Rough Intuitionistic Fuzzy Sets in a Lattice

Transcription

1 International Mathematical Forum, Vol. 6, 2011, no. 27, Rough Intuitionistic Fuzzy Sets in a Lattice K. V. Thomas Dept. of Mathematics BharataMata College, Thrikkakara Kochi, Kerala, India tkapiarumala@yahoo.co.in Latha S. Nair Dept. of Mathematics Marthanasius College, Kothamangalam Kochi, Kerala, India lathavichattu@gmail.com bstract In this paper basic notions of the rough intuitionistic fuzzy set theory will be given. In fact the paper concerns a relationship between rough intuitionistic fuzzy sets and lattice theory. We shall introduce the notion of rough intuitionistic fuzzy lattice (resp rough intuitionistic fuzzy ideals). We also defined intuitionistic fuzzy rough set corresponding to a rough set and given conditions for intuitionistic fuzzy rough lattice. We prove that an intuitionistic fuzzy rough set in pr(x) is an ) intuitionistic fuzzy rough lattice iff it s level rough set ( (α,β), (α,β) is a rough sub lattice of pr(x). Mathematics Subject Classification: 03B52, 03F55, 06D72 Keywords: Rough set, Intuitionistic fuzzy set (IFS),Rough intuitionistic fuzzy set(rifs), Intuitionistic fuzzy lattice (IFL), Rough intuitionistic fuzzy lattice (RIFL), Intuitionistic fuzzy rough set (IFRS), Intuitionistic fuzzy rough lattice (IFRL), Level rough sets 1 Introduction Rough set theory has been proposed by Pawlak [7] as a tool to conceptualize, organize and analyze various types of data in data mining. It has recently received wide attention in real life applications and theoretical research. This

2 1328 K. V. Thomas and L. S. Nair method is especially useful for dealing with uncertain and vague knowledge in information systems. The concept of a fuzzy set was introduced by Zadeh and it is now a rigorous area of research with applications in various fields. With the work of fuzzy sets, 1986 tanassov presented intuitionistic fuzzy sets which are very effective to ideal with vagueness. The concept of intuitionistic fuzzy sets is a generalization of one of the fuzzy sets. Recently many researchers applied the notion of intuitionistic fuzzy sets to relations, group theory, ring theory, lattice theory etc. In paper [9] we studied the rough sets corresponding to an ideal of a lattice and introduced rough sub lattice,rough ideals and their characterizations and in [8] we studied intuitionistic fuzzy sub lattices and ideals. This paper is a bland of the above papers where we studied relationship between rough sets and intuitionistic fuzzy sets. Here we give the rough approximation of an intuitionistic fuzzy set and introduced rough intuitionistic fuzzy sub lattices, ideals etc. lso we defined intuitionistic fuzzy rough sets, intuitionistic fuzzy rough sub lattices, and ideals and studied their properties. 2 Preliminaries In this section we recall the following definitions and results which are used in the sequel. Definition 2.1. [1] Let X be a non-empty set. n intuitionistic fuzzy set [IFS] of X is an object of the following form = { x, (x), ν(x) /x X} : X [0, 1] and ν : X [0, 1] define the degree of membership and where the degree of non membership of the element x X, 0 The set of all IFS s on X is denoted by IFS (X). Definition 2.2. [1]If = { x, (x), ν(x) /x X}andB = { x, B X} are any two IFS of X then, (i) B (ii) = B (iii) = { x, ν (x), (iv)[] ={ x, (x) (x)and ν(x) ν(x) x X. B B (x) =(x) and ν(x) =ν(x) x X. B B (x) /x X} (x), c (x) /x X}where c (x) =1 (x) (v) = { x, ν c(x), ν (vi) B = { x, B (x) /x X}where νc (x) =1 ν(x) (x), ν (x) /x X}where B (x)= min{(x), (x)}and ν (x)=max{ν(x), ν(x)} B B B B (vii) B = { x, (x), ν (x) /x X}where B B B (x) = max{(x), (x)}and B ν B (x) = min{ν (x), ν B (x)} (x)+ν (x) 1 (x), ν B (x) /x

