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Multi-fuzzy Sets: An Extension of Fuzzy Sets

Multi-fuzzy Sets: An Extension of Fuzzy Sets Fuzzy Inf. Eng.(2011) 1: 35-43 DOI 10.1007/s12543-011-0064-y ORIGINAL ARTICLE Sabu Sebastian· T.V. Ramakrishnan Received: 20 May 2010/ Revised: 20 January 2011/ Accepted: 15 Febuary 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T ,S,M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multi- fuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of theT andS operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory. Keywords Multi-fuzzy set · Intuitionistic fuzzy set · L-fuzzy set · Uncertainty · p- conjugate·T operation 1. Introduction In a couple of decades, several researchers studied various extensions and general- izations of Zadeh’s [11] fuzzy sets. To list a few, intuitionistic fuzzy sets [1], vague sets [4], rough sets [9], grey sets [3], L-fuzzy sets [5], interval valued fuzzy sets [10], type-2 fuzzy sets [8] etc. In this paper we discuss the concept of multi-fuzzy sets and its relation with Atanassov’s intuitionistic fuzzy sets. The membership function of a multi-fuzzy set is an ordered sequence of ordinary L-fuzzy membership functions. The notion of multi-fuzzy sets provides a new method to represent some problems, which are difficult to explain in other extensions of fuzzy set theory. For example, in a two dimensional image, colour of pixels cannot be characterized by a membership function of an ordinary fuzzy set, but it can be characterized by a three dimensional ,μ ,μ ); where μ , μ and μ are the membership functions membership function (μ r g b r g b Sabu Sebastian ()· T.V. Ramakrishnan Department of Mathematical Sciences, Kannur University, Mangattuparamba, Kannur-670567, Kerala, India email: sabukannur@gmail.com 36 Sabu Sebastian · T V Ramakrishnan(2011) of the primary colours red, green and blue respectively. So an image can be approxi- mated by a collection of ordered pixels with a multi-membership function (μ ,μ ,μ ). r g b Operations on these multi-membership functions produce colour modified images like black and white images, colour inverted images, gray colour images etc. In this paper we introduce some basic tools, which are very useful to the further study. Be- fore developing the notion of multi-fuzzy sets, we recall the following definitions and results from the literature. Definition 1 [7] A function t :[0, 1]× [0, 1] → [0, 1] is a t-norm if ∀a, b, c ∈ [0, 1], (i) t(a, 1) = a, (ii) t(a, b) = t(b, a), (iii) t(a, t(b, c)) = t(t(a, b), c), (iv) b ≤ c implies t(a, b) ≤ t(a, c). Similarly, t-conorm (s-norm) is a commutative, associative and non-decreasing map- ping s :[0, 1]× [0, 1] → [0, 1] that satisfies the boundary condition: s(a, 0) = a, for all a ∈ [0, 1]. Definition 2 [1] Let a (non-fuzzy) set X be fixed. An intuitionistic fuzzy set A in X is defined as an object of the following form: A = {x,μ (x),ν (x) : x ∈ X}, where the A A functions μ : X → [0, 1] and ν : X → [0, 1], define the degree of membership and A A the degree of non-membership of the element x ∈ X to the set A in X, respectively, and