A Note on Intuitionistic Fuzzy. Equivalence Relation


 Ειδοθεα Θεοδωρίδης
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1 International Mathematical Forum, 5, 2010, no. 67, A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar , Assam, India N. K. Sarma Dept. of Mathematics, Assam University Silchar , Assam, India. Abstract In this paper, some interesting properties of Intuitionistic fuzzy equivalence relation have been discussed. Also Intuitionistic fuzzy equivalence classes have been characterized with the help of (α, β)cut of Intuitionistic fuzzy relations and finally for any Intuitionistic fuzzy equivalence relation on a finite set, we have given an upper bound of the number of values that the degree of membership and nonmembership can assume. Keywords: Intuitionistic Fuzzy Set, Intuitionistic Fuzzy Relation, Intuitionistic Fuzzy Equivalence Relation, (α, β)cut of Intuitionistic Fuzzy Sets 1. Introduction The concept of Fuzzy relation on a set was defined by Zadeh [6]. Buhaescu[4] and Bustince[5] discussed some beautiful properties of Intuitionistic fuzzy relation. Banerjee and Basnet [3, 2] introduced and discussed the (α, β)cut of Intuitionistic Fuzzy Sets and Ideals of a ring. 2. Preliminaries Definition 2.1. Let E be a nonempty set. An Intuitionistic Fuzzy Set (IFS) A of E is an object of the form A = {< x, μ A (x), ν A (x) > x E}, where μ A : E [0, 1] and
2 3302 D. K. Basnet and N. K. Sarma ν A : E [0, 1] define the degree of membership and degree of nonmembership of the element x E respectively and for every x E, 0 μ A (x) + ν A (x) 1. Definition 2.2 If A = {< x, μ A (x), ν A (x) > x E} and B = {< x, μ B (x), ν B (x) > x E} be any two IFS of a set E then A B if and only if for all x E, μ A (x) μ B (x) and ν A (x) ν B (x) A = B if and only if for all x E, μ A (x) = μ B (x) and ν A (x) = ν B (x) A B = {< x, (μ A μ B )(x), (ν A ν B )(x) > x E}, where (μ A μ B )(x) = min { μ A (x), μ B (x)} and (ν A ν B )(x) = max { ν A (x), ν B (x)} A B = {< x, (μ A μ B )(x), (ν A ν B )(x) > x E}, where (μ A μ B )(x) = max{ μ A (x), μ B (x)} and (ν A ν B )(x) = min { ν A (x), ν B (x)} Also we see that a fuzzy set has the form {< x, μ A (x), μ c A(x) > x E}, where μ c A(x) = 1  μ A (x) 3. Intuitionistic Fuzzy Relation Definition 3.1. Let A be a nonempty set. Then an Intuitionistic fuzzy relation (IF relation) on A is an Intuitionistic fuzzy set {<(x, y), μ A (x, y), ν A (x, y) > (x, y) A A}, where μ A : A A [0, 1] and ν A : A A [0, 1]. Definition 3.2. An IF relation R = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) A A} is said to be reflexive if μ A (x, x) = 1 and ν A (x, x) = 0 for all x A. Also R is said to be symmetric if μ A (x, y) = μ A (y, x) and ν A (x, y) = ν A (y, x) for all x, y A. Definition 3.3. If R 1 = {<(x, y), μ 1 (x, y), ν 1 (x, y) > (x, y) A A} and R 2 = {<(x, y), μ 2 (x, y), ν 2 (x, y) > (x, y) A A} be two IF relations on A then Jcomposition denoted be R 1 οr 2 is defined by R 1 οr 2 = {<(x, y), (μ 1 ομ 2 )(x, y), (ν 1 ον 2 )(x, y) > (x, y) A A}, where (μ 1 ομ 2 )(x, y)=sup{min{μ 1 (x, z), μ 2 (z, y)}} z A and (ν 1 ον 2 )(x, y)=inf{max{ν 1 (x, z), ν 2 (z, y)}} z A Definition 3.4. An IF relation R on A is called transitive if RοR R. Definition 3.5. An IF relation R on A is called an Intuitionistic fuzzy equivalence relation if R is reflexive, symmetric and transitive. Definition 3.6. For any Intuitionistic fuzzy set A = {< x, μ A (x), ν A (x) > x X} of a set X, we define a (α,β)cut of A as the crisp subset {x X μ A (x) α, ν A (x) β} of X and it is denoted by C α, β (A). Theorem 3.7. Let R = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) X X} be a relation on a set X. Then A is an IF equivalence relation on X if and only if C α, β (R) is an equivalence relation on X, with 0 α, β 1 and α + β 1 Proof. We have C α, β (R) = {(x, y) X X μ R (x, y) α, ν R (x, y) β}. Since R is an IF equivalence relation so μ A (x, x) = 1 α and ν A (x, x) = 0 β, for all x X and so (x, x) C α, β (R) i.e., C α, β (R) is reflexive.
