Fuzzy Rough Relations T. K. Samanta and Biswajit Sarkar Department of Mathematics, Uluberia College, India - 711315. e-mail: [email protected] Department of Mathematics,Bantul Mahakali High School,India - 711312 e-mail: [email protected]
Abstract In this paper, the definition of fuzzy rough relation on a set will be introduced and then it would be proved that the collection of such relations is closed under different binary compositions such as, algebraic sum, algebraic product etc. Also the definitions of reflexive, symmetric and transitive fuzzy rough relations on a set are given and a few properties of them will be investigated. Lastly, we define a operation , which is a composition of two fuzzy rough relations, with the help of maxmin relation and thereafter it is shown that the collection of such relations is closed under the operation .
Keywords: Fuzzy Set, Rough Set, Fuzzy Rough Set, Fuzzy Rough Relation. 2010 Mathematics Subject Classification: 03E72, 03E02. 1 Introduction disciplines and real life problems are now Most of problems in engineering ,medical catching momentum. But, it is seen that all these science, economics , environments etc. have theories have their own difficulties, that is why in various uncertainties. To overcome these this paper we are going to study fuzzy rough uncertainties, some kind of theories were given relation depending upon fuzzy rough set, which is like theory of fuzzy set [4] , intuitionistic fuzzy another one new mathematical tool for dealing sets [6], rough sets [8] , soft sets [1] etc. , which with uncertainties. we can use as a mathematical tools for dealing In fact, there have been also attempts to with uncertainties. In 1965, Zadeh [4] initiated the “fuzzify” various mathematical structures like novel concept of fuzzy set theory, thereafter in topological spaces, groups, rings, etc. and also 1982, the concept of rough set theory was first concepts like relations, measure, probability, and given by Pawlak [8] and then in 1999, Molodtsov automata etc. In this paper we will develop the [1] initiated the concept of soft theory , all these concepts of relations with the help of fuzzy rough are used for modeling incomplete knowledge, set. The concept of fuzzy relation on a set was vagueness and uncertainties. In fact, all these defined by Zadeh [4, 5] and several authors have concepts having a good application in other considered it further, has generalized the concept by considering fuzzy relations on fuzzy sets and relation is also a set, i.e. a subset of a Cartesian developed the study of fuzzy graphs, obtaining product. So, let A = ( U , R ) be an beautiful analogs of several graph theoretical approximation space. Let X ⊆ U . A relation T properties. In 1971, fuzzy relations were applied on X is said to be a rough relation on X if to consider clustering analysis. In this paper, we T ≠ T, where T and T are lower and upper will introduce the definition of fuzzy rough approximation of T , respectively defined by relation on a set and then it would be proved that T x y U U x y X the collection of such relations is closed under = { ( , ) ∈ × [: , ] S ⊆ } different binary compositions such as, algebraic T={(,) xyUU∈× :[,] xyS ∩≠ X ϕ } sum, algebraic product etc. Also we define Definition 2.1 A fuzzy subset A of the universe reflexive, symmetric and transitive fuzzy rough U is characterized by a membership function relations on a set and investigate a few properties µ A : U → [ 1,0 ] . Then the product A × A is of them. Lastly, we define a operation with the defined by the membership function help of maxmin relation and thereafter it is shown µ(xy , )=min{ µ (), x µ ( y )} . A that the collection of such relations is closed under AA× AA the operation . fuzzy relation R on A is a fuzzy subset of A × A.