for every x ∈ X, μ (x)+ν (x) ≤ 1. A A Definition 3 [5] Let X be a nonempty ordinary set, L a complete lattice. An L-fuzzy set on X is a mapping A : X → L, that is the family of all the L-fuzzy sets on X is just L consisting of all the mappings from X to L. Definition 4 [6] A function c :[0, 1] → [0, 1] is called a fuzzy complement operation, if it satisfies the following conditions: (i) c(0) = 1 and c(1) = 0, (ii) for all a, b ∈ [0, 1],ifa ≤ b, then c(a) ≥ c(b). Definition 5 [6] A t-norm t and a t-conorm s are dual with respect to a fuzzy comple- ment operation c if and only if c(t(a, b)) = s(c(a), c(b)) and c(s(a, b)) = t(c(a), c(b)), for all a, b ∈ [0, 1]. 2. Multi-fuzzy Sets Throughout this paper, we will use the following notations. X, N, I and I stand for; a nonempty set called a universal set, the set of all natural numbers, the unit interval [0, 1] and the set of all functions from X to I respectively. I stands for I × I ×...× I (k-times), for any positive integer k. In this section we introduce the concept of multi- fuzzy set based on multi-membership functions. Fuzzy Inf. Eng. (2011) 1: 35-43 37 Definition 6 Let X be a nonempty set, N the set of all natural numbers and {L : i ∈ N} a family of complete lattices. A multi-fuzzy set A in X is a set of ordered sequences A = {x,μ (x),μ (x),...,μ (x),... : x ∈ X}, (1) 1 2 i whereμ ∈ L , for i ∈ N. Remark 1 The function μ = μ ,μ ,... is called a multi-membership function of A 1 2 multi-fuzzy set A. If the sequences of the membership functions have only k-terms (finite number of terms), k is called the dimension of A. Let L = [0, 1] (for i = 1, 2,..., k), then the set of all multi-fuzzy sets in X of dimension k is denoted by M FS(X). Example 1 Let L = [0, 1], for i ∈ N. In the following way, a fuzzy set can be represented as a multi-fuzzy set A = μ ,μ  of dimension 2. Let μ ,μ be linearly 1 2 1 2 dependent with the relation μ (x) + μ (x) = 1,∀x ∈ X. Then the multi-fuzzy set 1 2 represents an ordinary fuzzy set with membership value μ (x) and nonmembership value μ (x). If μ (x) + μ (x) ≤ 1,∀x ∈ X, then the multi-fuzzy set represents an 2 1 2 Atanassov intuitionistic fuzzy set withπ(x) = 1−μ (x)−μ (x) as the measure of non- 1 2 specificity of x. The multi-fuzzy set of dimension 3 withμ (x)+μ (x)+μ (x) = 1 also 1 2 3 represents the same intuitionistic fuzzy set. Uncertainty (non-specificity) depends on many noticed and unnoticed factors and each factor has a membership function. For this we define multi-fuzzy membership functions with linearly dependent infinite co- ordinates. That is, A = {x,μ (x),μ (x),... : x ∈ X, μ (x) = 1}, (2) 1 2 i i=1 where μ ∈ I (for i = 1, 2,...). Here μ (x) and μ (x) are the membership and non i 1 2 membership values of x in A and π(x) = μ (x),μ (x),... is the membership value 3 4 of non-specificity or non-determinacy (or uncertainty) of x. Clearly π(x) itself is a multi-fuzzy membership function of infinite dimension. Example 2 Using 3-dimensional multi-fuzzy membership functions, we can charac- terize the colour of a pixel in a colour image. Suppose an image is approximated by an m× n matrix of