3 A note on intuitionistic fuzzy equivalence relation 3303 Next let (x, y) C α, β (R), then μ R (x, y) α, ν R (x, y) β. But R is IF equivalence so μ R (y, x) = μ R (x, y) α and ν R (y, x) = ν R (x, y) β and hence (y, x) C α, β (R). Therefore C α, β (R) is symmetric. Finally let (x, y) C α, β (R) and (y, z) C α, β (R), then μ R (x, y) α, ν R (x, y) β and μ R (y, z) α, ν R (y, z) β min {μ R (x, y), μ R (y, z)} α and max{ν R (x, y), ν R (y, z)} β max{ min {μ R (x, y), μ R (y, z)}} α and min{max{ν R (x, y), ν R (y, z)}} β y y (μ R ομ R )(x, z) α and (ν R ον R )(x, z) β But R is IF equivalence relation so μ R (x, z) (μ R ομ R )(x, z) α and ν R (x, z) (ν R ον R )(x, z) β (x, z) C α, β (R), which shows that C α, β (R) is transitive. Conversely suppose that C α, β (R) is an equivalence relation on X. Taking α = 1 and β = 0 we get C 1, 0 (R) is equivalence and so a reflexive relation and so (x, x) C 1, 0 (R), for all x X. Thus μ R (x, x) 1, ν R (x, x) 0 and consequently μ R (x, x) = 1 and ν R (x, x) = 0. Therefore the IF relation R is reflexive. For any x, y X, let μ R (x, y) = α and ν R (x, y) = β. Then α + β 1 and so by hypothesis C α, β (R) is an equivalence and hence symmetric relation on X. Also (x, y) C α, β (R), so by symmetry (y, x) C α, β (R). Therefore μ R (y, x) α = μ R (x, y) and ν R (y, x) β = ν R (x, y). Similarly if μ R (y, x) = δ and ν R (y, x) = σ then (x, y) C δ, σ (R) and similarly as above we get μ R (x, y) δ = μ R (y, x) and ν R (x, y) σ = ν R (y, x) and hence μ R (x, y) = μ R (y, x) and ν R (x, y) = ν R (y, x). So the IF relation R is symmetric. Finally let x, y, z X and min{μ R (x, z), μ R (z, y)} = α and max{ ν R (x, z), ν R (z, y)} = β, then0 α, β 1 and α + β 1. Therefore C α, β (R) is an equivalence relation on X. Since μ R (x, y) α, μ R (z, y) α and ν R (x, z) β, ν R (z, y) β, so (x, z) C α, β (R) and (z, y) C α, β (R). As C α, β (R) is equivalence relation so by transitivity (x, y) C α, β (R).Therefore μ R (x, y) α and ν R (x, y) β μ R (x, y) α = min{μ R (x, z), μ R (z, y)} and ν R (x, y) β = max{ν R (x, z), ν R (z, y)}, z μ R (x, y) Sup{ min{μ R (x, z), μ R (z, y)}} z and ν R (x, y) Inf{max{ ν R (x, z), ν R (z, y)}} z μ R (x, y) (μ R ομ R )(x, y) and ν R (x, y) (ν R ον R )(x, y) μ R μ R ομ R and ν R ν R ον R, Which shows that the IF relation R is transitive and hence it is an IF equivalence relation. Definition 3.8. Let R = {<(x, y), μ R (x, y), ν R (x, y) > (x, y) X X} be an IF equivalence on a set X. Let a be any element of X. Then the IFS defined by ar = {< x, (aμ R )(x), (aν R )(x) > x X}, where (aμ R )(x) = μ R (a, x), (aν R )(x) = ν R (a, x) x X, is called an IF equivalence class of a with respect to R. Theorem 3.9. Let R = {<(x, y), μ R (x, y), ν R (x, y) > (x, y) X X} be an IF