2 Preliminaries ei ,.. µ R ( x , y ) ≤ µ A × A ( x , y ) ∀ x , y ∈ U. Let A = ( U , R ) be approximation space. There are three kind of fuzzy relations mainly, The product product space is also an which are as follows : approximation space A 2 = (U 2 , S ) , where the 1. Reflexive Fuzzy Relation : A fuzzy relation indiscernibility relation S ⊆ U 2 is defined by R on A is said to be reflexive iff
µ (xx , )=1 ∀ xU ∈ such that µ A ( x ) > 0 . (( x 1 , y 1 (,) x 2 , y 2 )) ∈ S if and only if R 2. Symmetric Fuzzy Relation : A fuzzy relation ( x 1 , x 2 )∈ R and ( y 1 , y 2 )∈ R , for R on A is said to be symmetric iff x1 , x 2 , y 1 , y 2∈ U . It can be easily verified that µ ( x , y ) = µ ( y , x ) ∀ x , y ∈ U . S is an equivalence relation on U . The elements R R 3. Transitive Fuzzy Relation : A fuzzy relation ( x 1 , y 1 ) and ( x 2 , y 2 ) are indiscernible in R on A is said to be transitive iff R ⊇ R R S if and only if the elements x1 , x 2 are where R R is defined by the membership indiscernible in R and so are y , y in R . 1 2 function This implies that the equivalence class containing µRR (xy , )=maxmin{ µ R (,), xu µ R (, uy ) } the element ( x , y ) with respect to S , denoted u∈ U Definition 2.2 Let A = ( U , R ) be an by [ x , y ] S , should be equal to the Cartesian approximation space and X ( ⊆ U ) be a rough product of [ x ] R and [ y ] R . set in A . Let Y be a fuzzy set in U with The concepts of rough set can be easily membership grade µ : U → [ 0 1, ] . Then Y is extended to a relation, mainly due to the fact that a Y said to be fuzzy rough set in A if the following conditions holds : µ'xy, = µ∧ µ xy , () ( R1 R 2 ) () i)( µ Y ( x ) =1 ∀ x ∈ X
()iiµ Y ()=0 x∀ xUX ∈ ( \ ) = minµxy , , µ xy , . { R1()() R 2 } (iii ) 0 < µ Y ( x ) < 1 ∀ x ∈ ( X \ X ) Since µ, µ are two FR relations, 3 Fuzzy Rough Relation R1 R 2 Definition 3.1 Let A = ( U , R ) be an µxy, =1= µ xy , R1( ) R 2 ( ) approximation space and X ( ⊆ U ) be a rough ∀( xy, ) ∈ X × X . set in A. Let Y be a fuzzy set in A Therefore characterized by the membership function minµR()()xy , , µ R xy , = 1 µ Y :U → [0,1] . Then the product Y × Y is { 1 2 } defined by the membership function ∀( xy, ) ∈ X × X
µYY× (xy , )=min{ µ Y ( x ), µ Y () y } ⇒ µ 'xy( ,) =1∀( xyXX , ) ∈ × .
∀(xy , ) ∈ U × U . Also, since µ, µ are two FR relations, R1 R 2 A fuzzy rough relation R on Y is a fuzzy rough µxy, =0= µ xy , R1( ) R 2 ( ) subset of Y× Y , ∀( xy,[) ∈× UU \ XX × ] i.e. µR(,)xy≤ µ Y× Y (,) xy ∀(xy , ) ∈ U × U . So, Satisfying the followings : µ 'xy( ,=0) ∀( xy ,[) ∈× UUXX \ × ] .
()iµ R (,)=1 xy∀ (,) xyXX ∈ × Again, since
0<µR( xy ,,) µ R ( xy ,<1) ()iiµ R (,)=0 xy∀ (,)[ xy ∈× UUXX \ × ] 1 2 ()0 µR( xy,) = 1= µ R ( xy , ) 1 2 Proposition 3.5 ( Algebraic Sum :) Let µ , R1 ∀( xy, ) ∈ XX × . µ be two FR relation on X (⊆ U ) and µ is R 2 Therefore pre-defined membership grade on U. Then maxµR()()xy , , µ R xy , = 1 { 1 2 } µ R ⊕µ R is FR relation, where µ⊕ µ is 1 2 R1 R 1 ∀( xy, ) ∈ X × X defined by ⇒ µ 'xy( ,) =1∀( xyXX , ) ∈ × . µµ⊕xy,= µ xy , + µ xy , ( RR1 1) ()()() R 1 R 2 Also, since µ R , µ R are two FR relations, −µ( xy,) µ ( xy ,,) ∀∈×( xy , ) UU 1 2 R1 R 2 Proof : Obvious. µxy, =0= µ xy , R1( ) R 2 ( ) Definition 3.6 An FR relation µ on R 