pixels with multi-fuzzy membership function (μ ,μ ,μ ). The r g b membership values μ (x),μ (x),μ (x) being the normalized red value, green value r g b and blue value of the pixel x ∈ X respectively, where X is the set consisting of the mn pixels. So the colour image can be approximated by the collection of pixels with the multi-membership function (μ ,μ ,μ ) and it can be represented as a multi-fuzzy set r g b A = {x,μ (x),μ (x),μ (x) : x ∈ X}. r g b Construct 3 different gray images of the colour image A as follows. Let x be an arbitrary pixel of the image, for i, j = 1, 2, 3; a ≥ 0 and a + a + a = 1. (i, j) (i,1) (i,2) (i,3) Define gray values h (x), h (x) and h (x)of x as 1 2 3 h (x) = a μ (x)+ a μ (x)+ a μ (x). i (i,1) r (i,2) g (i,3) b 38 Sabu Sebastian · T V Ramakrishnan(2011) The fuzzy sets H = {x, h (x) : x ∈ X}; (for i = 1, 2, 3) are the gray images (with i i different gray tones) of the colour image A. If the coefficient matrix of h (x), for i = 1, 2, 3 is invertible, then using matrix inversion we can reconstruct the original colour image from the 3 gray images. Let PA(x) = H(x) be the matrix representation of the pixel x ∈ X of the 3 gray images, where ⎡ ⎤ ⎢ ⎥ a a a ⎢ (1,1) (1,2) (1,3)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ P = a a a , ⎢ (2,1) (2,2) (2,3)⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ a a a (3,1) (3,2) (3,3) A(x) = (μ (x),μ (x),μ (x)) , r g b and H(x) = (h (x), h (x), h (x)) . 1 2 3 −1 If P is invertible, then we have A(x) = P H(x). Hence we can reconstruct the colour membership value μ (x) = (μ (x),μ (x),μ (x)) of the pixel x from 3 linearly A r g b independent gray values. We hope that this technique is useful in image processing. Definition 7 Let{L : i ∈ N} be a family of complete lattices; A = {x,μ (x),...,μ (x),... : x ∈ X,μ ∈ L , i ∈ N}, 1 i i B = {x,ν (x),...,ν (x),... : x ∈ X,ν ∈ L , i ∈ N} 1 i i be multi-fuzzy sets in a nonempty set X. We define the following relations and opera- tions: (a) A B if and only if μ (x) ≤ ν (x),∀x ∈ X and for all i ∈ N; i i (b) A = B if and only if μ (x) = ν (x),∀x ∈ X and for all i ∈ N; i i (c) A B = {x,μ (x)∨ν (x),μ (x)∨ν (x),...,μ (x)∨ν (x),... : x ∈ X}; 1 1 2 2 i i (d) A B = {x,μ (x)∧ν (x),μ (x)∧ν (x),...,μ (x)∧ν (x),... : x ∈ X}. 1 1 2 2 i i X X k Remark 2 Let L = L, for i = 1,..., k. L = (L ) . The set of all multi-fuzzy membership functions of multi-fuzzy sets in X of dimension k does not form a vector space over the lattice L with respect to the operations and∧, where l∧ A = {x, l∧ μ (x), l ∧ μ (x),..., l ∧ μ (x),... : x ∈ X}, for any l ∈ L. So the name vector valued 1 2 i membership is not appropriate for multi-membership function. 