several model which generate orthopairs (subset pairs) Boolean rough sets Pawlak, VPRS, decision theoretic, ... Twofold sets Shadowed sets A IM ( OF THIS PRESENTATION ) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets Davide Ciucci (DISCo) Orthopairs RST 6 / 33
Introduction Models Mathematical aspects Models Several model which generate orthopairs (subset pairs) Boolean rough sets Pawlak, VPRS, decision theoretic, ... Twofold sets Shadowed sets A IM ( OF THIS PRESENTATION ) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets Davide Ciucci (DISCo) Orthopairs RST 6 / 33
Introduction Models Mathematical aspects Models Several model which generate orthopairs (subset pairs) Boolean rough sets Pawlak, VPRS, decision theoretic, ... Twofold sets Shadowed sets A IM ( OF THIS PRESENTATION ) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets Davide Ciucci (DISCo) Orthopairs RST 6 / 33
Introduction Models Mathematical aspects Models Several model which generate orthopairs (subset pairs) Boolean rough sets Pawlak, VPRS, decision theoretic, ... Twofold sets Shadowed sets A IM ( OF THIS PRESENTATION ) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets Davide Ciucci (DISCo) Orthopairs RST 6 / 33
Introduction Models Mathematical aspects Models Several model which generate orthopairs (subset pairs) Boolean rough sets Pawlak, VPRS, decision theoretic, ... Twofold sets Shadowed sets A IM ( OF THIS PRESENTATION ) briefly review these models study properties and define operations on orthopairs report the obtained results on rough sets Davide Ciucci (DISCo) Orthopairs RST 6 / 33
Introduction Models Mathematical aspects Outline Introduction 1 Models 2 Mathematical aspects 3 On three values and orthopairs Operations on orthopairs Operations on rough sets Davide Ciucci (DISCo) Orthopairs RST 7 / 33
Introduction Models Mathematical aspects Pawlak Rough Sets Definition An approximation space is a pair ( X , E ) with X a set of objects and E an equivalence (reflexive, simmetric, transitive) relation on X . Equivalence classes are denoted as [ x ] E . Rough membership function ∀ H ⊆ X µ H ( y ) := | [ y ] E ∩ H | µ H : X �→ [ 0 , 1 ] | [ y ] E | Davide Ciucci (DISCo) Orthopairs RST 8 / 33
Introduction Models Mathematical aspects Pawlak Rough Sets Definition An approximation space is a pair ( X , E ) with X a set of objects and E an equivalence (reflexive, simmetric, transitive) relation on X . Equivalence classes are denoted as [ x ] E . Rough membership function ∀ H ⊆ X µ H ( y ) := | [ y ] E ∩ H | µ H : X �→ [ 0 , 1 ] | [ y ] E | Davide Ciucci (DISCo) Orthopairs RST 8 / 33
Introduction Models Mathematical aspects Pawlak rough sets Lower approximation L ( H ) = { y ∈ X : [ y ] E ⊆ H } Upper approximation U ( H ) = { y ∈ X : y ∈ X : [ y ] E ∩ H � = ∅} E ( H ) = U c ( H ) Exterior region ( L ( H ) , E ( H )) is an orthopair Davide Ciucci (DISCo) Orthopairs RST 9 / 33
Introduction Models Mathematical aspects Pawlak rough sets Lower approximation L ( H ) = { y ∈ X : [ y ] E ⊆ H } Upper approximation U ( H ) = { y ∈ X : y ∈ X : [ y ] E ∩ H � = ∅} E ( H ) = U c ( H ) Exterior region ( L ( H ) , E ( H )) is an orthopair Davide Ciucci (DISCo) Orthopairs RST 9 / 33
Introduction Models Mathematical aspects Pawlak rough sets Lower approximation L ( H ) = { y ∈ X : [ y ] E ⊆ H } Upper approximation U ( H ) = { y ∈ X : y ∈ X : [ y ] E ∩ H � = ∅} E ( H ) = U c ( H ) Exterior region ( L ( H ) , E ( H )) is an orthopair Davide Ciucci (DISCo) Orthopairs RST 9 / 33
Introduction Models Mathematical aspects Pawlak rough sets Lower approximation L ( H ) = { y ∈ X : [ y ] E ⊆ H } Upper approximation U ( H ) = { y ∈ X : y ∈ X : [ y ] E ∩ H � = ∅} E ( H ) = U c ( H ) Exterior region ( L ( H ) , E ( H )) is an orthopair Davide Ciucci (DISCo) Orthopairs RST 9 / 33
Introduction Models Mathematical aspects Twofold sets Given a universe X , a multivalued attribute a with domain D Γ a ( x ) ⊆ D set of possible values for a of x , Γ a : X �→ P ( D ) Let E ⊆ D a subset of values The ill-known set Γ − 1 a ( E ) ⊆ X is approximated by the pair � Γ − 1 a ∗ ( E ) , Γ − 1 ∗ ( E ) � a Γ − 1 a ∗ ( E ) := { x ∈ X , Γ a ( x ) ⊆ E } Γ − 1 ∗ ( E ) := { x ∈ X , Γ a ( x ) ∩ E � = ∅} a D. Dubois and H. Prade Twofold fuzzy sets and rough sets – some issues in knowledge representation Fuzzy Sets and Systems 23 (1987), 3–18 Davide Ciucci (DISCo) Orthopairs RST 10 / 33
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