4 3304 D. K. Basnet and N. K. Sarma equivalence relation on a set X. Let a be any element of X. Then for 0 α, β 1 and α + β 1, C α, β (ar) = [a], the equivalence class of a with respect to the equivalence relation C α, β (R) in X. Proof. We have [a] = {x X (a, x) C α, β (R)} = {x X μ R (a, x) α and ν R (a, x) β} = {x X (aμ R )(x) α and (aν R )(x) β} = C α, β (ar). With the help of this result we are now giving an alternative proof of a well known result of equivalence relation as follows. Theorem Let R = {<(x, y), μ R (x, y), ν R (x, y) > (x, y) X X} be an IF equivalence relation on a set X. Then [a] = [b] if and only if (a, b) C α, β (R), where [a], [b] are equivalence classes of a and b with respect to the equivalence relation C α, β (R) in X for 0 α, β 1 and α + β 1 Proof. Let [a] = [b] then by theorem 3.9, C α, β (ar) = C α, β (br) {x X (aμ R )(x) α, (aν R )(x) β} = {x X (bμ R )(x) α, (bν R )(x) β} As the two sets on both sides of the equality are nonempty let x C α, β (ar) = C α, β (br) (aμ R )(x) α, (aν R )(x) β and (bμ R )(x) α, (bν R )(x) β μ R (a, x) α, ν R (a, x) β and μ R (b, x) α, ν R (b, x) β min{μ R (a, x), μ R (x, b)} α and max{ν R (a, x), ν R (x, b)} β sup{ min{μ R (a, x), μ R (x, b)}} α and inf{max{ν R (a, x), ν R (x, b)}} β x x (μ R ομ R )(a, b) α and (ν R ον R )(a, b) β μ R (a, b) (μ R ομ R )(a, b) α and ν R (a, b) (ν R ον R )(a, b) β (a, b) C α, β (R) Conversely let (a, b) C α, β (R) μ R (a, b) α and ν R (a, b) β... (i) Let x C α, β (ar), then (aμ R )(x) α, (aν R )(x) β μ R (a, x) α, ν R (a, x) β min{μ R (b, a), μ R (a, x)} α and max{ν R (b, a), ν R (a, x)} β, using (i) sup{min{μ R (b, a), μ R (a, x)}} α and inf{max{ν R (b, a), ν R (a, x)}} β a a (μ R ομ R )(b, x) α and (ν R ον R )(b, x) β μ R (b, x) (μ R ομ R )(b, x) α and ν R (b, x) (ν R ον R )(b, x) β (bμ R )(x) α and (bν R )(x) β x C α, β (br) C α, β (ar) C α, β (br) Similarly we can show that C α, β (br) C α, β (ar) Hence C α, β (ar) = C α, β (br) i.e., [a] = [b] and this completes the proof. Theorem The intersection of two IF equivalence relations on a set is again an IF equivalence relation on the set. Proof. Let A = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) X X} and B = {<(x, y), μ B (x, y), ν B (x, y) > (x, y) X X} be two IF equivalence relations on a set X. Now for any 0 α, β 1 and α + β 1. We have C α, β (A B) = C α, β (A)