3 Rough intuitionistic fuzzy sets in a lattice 1329 Definition 2.3. [8]Let L be a lattice and = { x, (x), ν(x) /x L} be an IFS of L. Then is called an Intuitionistic fuzzy sublattice(ifl) of L if the following conditions are satisfied: (i) (x y) min{(x), (y)}, (ii) (x y) min{(x), (y)} (iii) ν(x y) max{ν(x), ν(y)}, (iv) ν(x y) max{ν(x), ν(y)} x, y L. The set of all IFL s of L is denoted as IFL (L). Definition 2.4. [8]n IFS of L is called an intutionistic fuzzy ideal (IFI) of L, if the following conditions are satisfied: (i) (x y) min{(x), (y)},(ii) (x y) max{(x), (y)} (iii) ν(x y) max{ν(x), ν(y)}, (iv) ν(x y) min{ν(x), ν(y)} x, y L The set of all IFI s of L is denoted as IFI (L). Theorem 2.5. [8] If and B are two IFL s (IFI s) of a lattice L then B is an IFL (IFI) of L. Definition 2.6. [9] Let U is a nonempty set and θ an equivalence relation on U. Then the pair (U,θ ) is called an approximation space. Definition 2.7. [9] Let (U,θ ) be an approximation space and X any nonempty subset of U.Then the sets, pr(x) ={x U/[x] θ X} and pr(x) ={x U/[x] θ X φ} are called the lower and upper rough approximations of the set X. Then pr(x) =(pr(x), pr(x)) is called a rough set in (U,θ ). Definition 2.8. [9] rough set pr(x) =(pr(x), pr(x)) in a lattice L is called a rough lattice if both pr(x), and pr(x)) are sublattices of L. 3 Rough Intuitionistic Fuzzy Set (RIFS) In this section we define Rough intuitionistic fuzzy set and defined operations on them. Rough intuitionistic fuzzy lattices (RIFL) and ideals (RIFI) are introduced and certain properties of them are studied. Definition 3.1. Let = { x, (x), ν(x) /x X} be an IFS of X.Then the rough intuitionisticfuzzyset of (RIFS) denoted as pr() =(pr(), pr()) and defined as pr = { x, (x),ν (x)} where (x) = (x ) and ν (x) = ν(x ) x [x] θ pr = { x, (x), ν (x)} where (x) = x [x] θ (x ) and ν (x) = ν(x ) x [x] θ x [x] θ Where θ is an equivalence relation on X and [x] θ is the equivalence class containing x.

4 1330 K. V. Thomas and L. S. Nair Definition 3.2. Letpr() =(pr(), pr()), and pr(b) =(pr(b), pr(b)) are two RIFS, Then (1)pr() pr(b) =(pr() pr(b), pr() pr(b)) (2)pr() pr(b) =(pr() pr(b), pr() pr(b)) (3)pr() pr(b) if and only if pr() pr(b), pr() pr(b) (4)The complement of pr() denoted as pr c () =(pr c (), pr c ()) Proposition 3.3. For every approximation space (X,θ ) and for every IFS s, B X, we have, (1) pr() pr() (2) If B then pr() pr(b) and pr() pr(b) (3) pr c () =(pr c (), pr c ()) (4) pr[pr()] = pr() (5) pr[pr()] = pr() (6) pr[pr()] = pr() (7) pr[pr()] = pr() (8) pr() =[pr( c )] c (9) pr() =[pr( c )] c (10) pr( B) =pr() pr(b) (11) pr( B) pr() pr(b) (12) pr( B) pr() pr(b) (13) pr( B) =pr() pr(b) Proof. Follow from Definitions 2.2, and 3.2 Proposition 3.4. Let L be a lattice and is an IFL (IFI) of L. Then pr() and pr() are also IFL s (IFI s) of L. Proof. We will prove the case of IFL. Proof for IFI is similar.we have (x y) = (x y ) (x ), (y )}], since IFL of L. min{ x y [x y] θ x [x] θ (x ), y [y] θ [min{ x y [x y] θ (y )} = min{ (x), (y)}.i.e (x y) min{ (x), (y)}. Similarly (x y) = (x y ) [min{(x ), (y )}], since x y [x y] θ x y [x y] θ IFL of L. min{ (x ), (y )} = min{ (x), (y)}.i.e (x y) min{ (x), (y)}. x [x] θ Now ν (x y) = y [y] θ ν(x y ) x y [x y] θ [max{ν(x ), ν(y )}], since x y [x y] θ IFL of L. max{ ν(x ), ν(y )} = max{ν (x),ν (y)}.i.e ν (x y) max{ν (x),ν (y)}. x [x] θ y [y] θ Similarly ν (x y) max{ν (x),ν (y)}.hence pr() is an IFL of L. Now (x y) = (x y ) [min{(x ), (y )}], since x y [x y] θ x y [x y] θ IFL of L.