3.T ,S andM Operations In this section we introduce the operations T , S and M on multi-fuzzy sets. From now onwards, A, B and C will denote the following multi-fuzzy sets in X of dimension k (where k is a fixed positive integer) and L = [0, 1], for i = 1,..., k. A={x,μ (x),μ (x),...,μ (x) : x ∈ X}, (3) 1 2 k B={x,ν (x),ν (x),...,ν (x) : x ∈ X}, (4) 1 2 k C={x,γ (x),γ (x),...,γ (x) : x ∈ X}. (5) 1 2 k Fuzzy Inf. Eng. (2011) 1: 35-43 39 Definition 8 Let A and B be multi-fuzzy sets in X. The T operations, S operations andM operations can be defined as: T (A, B)={x, t (μ (x),ν (x)), t (μ (x),ν (x)),..., t (μ (x),ν (x)),... : x ∈ X}, 1 1 1 2 2 2 n n n S(A, B)={x, s (μ (x),ν (x)), s (μ (x),ν (x)),..., s (μ (x),ν (x)),... : x ∈ X}, 1 1 1 2 2 2 n n n M(A, B)={x, m (μ (x),ν (x)), m (μ (x),ν (x)),..., m (μ (x),ν (x)),... : x ∈ X}, 1 1 1 2 2 2 n n n where n is a positive integer; t ,s and m are different t-norm, s-norm and aggrega- i j l tion operation [6] from I × I into I respectively, for 1 ≤ i, j, l ≤ k. Example 3 Let A and B be multi-fuzzy sets in X of dimension k. (a) A• B = {x,μ (x)·ν (x),μ (x)·ν (x),...,μ (x)·ν (x) : x ∈ X}. 1 1 2 2 k k (b) A+ B = {x,μ (x)+ν (x)−μ (x)·ν (x),...,μ (x)+ν (x)−μ (x)·ν (x) : x ∈ X}. 1 1 1 1 k k k k (c) A⊕B = {x, min(1,μ (x)+ν (x)),..., min(1,μ (x)+ν (x)) : x ∈ X}. 1 1 k k Example 4 Let A and B be multi-fuzzy sets in X of dimension k. Hamacher T operations andS operations of A and B are: T (A, B) = {x, t (μ (x),ν (x)),..., t (μ (x),ν (x)) : x ∈ X}, (6) r 1 1 r k k 1 k ab where t (a, b) = , r > 0; r+(1−r)(a+b−ab) S (A, B) = {x, s (μ (x),ν (x)),..., s (μ (x),ν (x)) : x ∈ X}, (7) r 1 1 r k k 1 k a+b+(r−2)ab where s (a, b) = , r > 0. 1+(r−1)ab Theorem 1 Let A, B and C be multi-fuzzy sets in X of dimension k. (a) T (A, 1) = A, where 1 = {x, 1, 1,..., 1 : x ∈ X}. (b) T (A, B) = T (B, A). (c) T (A,T (B, C)) = T (T (A, B), C). (d) B C impliesT (A, B) T (A, C). Proof (a) T (A, 1)={x, t (μ (x), 1), t (μ (x), 1),..., t (μ (x), 1) : x ∈ X} 1 1 2 2 k k ={x,μ (x),μ (x),...,μ (x) : x ∈ X} 1 2 k = A. Proofs of (b), (c) and (d) are similar to (a). Remark 3 Similarly S operations are commutative, associative and non-decreasing operations withS(A, 0) = A, where 0 = {x, 0, 0,..., 0 : x ∈ X}. In the remaining part, we will use t and s as different t-norm and s-norm re- i j spectively, for 1 ≤ i, j ≤ k. Here we introduce the notion of p-conjugates of T and S Operations, which are very useful links between multi-fuzzy sets and Atanassov intuitionistic fuzzy sets. 40 Sabu Sebastian · T V Ramakrishnan(2011) Definition 9 Let A and B be multi-fuzzy sets in X of dimension k and let s-norm s be the dual of t-norm t with respect to the fuzzy complement operation c. For a fixed p ∈{1, 2,..., k}, the p-conjugate of T (A, B) with respect to c is the multi-fuzzy set T (A, B) obtained from T (A, B) by replacing t (μ (x),ν (x)) as s (μ (x),ν (x)) and p p p p p p the remaining co-ordinates kept to be the same as inT (A, B). That is, T (A, B) = {x, t (μ (x),ν (x)),..., t (μ (x),ν (x)), s (μ (x),ν (x)), 1 1 1 p−1 p−1 p−1 p p p t (μ (x),ν (x)),..., t (μ (x),ν (x)) : x ∈ X}. p+1 p+1 p+1 k k k Definition 10 Let A, B be multi-fuzzy sets in X of dimension k and let t-norm t be the dual of s-norm s with respect to the fuzzy complement operation c. For a fixed p ∈{1, 2,..., k}, the p-conjugate of S(A, B) with respect to c is the multi-fuzzy set S (A, B) obtained from S(A, B) by replacing s (μ (x),ν (x)) as t (μ (x),ν (x)) and p p p p p p the remaining co-ordinates kept to be the same as inS(A, B). That is, S (A, B) = {x, s (μ (x),ν (x)),..., s (μ (x),ν (x)), t (μ (x),ν (x)), 1 1 1 p−1 p−1 p−1 p p p s (μ (x),ν (x)),..., s (μ (x),ν (x)) : x ∈ X}. p+1 p+1 p+1 k k k Example 5 Let A and B be multi-fuzzy sets in X of dimension 2, c be the classical fuzzy complement. Let S(A, B) = {x,μ (x)+ν (x)−μ (x)·ν (x),μ (x)+ν (x)−μ (x)·ν (x) : x ∈ X}. 