5 A note on intuitionistic fuzzy equivalence relation 3305 C α, β (B) (by theorem 3.6 of [3] ). By theorem 3.7, C α, β (A) and C α, β (B) are equivalence relations on X and so being intersection of two equivalence relations C α, β (A B) is also an equivalence relation on X and again by theorem 3.7, A B is an IF equivalence relation on X. Note However the union of two IF equivalence relation on a set is not necessarily an IF equivalence relation on the set as shown by the following example: Let x = {a, b, c} and A = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) X X} and B = {<(x, y), μ B (x, y), ν B (x, y) > (x, y) X X} be two IFS on X, where μ A (a, a) = μ A (b, b) = μ A (c, c) = 1, μ A (a, b) = μ A (b, a) = μ A (a, c) = μ A (c, a) = 0.2 μ A (b, c) = μ A (c, b) = 0.7, ν A (a, a) = ν A (b, b) = ν A (c, c) = 0, ν A (a, b) = ν A (b, a) = ν A (a, c) = ν A (c, a) = 0.6 ν A (b, c) = ν A (c, b) = 0.1 and μ B (a, a) = μ B (b, b) = μ B (c, c) = 1, μ B (a, b) = μ B (b, a) = μ B (b, c) = μ B (c, b) = 0.4 μ B (a, c) = μ B (c, a) = 0.6, ν B (a, a) = ν B (b, b) = ν B (c, c) = 0, ν B (a, b) = ν B (b, a) = ν B (b, c) = ν B (c, b) = 0.5 ν B (a, c) = ν B (c, a) = 0.3 Then it is easy to check that A and B are IF equivalence relation on X. Now A B = {<(x, y), (μ A μ B ) (x, y), (ν A ν B )(x, y) > (x, y) X X} and it is not transitive as shown below: {(μ A μ B ) ο (μ A μ B )}(a, b) = Sup{ min{(μ A μ B )(a, a), (μ A μ B )(a, b)}, min{(μ A μ B )(a, b), (μ A μ B )(b, b)}, min{(μ A μ B )(a, c), (μ A μ B )(c, b)}} = Sup{ min{1, 0.4}, min{0.4, 1}, min{0.6, 0.7}} = = max{μ A (a, b), μ B (a, b)} = (μ A μ B )(a, b) This shows that A B is not an IF equivalence relation on X. Before proceeding further we observe the followings: Let A = {<(x, y), μ A (x, y), ν A (x, y) > (x, y) X X} be an IF equivalence relation on a set X = {a, b, c}. So μ A (a, a) = μ A (b, b) = μ A (c, c) = 1, μ A (a, b) = μ A (b, a), μ A (a, c) = μ A (c, a) and μ A (b, c) = μ A (c, b). Without loss of generality let μ A (a, b) μ A (b, c) μ A (c, a). By symmetry and transitivity we have μ A (a, b) (μ A ομ A )(a, b) = Sup{ min{μ A (a, a), μ A (a, b)}, min{μ A (a, b), μ A (b, b)}, min{μ A (a, c), μ A (c, b)}} min{μ A (a, c), μ A (c, b)} = μ A (c, b) = μ A (b, c) Similarly μ A (b, c) (μ A ομ A )(b, c) = Sup{min{μ A (b, a), μ A (a, c)}, min{μ A (b, b), μ A (b, c)}, min{μ A (b, c), μ A (c, c)}} min{μ A (b, a), μ A (a, c)} = μ A (b, a) = μ A (a, b) Therefore μ A (a, b) = μ A (b, c). This shows that #(Imμ A ) 3. Similarly we can show #(Imν A ) 3. Next let X = {a, b, c, d} and A = {<(x, y), μ A (x, y), ν A (x, y)> (x, y) X X} be an IF equivalence relation on X. We have μ A (a, a)=μ A (b, b) = μ A (c, c) = μ A (d, d) = 1, μ A (a, b) = μ A (b, a), μ A (a, c) = μ A (c, a), μ A (a, d) = μ A (d, a), μ A (b, c) = μ A (c, b), μ A (b, d) = μ A (d, b), μ A (c, d) = μ A (d, c). Without loss of generality let μ A (a, b) μ A (a, c) μ A (a, d) μ A (b, c) μ A (b, d) μ A (c, d).