5 Rough intuitionistic fuzzy sets in a lattice 1331 min{ (x ), (y )} since x and y vary independently x [x] θ y [y] θ min{ (x), (y)}.i.e (x y) min{ (x), (y)}. Similarly (x y) min{ (x), (y)} lso ν (x y) = ν(x y ) [max{ν(x ), ν(y )}], since x y [x y] θ x y [x y] θ IFL of L. max{ ν(x ), ν(y )} since x and y vary independently x [x] θ y [y] θ max{ν (x), ν (y)}.i.e ν (x y) max{ν (x), ν (y)}. Similarly ν (x y) max{ν (x), ν (y)}.hence pr() is an IFL of L. Definition 3.5. rough intuitionistic fuzzy set pr() of L is called a rough intuitionistic fuzzy lattice (RIFL)[rough intuitionistic fuzzy ideal (RIFI)] if bothpr() and pr() are IFL s (IFI s) of L. Theorem 3.6. If is an IFL (IFI) of L then pr() is a RIFL (RIFI) of L. Proof. Follow from Proposition 3.4. Theorem 3.7. If pr () and pr (B) are RIFL s (RIFI s), then pr () pr (B) is also a RIFL (RIFI). Proof. We have pr () pr (B)= (pr() pr(b), pr() pr(b)).since pr() and pr(b) are RIFL s (RIFI s) we have pr(), pr(), pr(b), andpr(b) are IFL s (IFI s).then pr() pr(b)andpr() pr(b) are IFL s (IFI s) by Theorem 2.5. So pr() pr (B) is a RIFL (RIFI) by Def 3.5. Remark 3.8. The union of two RIFI s need not be a RIFI. Consider the lattice L= {1, 2, 3, 4, 6, 12} of divisors of 12. Let θ ={1,2),(3,6),(4),(12)}be the equivalence class. We define = { x, (x), ν(x) /x L} by 1,.7,.2, 2,.5,.5, 3,.6,.3, 4,.4,.5, 6,.5,.5, 12,.4,.5 and B = { x, (x), ν(x) /x L} by 1,.6,.2, 2,.6,.4, 3,.5,.5, 4,.5,.4, 6,.4, B B.5, 12,.4,.5. Here and B are IFI s of L.Nowpr() =(pr(), pr()), where pr = { x, (x),ν (x)} is 1,.5,.5, 2,.5,.5, 3,.5,.5, 4,.4,.5, 6,.5,.5, 12,.4,.5 and pr = { x, (x), ν (x)} is 1,.7,.2, 2,.7,.2, 3,.6,.3, 4,.4,.5, 6,.6,.3, 12,.4,.5.lsopr(B) =(pr(b), pr(b)), where prb = { x, B (x),ν B (x)} is 1,.6,.4, 2,.6,.4, 3,.4,.5, 4,.5,.4, 6,.4,.5, 12,.4,.5 and prb = { x, B (x), ν B (x)} is 1,.6,.2, 2,.6,.2, 3,.5,.5, 4,.5,.4, 6,.5,.5, 12,.4,.5.Clearly pr() and pr(b) are RIFI s. Now pr () pr (B) = (pr() pr(b), pr() pr(b)) is given by,pr() pr(b) = 1,.6,.4, 2,.6,.4, 3,.5,.5, 4,.5,.4, 6,.5,.5, 12,.4,.5. pr() pr(b)) = 1,.7,.2, 2,.7,.2, 3,.6,.3, 4,.5,.4, 6,.6,.3, 12,.4,.5. Here pr() pr(b) (3 4) = pr() pr(b) (12) =.4 min{ pr() pr(b) (3), pr() pr(b) (4)} =.5. Hence pr () pr (B) is not an RIFI.

