A Note on Intuitionistic Fuzzy. Equivalence Relation

Transcript

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 dkbasnet@rediffmail.com N. K. Sarma Dept. of Mathematics, Assam University Silchar , Assam, India. Pkanta_naba@rediffmail.com 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 J-composition 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 n-1 } {a n }. When restricted to Y = {a 1, a 2, a 3,..., a n-1 }, 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 n-1, 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 n-1, a n-1 ). 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 n-1, 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 n-2, a n ) = μ A (a n-2, a n-1 ) 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 n-1, a n ) and μ A (a n, a n ) all the elements except μ A (a n-1, a n ) coincide with some of the n 1 elements of Imμ A restricted to Y = {a 1, a 2, a 3,..., a n-1 }. 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