6 3306 D. K. Basnet and N. K. Sarma Now by transitivity min{μ A (a, c), μ A (c, b)} μ A (a, b) μ A (a, c) μ A (a, b), which gives μ A (a, b) = μ A (a, c). Similarly we can show that μ A (a, b) = μ A (a, d), μ A (b, c) = μ A (b, d). Therefore #(Imμ A ) 4. Similarly we can show that #(Imν A ) 4. The above observations actually lead to the following theorem. Theorem Let X = {a 1, a 2, a 3,..., a n } be a set having n elements and A = {<(x, y), μ A (x, y), ν A (x, y)> (x, y) X X} be an IF equivalence relation on X. Then #(Imμ A ) n and #(Imν A ) n. Proof. We shall prove the first result only and the second will follow similarly. We prove the result by induction on n. For n = 1 the result is trivial [for n = 3 and n = 4 the result is true by the above observations]. Suppose the result is true for any set with n 1 elements. Now X ={a 1, a 2, a 3,..., a n1 } {a n }. When restricted to Y = {a 1, a 2, a 3,..., a n1 }, A is also an IF equivalence relation on this set. By induction hypothesis #(Imμ A ) n 1 for the set Y with n 1 elements. For the IF equivalence relation A in X, in addition to these elements in Imμ A there can be only the following elements: μ A (a 1, a n ), μ A (a 2, a n ), μ A (a 3, a n ),..., μ A (a n1, a n ) and μ A (a n, a n ). Out of these, μ A (a n, a n ) = 1 = μ A (a 1, a 1 ) = μ A (a 2, a 2 ) =... = μ A (a n1, a n1 ). Without loss of generality let μ A (a 1, a n ) μ A (a 2, a n ) μ A (a 3, a n )... μ A (a n1, a n ). By transitivity min{μ A (a 1, a n ), μ A (a n, a 2 )} μ A (a 1, a 2 ) μ A (a 1, a n ) μ A (a 1, a 2 ) Also min{μ A (a 1, a 2 ), μ A (a 2, a n )} μ A (a 1, a n ) μ A (a 1, a 2 ) μ A (a 1, a n ) [as μ A (a 1, a n ) μ A (a 2, a n )] Therefore μ A (a 1, a n ) = μ A (a 1, a 2 ). Similarly, min{μ A (a 2, a n ), μ A (a n, a 3 )} μ A (a 2, a 3 ) μ A (a 2, a n ) μ A (a 2, a 3 ) Also min{μ A (a 2, a 3 ), μ A (a 3, a n )} μ A (a 2, a n ) μ A (a 2, a 3 ) μ A (a 2, a n ) [as μ A (a 2, a n ) μ A (a 3, a n )] Therefore μ A (a 2, a n ) = μ A (a 2, a 3 ). Proceeding in this way we can show that μ A (a 3, a n ) = μ A (a 3, a 4 ), μ A (a 4, a n ) = μ A (a 4, a 5 ),..., μ A (a n2, a n ) = μ A (a n2, a n1 ) Thus out of the n elements μ A (a 1, a n ), μ A (a 2, a n ), μ A (a 3, a n ),..., μ A (a n1, a n ) and μ A (a n, a n ) all the elements except μ A (a n1, a n ) coincide with some of the n 1 elements of Imμ A restricted to Y = {a 1, a 2, a 3,..., a n1 }. Hence Imμ A on X can have a maximum of n elements. This completes the proof. References [1] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986), no. 1, [2] B. Banerjee and D. K. Basnet, Intuitionistic Fuzzy Subrings and Ideals, J. of Fuzzy Mathematics, Vol. 11, No. 1, 2003, [3] D. K. Basnet, (α, β)cut of Intuitionistic fuzzy ideals (submitted)
7 A note on intuitionistic fuzzy equivalence relation 3307 [4] T. T. Buhaescu, Some observationson Intuitionistic fuzzy relations, Itimerat Seminar on Functional Equations, [5] H. Bustince, Conjuntos Intuicionistas e Intervalo valorados Difusos: Propiedades y Construccion Relatciones Intuicionistas Fuzzy. Thesis, Univerrsidad Publica de Navarra,(1994). [6] L. A. Zadeh, Similarity relations and fuzzy orderings, Information Sci. 3 (1971), Received: July, 2010
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