1 1 1 1 2 2 2 2 Then S (A, B) = {x,μ (x)+ν (x)−μ (x)·ν (x),μ (x)·ν (x) : x ∈ X}. 1 1 1 1 2 2 Remark 4 q-conjugate of T (A, B) with respect to the fuzzy complement operation (c,d) d is denoted byT (A, B), for every p, q∈{1, 2,..., k}. That is (p,q) (c,d) d c T (A, B) = T (T (A, B)). q p (p,q) Theorem 2 Let A, B be multi-fuzzy sets in X of dimension k and let T , S be fixed operators. If p , p and p are distinct integers in{1, 2,..., k}, then 1 2 3 (c,c) (a) T (A, B) = T (A, B); (p,p) (c ,c ) (c ,c ) 1 2 2 1 (b) T (A, B)=T (A, B); (p ,p ) (p ,p ) 1 2 2 1 ((c ,c ),c ) (c ,(c ,c )) 1 2 3 1 2 3 (c) T (A, B)=T (A, B); ((p ,p ),p ) (p ,(p ,p )) 1 2 3 1 2 3 where c , c , c and c are fuzzy complement operations. 1 2 3 Proof (a)T (A, B) is obtained fromT (A, B) by replacing t (μ (x),ν (x)) as s (μ (x), p p p p p (c,c) ν (x)) and remaining coordinates are unchanged. Similarly T (A, B) is obtained (p,p) fromT (A, B) by replacing s (μ (x),ν (x)) as t (μ (x),ν (x)) and remaining coordi- p p p p p p (c,c) nates are unchanged. HenceT (A, B) = T (A, B). (p,p) Fuzzy Inf. Eng. (2011) 1: 35-43 41 1 th (b) The operation T makes a change in the p co-ordinate only. Similar change may be obtained in the case of T also. Since p  p , the sequential order of the 1 2 operatorsT s will not effect their compositions. (c) Similar to the proofs of (a) and (b). Remark 5 • Similar results are obtained when replaceT byS. (c ,...,c ) (c ,...,c ) 1 m 1 m−1 • In generalT (A, B)bethe p conjugates ofT (A, B) with respect (p ,...,p ) (p ,...,p ) 1 m 1 m−1 to the complement operation c , for p ,..., p ∈{1, 2,..., k} and for any positive m 1 m integer m ≥ 2. (c ,...,c ) (c ,...,c ) 1 m 1 m−1 • SimilarlyS (A, B) be the p conjugates ofS (A, B) with respect (p ,...,p ) (p ,...,p ) 1 m 1 m−1 to the complement operation c , for p ,..., p ∈{1, 2,..., k} and for any positive m 1 m integer m ≥ 2. • For simplicity we avoid the superscript of the p-conjugate operators. Theorem 3 Let A and B be multi-fuzzy sets in X of dimension k. (a) T (A, B) = S(A, B). (1,2,...,k) (b) S (A, B) = T (A, B). (1,2,...,k) Proof (a) Let T (A, B) = {x, t (μ (x),ν (x)), t (μ (x),ν (x)),..., t (μ (x),ν (x)) : x ∈ X} 1 1 1 2 2 2 k k k and s (μ (x),ν (x)) be the dual of t (μ (x),ν (x)), for p = 1, 2,..., k. Therefore, 1 to p p p p p p k conjugates ofT (A, B)is {x, s (μ (x),ν (x)), s (μ (x),ν (x)),..., s (μ (x),ν (x)) : x ∈ X} = S(A, B). 1 1 1 2 2 2 k k k (b) Similar to the proof of (a). 