1 ∀( xy,[) ∈× UU \ XX × ] X( ⊆ U ) is said to be reflexive FR relation if So, µ R ( xy,) = 1 ∀ xU ∈ such that µ 'xy( ,=0) ∀( xy ,[) ∈× UUXX \ × ] 1 Again, since µ ( x ) > 0 where µ : U → [ 1,0 ] is pre- 0<µxy ,, µ xy ,<1 defined membership function on U. R1( ) R 2 ( ) Definition 3.7 An FR relation µ on R 1 ∀( xy,) ∈× [ XXXX \ × ] X( ⊆ U ) is said to be reflexive FR relation of ⇒ 0 µ∨ µ are reflexive. µµRR⊕()xy, = µ R (,) xy + µ R (,) xy R1 R 2 ( 12) 1 2 −µ(,)xy µ (,),(,) xy ∀∈× xyUU . R1 R 2 Let µ ' = µ R ∧µ R . Since µ R , µ R are 1 2 1 2 Proof : Obvious. reflexive on X , 4 Composition of two Fuzzy Rough relations µ( xx,) =1= µ ( xx , ) ∀ x Let µ, µ be two FR relations on R1 R 2 R1 R 2 with µ ( x ) > .0 So, X( ⊆ U ) . Then composition of these two FR µ′(,)=minxx µ (,), xx µ (,) xx =1 relations, denoted by µ µ , is defined by { R1 R 2 } R1 R 2 ∀ x with µ ( x ) > .0 µ µ x, y ( R1 R 2 ) () Thus, µ′ is reflexive on X . = max minµR (xu , ) , µ R ( uy , ) Similarly, µ∨ µ x, x u∈ U { 1 2 } ( R1 R 2 ) () ∀x, y ∈ U . =maxµ (xx , ), µ (, xx )=1 { R1 R 2 } Proposition 4.1 Composition of two FR relations ∀x with µ ( x ) > 0 . on X( ⊆ U ) is associative. Hence, µ ∨ µ and µ ∧ µ are reflexive Proof : Obvious. R1 R 2 R1 R2 Proposition 4.2 Let µ, µ be two FR FR relations. R1 R 2 Proposition 3.11 ( Algebraic Product :) Let relations on X( ⊆ U ) and µ be pre-defined , be two reflexive FR relation on µ R µ R membership grade on U. Then µ µ is 1 2 R1 R 2 X( ⊆ U ) and µ is pre-defined membership FR relation. grade on U. Then µ⋅ µ is reflexive FR Proof : Since µ, µ are two FR relations R1 R 2 R1 R 2 relation, where µ⋅ µ is defined by µ(,)=1=xy µ ( xy ,) R1 R 2 R1 R 2 ∀(,)xy ∈ X × X . Let ( x , y ) ∈ X × X. Now, = max max minµR()()xu , , µ R ux , , u u≠ x { 1 2 } µ µ x, y ( R1 R 2 ) () max minµR()()xu , , µ R ux , u= x { 1 2 } } =maxminµ (xu ,), µ ( uy , ) { R1 R 2 } u∈ U = 1 [ since µ, µ are reflexive R1 R 2 = 1. This holds for all ( x , y ) ∈ X × X. FR relations ] Let xy,∈ UU × \ XX × . So, Hence µ µ is reflexive FR relation. () R1 R 2 µR(,)=0=xy µ R (,) xy Then Definition 4.4 An FR relation µ on 1 2 R 1 µ µ x, y X( ⊆ U ) is said to be symmetric FR relation if ( R1 R 2 ) () µ(,)=xy µ (,) yx ∀ x , y ∈ U . R1 1 =maxminµ (xu ,), µ ( uy , ) = 0 { R1 R 2 } u∈ U Definition 4.5 An FR relation µ on R 1 Again, since 0<µR (,),xy µ R (,)<1 xy 1 2 X( ⊆ U ) is said to be transitive FR relation if ∀(,)[xy ∈× XX \ XX × ] , µ⊇ µ µ , where '′ is defined by R1 R 1 R 1 ⇒ 0 Proposition 4.3 Let µR, µ R be two reflexive transitive FR relation then µ is weakly 1 2 R 1 FR relations on X( ⊆ U ) and µ be pre-defined reflexive FR relation. membership grade on U. Then µ µ is Proof : Since µ is symmetric, R1 R 2 R 1 also reflexive FR relation. µ(,)=xy µ (,) yx∀ xyU , ∈ . R1 R 1 Proof : By proposition 4.1, µ µ is a FR R1 R 2 Again, since µ is transitive, R 1 relation on X . So, it suffices to show that µ(,)xy≥ max min µ (,), xu µ (,) uy µ µ is a reflexive FR relation on X . R1{ R 1 R 1 } R1 R 2 u∈ U Taking y = x , we get We see that, µ(,)xx≥ max min µ (,), xu µ (,) ux µ µ x, x R1{ R 1 R 1 } ( R1 R 2 ) () u∈ U = max minµR (xu , ) , µ R ( xu , ) =maxminµR (xu ,), µ R ( ux ,) u∈ U { 1 1 } u∈ U { 1 2 } = maxµ R (x , u ) u∈ U 1 Therefore, µ(,)xx≥ µ (,) xy for all followings : R1 R 1 ()iµ ()= x α y ∈ U where µ ( x ) > 0. R x This shows that µ is weakly reflexive FR ()iiµR ()= y µ R () x R 1 x y ()iiiµ ()>0, y µ ()>0 z relation on X . Rx R y Proposition 4.7 Let µ, µ be w-reflexive ⇒ µ (z ) >0 R1 R 2 Rx FR relations then µR∨ µ R ⊆ µµ RR . ()ivµR ()=0 u ⇒ µµϕ R∧ R =. 