6 1332 K. V. Thomas and L. S. Nair Remark 3.9. Every RIFI is a RIFL.But the converse is not true. Consider the lattice and the equivalence relation given in the Result 3.8.Let B = { x, (x), ν(x) /x L} be given by, 1,.2,.7, 2,.4,.4, 3,.2,.5, 4,.3,.6, 6,.5,.5, B B 12,.6,.3.Now pr(b) =(pr(b), pr(b)), where prb = { x, B (x),ν B (x)} is 1,.2,.7, 2,.2,.7, 3,.2,.5, 4,.3,.6, 6,.2,.5, 12,.6,.3.andprB = { x, B (x), ν B (x)} is 1,.4,.4, 2,.4,.4, 3,.5,.5, 4,.3,.6, 6,.5,.5, 12,.6,.3.It is easily verified that pr(b) is a RIFL, But not RIFI, because B (4 6) = B (2) =.2 max{ B (4), B (6)} =.3 Remark If pr () is a RIFI and pr (B) is a RIFL.Then pr prb is a RIFL, but not a RIFI. Consider the the RIFI pr() given in Result 3.8 and the RIFL pr(b) given in Result 3.9. Then pr () pr (B) = (pr() pr(b), pr() pr(b)) is given by, pr() pr(b)= 1,.2,.7, 2,.2,.7, 3,.2,.5, 4,.3,.6, 6,.2,.5, 12,.4,.5 and pr() pr(b)= 1,.4,.4, 2,.4,.4, 3,.5,.5, 4,.3,.6, 6,.5,.5, 12,.4,.5. Here it is easily verified that pr () pr (B) is a RIFL, but not a RIFI because B (4 6) = B (2) =.2 max{ B (4), B (6)} =.3 4 Intuitionistic Fuzzy Rough Set (IFRS) In this section we introduce intuitionistic fuzzy rough sublattices and ideals, and certain characterization of intuitionistic fuzzy rough sublattice (ideal) in terms of level rough set is given. Definition 4.1. Let pr() is a rough set in (U, θ ) then an intuitionistic fuzzy rough set (IFRS)pr() =(pr(), pr()), in pr(x) is obtained by the maps pr() : pr(x) [0, 1] and ν pr() : pr(x) [0, 1] where 0 pr() +ν pr() 1 and pr() : pr(x) [0, 1] and ν pr() : pr(x) [0, 1] where0 pr() + ν pr() 1 and also with property pr() (x) pr() (x) x pr(x) and ν pr() (x) ν pr() (x) x pr(x). Definition 4.2. Let pr(x) be a rough set and pr() =(pr(), pr()), is an IFRS in pr(x). Then we can define an interval valued intuitionistic fuzzy set = { x, [ pr() (x), pr() (x)], [ν pr() (x), ν pr() (x)] } where, pr() (x) = pr() (x)ifx pr(x) =0ifx Âpr(X), and ν pr()(x) =ν pr() (x)ifx pr(x) =1ifx Âpr(X), where Âpr(X) =pr(x) pr(x) and we denote (x) =[ pr() (x), pr() (x)] and ν (x) =[ν pr() (x), ν pr() (x)].

7 Rough intuitionistic fuzzy sets in a lattice 1333 Definition 4.3. The family of all closed subintervals [0,1] is denoted by D [0,1].If D 1 =[a, b],d 2 =[c, d] then max{d 1,D 2 } =[a c, b d] and min{d 1,D 2 } = [a c, b d]. Definition 4.4. Let pr(x) be a rough lattice andpr() =(pr(), pr()) is an IFRS in pr(x).then pr() is called a intuitionistic fuzzy rough sublattice (IFRL) if x pr(x) the following hold. (i) (x y) min{ (x), (y)}, (ii) (x y) min{ (x), (y)} (iii) ν (x y) max{ ν (x), ν (y)}, (iv) ν (x y) max{ ν (x), ν (y)} x, y L. If conditions (ii) and (iv) are replaced by (x y) max{ (x), (y)} and ν (x y) min{ ν (x), ν (y)}. Then pr () is called an intuitionistic fuzzy rough ideal (IFRI). Definition 4.5. Let pr(x) be a rough lattice and pr() =(pr(), pr()) a IFRS in pr(x). Then we define (α,β) = {x pr(x)/ pr() (x) α and ν pr() (x) β} and (α,β) = {x pr(x)/ pr() (x) α and ν pr() (x) β}.then ( (α,β), (α,β) ) is called a level rough set. Theorem 4.6. Let pr(x) be a rough lattice and pr() =(pr(), pr()) is an IFRS in pr(x).then pr() is an IFRL iff ( (α,β), (α,β) ) is a rough sublattice of pr(x). Proof. First assume that ( (α,β), (α,β) ) is a rough sublattice in pr(x). We have to prove that pr() is a IFRL of pr(x).set min{ (x), (y)} =[α 0,α 1 ] and max{ ν (x), ν (y)} =[β 0,β 1 ]. Then min { pr() (x), pr() (y)} = α 0 and min { pr() (x), pr() (y)} = α 1 max{ν pr() (x),ν pr() (y)} = β 0 and max {ν pr() (x), ν pr() (y)} = β 1. Then pr() (x) α 1, pr() (y) α 1,ν pr() (x) β 0,ν pr() (y) β 0.Hence x, y (α1,β 0 ) x y, x y (α1,β 0 ). So pr() (x y) α 1, pr() (x y) α 1,ν pr() (x y) β 0,ν pr() (x y) β 0. Letx, y pr(x) either x or y Âpr(X) or x and y / Âpr(X).If x or y Âpr(X) then α 0 = 0 and β 1 = 1. So that pr() (x y) 0=α 0, pr() (x y) 0=α 0, ν pr() (x y) 1=β 1, ν pr() (x y) 1=β 1,. If x and y / Âpr(X), then pr()(x) = pr() (x), pr() (y) = pr() (y), ν pr() (x) = ν pr() (x), ν pr() (y) =ν pr() (y). So min { pr() (x), pr() (y)} = α 0 and max {ν pr() (x),ν pr() (y)} = β 1 pr() (x) α 0, pr() (y) α 0,ν pr() (x) β 1,ν pr() (y) β 1 x, y (α0,β 1 ) x y, x y (α0,β 1 ), since (α0,β 1 ) is a sublattice. So pr() (x y) α 0, pr() (x y)} α 0,ν pr() (x y) β 1,ν pr() (x y) β 1. Hence (x y) =[ pr() (x y), pr() (x y)] [α 0,α 1 ] = min{ (x), (y)} (x y) =[ pr() (x y), pr() (x y)] [α 0,α 1 ] = min{ (x), (y)}