4. Relation Between Intuitionistic Fuzzy Operations andT, S, M Operations In this section we study the multi-fuzzy sets of dimension 2 (that is k = 2). Through- out this section conjugate means 2-conjugate with respect to classical fuzzy comple- ment operation and intuitionistic fuzzy set means an Atanassov intuitionistic fuzzy set. Here triangular norms t ≡ t ≡ t and s ≡ s ≡ s. 1 2 1 2 Remark 6 A multi-fuzzy set A = {x,μ (x),μ (x) : x ∈ X}, is called an intuitionistic 1 2 fuzzy set, if μ (x) + μ (x) ≤ 1 and if μ (x) and μ (x) represent the membership 1 2 1 2 and nonmembership values of x in A. Therefore every intuitionistic fuzzy set is a multi-fuzzy set, but the converse need not be true. For example, the multi-fuzzy set A = {x, 1, 1 : x ∈ X} is not an intuitionistic fuzzy set. Remark 7 If A and B are multi-fuzzy sets in X of dimension 2, then T (A, B) = S (A, B) andS (A, B) = T (A, B). 2-conjugates ofT andS operations on Atanassov 2 1 2 intuitionistic fuzzy sets are intuitionistic operations. 42 Sabu Sebastian · T V Ramakrishnan(2011) Theorem 4 Let A = {x,μ (x),μ (x) : x ∈ X} and B = {x,ν (x),ν (x) : x ∈ 1 2 1 2 X} be multi-fuzzy sets in X of dimension 2 and let s, t be dual with respect to the classical fuzzy complement operation. If μ (x) + μ (x) ≤ 1, ν (x) + ν (x) ≤ 1, then 1 2 1 2 the conjugates of T (A, B)={x, t(μ (x),ν (x)), t(μ (x),ν (x)) : x ∈ X}, (8) 1 1 2 2 S(A, B)={x, s(μ (x),ν (x)), s(μ (x),ν (x)) : x ∈ X} (9) 1 1 2 2 are intuitionistic fuzzy operations. Proof Let T (A, B) = {x, t(μ (x),ν (x)), s(μ (x),ν (x)) : x ∈ X} 2 1 1 2 2 and S (A, B) = {x, s(μ (x),ν (x)), t(μ (x),ν (x)) : x ∈ X} 2 1 1 2 2 be the conjugates of T (A, B) andS(A, B), respectively. Since μ (x) ≤ 1−μ (x), 2 1 ν (x) ≤ 1−ν (x), 2 1 t(μ (x),ν (x)) = 1− s(1−μ (x), 1−ν (x)) 1 1 1 1 and monotonicity of s,wehave t(μ (x),ν (x))+ s(μ (x),ν (x))≤ t(μ (x),ν (x))+ s(1−μ (x), 1−ν (x)) 1 1 2 2 1 1 1 1 = t(μ (x),ν (x))+ (1− t(μ (x),ν (x))) 1 1 1 1 = 1. Since t-norms and s-norms are non-negative functions, we have 0 ≤ t(μ (x),ν (x))+ s(μ (x),ν (x)) ≤ 1. 1 1 2 2 Similarly 0 ≤ s(μ (x),ν (x))+ t(μ (x),ν (x)) ≤ 1. 1 1 2 2 Therefore,T (A, B) andS (A, B) are intuitionistic fuzzy operations. 2 2 Example 6 Let A and B be multi-fuzzy sets in X of dimension 2 (that is A, B ∈ M FS(X)) withμ (x)+μ (x) ≤ 1,ν (x)+ν (x) ≤ 1. Then the intuitionistic fuzzy op- 1 2 1 2 erations ‘ ’,‘ ’,‘·’,‘+’, ‘ &’,‘ $’, ‘ #’ (defined by Atanassov [2]) can be expressed as the operations in M FS(X). (a) A B = {x, max(μ (x),ν (x)), min(μ (x),ν (x)) : x ∈ X} = S (A, B) 1 1 2 2 2 with s ≡ max. (b) A B = {x, min(μ (x),ν (x)), max(μ (x),ν (x)) : x ∈ X} = T (A, B) 1 1 2 2 2 with t ≡ min. Fuzzy Inf. Eng. (2011) 1: 35-43 43 (c) A· B = {x,μ (x)·ν (x),μ (x)+ν (x)−μ (x)·ν (x) : x ∈ X} = T (A, B) 1 1 2 2 2 2 2 with t as the algebraic product. (d) A+ B = {x,μ (x)+ν (x)−μ (x)·ν (x),μ (x)·ν (x) : x ∈ X} = S (A, B) 1 1 1 1 2 2 2 with s as the algebraic sum. μ (x)+ν (x) μ (x)+ν (x) 1 1 2 2 (e) A&B = {x, ,  : x ∈ X} = M(A, B) 2 2 with m ≡ m as the algebraic mean. 1 2 (f) A$B = {x, μ (x)ν (x), μ (x)ν (x) : x ∈ X} = M(A, B) 1 1 2 2 with m ≡ m as the geometric mean. 1 2 2μ (x)ν (x) 2μ (x)ν (x) 1 1 2 2 (g) A#B = {x, ,  : x ∈ X} = M(A, B) μ (x)+ν (x) μ (x)+ν (x) 1 1 2 2 μ (x)ν (x) i i with m ≡ m as the harmonic mean, for which = 0, 1 2 μ (x)+ν (x) i i if μ (x) = ν (x) = 0, i = 1, 2. i i 5. Concluding Remarks If k=1 and L = [0, 1], then the multi-fuzzy set becomes an ordinary fuzzy set. An or- dinary L-fuzzy set is a multi-fuzzy set of dimension