1 2 12 x x y Proof : We see that , Proof : We see that µ µ x, y µR (,)xy≤ min{ µ (), x µ () y} ∀ xyU , ∈ ( R1 R 1 ) () 1 For each x∈ U , with µ (x ) > 0 , we defined =maxminµR (xu ,), µ R ( uy , ) µ()=y µ (,) xy∀ yU ∈ . We note u∈ U { 1 1 } R x R 1 that, µ()=y µ (,) x y ≥ minµ (,),xx µ (,) xy Rx R 1 { R1 R 2 } ≤min{ µ (),()x µ y} ≤∀∈ µ () y yU . ≥ minµ (),x µ ( x ,) y R2 { } i.e. µ()y≤ µ () y ∀ yU ∈ . Rx [ as µ R is w – reflexive ] 1 Thus the fuzzy rough set determined by µ is a R x Again, fuzzy rough subset of X . µ(,)xy≤ min µµµ (), xy () ≤ () x R2 { } ()iµ ()= x µ (,)= xx α [ since µ is Rx R 1 R1 So, µ µxy,≥ µ xy , . ( R1 R 2) ()() R 2 similitude of order α ] Therefore, µ⊆ µ µ Similarly, we can ()iiµ ()= y µ (,) xy R2 R 1 R 2 Rx R x show that µ⊆ µ µ . =µ (,)=y x µ () x R1 R 1 R 2 Rx R y Hence, µR∨ µ R ⊆ µµ RR . [ as µ is symmetric ] 1 2 12 R1 Definition 4.8 An FR relation is on X⊆ U is (iii ) We suppose that µ (y ) >0 and ( ) Rx similitude said to be a FR relation if it is reflexiv, µ (z ) >0 R y symmetric and transitive. i.e. µ(,),xy µ (,)>0 yz ( 1 ) Theorem 4.9 Let X be a fuzzy rough subset of R1 R 1 By transitivity of µ , universe U where µ be a membership function R1 on U . Also let µ R , a similitude FR relation of 1 µ(,)xz≥ max min µ (,), xu µ (,) uz R1{ R 1 R 1 } order α . Then for each x∈ U , with µ (x )>0 u∈ U Claim: , ∃ a fuzzy rough subset of X determined by maxminµ (,),xu µ (,) uz >0 . membership function µ satisfying the { R1 R 1 } R x u∈ U Thus, µ∧ µ= ϕ . For u= y(∈ U ) , it becomes Rx R y This completes the proof. maxmaxmin µR()()xu , , µ R ux , , u u≠ y { 1 1 } References [1] D. Molodtsov , Soft Set Theory - First Results, max minµR()()xu , , µ R ux , > 0 u= y { 1 1 } } Computer math. Applic , 37 94/5 (1999 ) 19 – 31 , [ by )1( ] [2] M. K. Chakraborty, Mili Das, On Fuzzy So, our claim is justified. Equivalence I , Fuzzy Sets and Syst, (1983 ) 185 – Thus, µ()=z µ ( x ,)>0. z 193. Rx R 1 [3] Maria Do Nicoletti , Joaquim Quinteiro (iv ) Let µ R (u )=0. We need to show that x Uchôa, Margarete T. Z. Baptistini, Rough µ∧ µ= ϕ . If possible, let Rx R y Relation Properties , Int. J. Appl. Math. Comput. Sci., 2001,Vol. 11, No.3, 621 – 635 . µ∧ µ ()z > 0 ( Rx R y ) [4] L. A. Zadeh Fuzzy sets , Information and control 8 (1965 ) 338 – 353 . ⇒ minµ (),z µ () z >0 { Rx R y } [5] L. A. Zadeh , Similarity relations and fuzzy So, µ(),z µ ()>0. z Rx R y orderings , Inform. Sci. 3, 177 – 200 (1971). i.e. µ(xz ,), µ (,)>0. yz Now, [6] K. Atanassov , Intutionistic Fuzzy Sets , Fuzzy R1 R 1 Sets and Systems Vol.64 , No. 2 , (1986) 87 – 96 . µR(xu , )≥ maxmin µ R (,), xt µ R (, tu ). [7] Z. Pawlak , Rough Relations , Techn. Rep. 1t∈ U { 1 1 } Taking t = z we have, No. 435, Institute of Computer Science , Polish µ(x ,)= u µ ()>0 u , Academy of Science , warsaw. 1981. R1 R x [8] Z. Pawlak , Rough Sets , Int. J. Inf. Comp. This contradicts µ (u )=0. Rx Sci. Vol. – 11 , No. 5 , (1982) 341 – 356 .