8 1334 K. V. Thomas and L. S. Nair ν (x y) =[ν pr() (x y), ν pr()(x y)] [β 0,β 1 ] = max{ ν (x), ν (y)} ν (x y) =[ν pr() (x y), ν pr()(x y)] [β 0,β 1 ] = max{ ν (x), ν (y)}. So pr() is a IFRL. Conversely, assume that pr() is an IFRL of pr(x). We have to prove that (α,β) and (α,β) ) are sublattices of L. Let x,y (α,β), Then pr() (x) α, pr() (y) α, ν pr() (x) β,ν pr() (y) β. So min{ (x), (y)} [α, min{ pr() (x), pr() (y)}] and max{ ν (x), ν (y)} [max{ν pr() (x),ν pr() (y)},β]. Hence (x y) [α, min{ pr() (x), pr() (y)}], (x y) [α, min{ pr() (x), pr() (y)}], ν (x y) [max{ν pr() (x),ν pr() (y)},β] and ν (x y) [max{ν pr() (x),ν pr() (y)},β]. From these inequalities we get pr() (x y) α, pr() (x y) α, ν pr() (x y) β,ν pr() (x y) β. Since x y, x y pr(x) we have pr() (x y) = pr() (x y), pr() (x y) = pr() (x y), ν pr() (x y) =ν pr() (x y), ν pr() (x y) =ν pr() (x y). Hence x y, x y (α,β). Thus (α,β) is a sublattice.similarly if x,y (α,β) α, ν pr() (x) β,ν pr() (y) β. So min{ (x y) [0,α], [β,1]. Hence then pr() (x) α, pr() (y) (x), (y)} [0,α] and max{ ν (x), ν (y)} (x y) [0,α], ν (x y) [β,1], ν (x y) [β,1]. Hence pr() (x y) α, pr() (x y) α, ν pr() (x y) β,ν pr() (x y) β.so x y, x y (α,β).thus (α,β) is a sublattice. References [1] K.T. tanassov, Intutionistic fuzzy sets,fuzzy Sets and Systems,20(1), 1986, [2] R Biswas and S. Nanda, Rough groups and rough sub groups. Bull. Polish cad,sci.math. 42(1994), [3] G. Brikhoff., Lattice theory, Published by merican Mathematical Soceity,Providence, Rhode Island [4] B. Davvaz, Roughness in Rings. Information Sciences Vol.164.(2004) [5] D Dubois and H Prade, Rough fuzzy sets and fuzzy rough sets. International Journal of General Systems17(1990) [6] N Kuroki,Rough ideals in semi groups. Information Sciences 100(1997)pp [7] Z Pawlak, Rough sets, Int J Inf Comp Sci 11(1982) pp [8] K.V Thomas and Latha.S.Nair, Intuitionistic Fuzzy Sublattices and Ideals.Fuzzy Information and engg (To appear). [9] K.V Thomas and Latha.S.Nair, Rough Ideals in a Lattice,International journal of Fuzzy systems and Rough systems.(to appear). [10] L. Zadeh,Fuzzy Sets, Information and control,8(1965)

9 Rough intuitionistic fuzzy sets in a lattice 1335 Received: November, 2010