k=1. By Remark 4 we conclude that every intuitionistic fuzzy set is a multi-fuzzy set and operations in intuitionistic fuzzy sets (see [2]) are certain operation in multi-fuzzy sets. Hence the theory of multi-fuzzy sets is an extension of fuzzy set theory, L-fuzzy set theory and intuition- istic fuzzy set theory. Acknowledgments The authors are very grateful to Prof. T. Thrivikraman and referees for their very con- structive comments. The first author is thankful to the University Grants Commission of India for awarding teacher fellowship under the Faculty Development Programme (No. F.FIP/XI Plan/KLKA 001 TF 01). References 1. Atanassov K T (1986) Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20(1): 87-96 2. Atanassov K T (1999) Intuitionistic Fuzzy Sets. Heidelberg: Springer-Physica Verlag, Vol. 35 3. Deng J L (1989) Introduction to grey system theory. Journal of Grey Systems 1: 1-24 4. Gau W L, Buehrer D J (1993) Vague sets. IEEE Trans. on Systems, Man, and Cybernetics 23(2): 610-614 5. Goguen J A (1967) L-fuzzy sets. Journal of Mathematical Analysis and Applications 18(1): 145-174 6. Klir G J, Yuan B (1995) Fuzzy sets and fuzzy logic: Theory and applications. New Delhi: Prentice- Hall 7. Menger K (1942) Statistical metrics. Proceedings of the National Academy of Sciences of the United States of America 28: 535-537 8. Mizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type-2. Information and Control 31: 312-340 9. Pawlak Z (1982) Rough sets. International Journal of Computer and Information Sciences 11(5): 341-356 10. Sambuc R (1975) Fonctionsφ-floues. Applications I’Aide au Diagnostic en Pathologie Thyroidienne, Ph.D Thesis, Univ. Marseille, France 11. Zadeh L A (1965) Fuzzy sets. Information and Control 8(3): 338-353 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Fuzzy Information and Engineering Taylor & Francis

Multi-fuzzy Sets: An Extension of Fuzzy Sets

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Fuzzy Inf. Eng.(2011) 1: 35-43 DOI 10.1007/s12543-011-0064-y ORIGINAL ARTICLE Sabu Sebastian· T.V. Ramakrishnan Received: 20 May 2010/ Revised: 20 January 2011/ Accepted: 15 Febuary 2011/ © Springer-Verlag Berlin Heidelberg and Fuzzy Information and Engineering Branch of the Operations Research Society of China Abstract In this paper we propose a method to construct more general fuzzy sets using ordinary fuzzy sets as building blocks. We introduce the concept of multi-fuzzy sets in terms of ordered sequences of membership functions. The family of operations T ,S,M of multi-fuzzy sets are introduced by coordinate wise t-norms, s-norms and aggregation operations. We define the notion of coordinate wise conjugation of multi- fuzzy sets, a method for obtaining Atanassov’s intuitionistic fuzzy operations from multi-fuzzy sets. We show that various binary operations in Atanassov’s intuitionistic fuzzy sets are equivalent to some operations in multi-fuzzy sets like M operations, 2-conjugates of theT andS operations. It is concluded that multi-fuzzy set theory is an extension of Zadeh’s fuzzy set theory, Atanassov’s intuitionsitic fuzzy set theory and L-fuzzy set theory. Keywords Multi-fuzzy set · Intuitionistic fuzzy set · L-fuzzy set · Uncertainty · p- conjugate·T operation 1. Introduction In a couple of decades, several researchers studied various extensions and general- izations of Zadeh’s [11] fuzzy sets. To list a few, intuitionistic fuzzy sets [1], vague sets [4], rough sets [9], grey sets [3], L-fuzzy sets [5], interval valued fuzzy sets [10], type-2 fuzzy sets [8] etc. In this paper we discuss the concept of multi-fuzzy sets and its relation with Atanassov’s intuitionistic fuzzy sets. The membership function of a multi-fuzzy set is an ordered sequence of ordinary L-fuzzy membership functions. The notion of multi-fuzzy sets provides a new method to represent some problems, which are difficult to explain in other extensions of fuzzy set theory. For example, in a two dimensional image, colour of pixels cannot be characterized by a membership function of an ordinary fuzzy set, but it can be characterized by a three dimensional ,μ ,μ ); where μ , μ and μ are the membership functions membership function (μ r g b r g b Sabu Sebastian ()· T.V. Ramakrishnan Department of Mathematical Sciences, Kannur University, Mangattuparamba, Kannur-670567, Kerala, India email: sabukannur@gmail.com 36 Sabu Sebastian · T V Ramakrishnan(2011) of the primary colours red, green and blue respectively. So an image can be approxi- mated by a collection of ordered pixels with a multi-membership function (μ ,μ ,μ ). r g b Operations on these multi-membership functions produce colour modified images like black and white images, colour inverted images, gray colour images etc. In this paper we introduce some basic tools, which are very useful to the further study. Be- fore developing the notion of multi-fuzzy sets, we recall the following definitions and results from the literature. Definition 1 [7] A function t :[0, 1]× [0, 1] → [0, 1] is a t-norm if ∀a, b, c ∈ [0, 1], (i) t(a, 1) = a, (ii) t(a, b) = t(b, a), (iii) t(a, t(b, c)) = t(t(a, b), c), (iv) b ≤ c implies t(a, b) ≤ t(a, c). Similarly, t-conorm (s-norm) is a commutative, associative and non-decreasing map- ping s :[0, 1]× [0, 1] → [0, 1] that satisfies the boundary condition: s(a, 0) = a, for all a ∈ [0, 1]. Definition 2 [1] Let a (non-fuzzy) set X be fixed. An intuitionistic fuzzy set A in X is defined as an object of the following form: A = {x,μ (x),ν (x) : x ∈ X}, where the A A functions μ : X → [0, 1] and ν : X → [0, 1], define the degree of membership and A A the degree of non-membership of the element x ∈ X to the set A in X, respectively, and for every x ∈ X, μ (x)+ν (x) ≤ 1. A A Definition 3 [5] Let X be a nonempty ordinary set, L a complete lattice. An L-fuzzy set on X is a mapping A : X → L, that is the family of all the L-fuzzy sets on X is just L consisting of all the mappings from X to L. Definition 4 [6] A function c :[0, 1] → [0, 1] is called a fuzzy complement operation, if it satisfies the following conditions: (i) c(0) = 1 and c(1) = 0, (ii) for all a, b ∈ [0, 1],ifa ≤ b, then c(a) ≥ c(b). Definition 5 [6] A t-norm t and a t-conorm s are dual with respect to a fuzzy comple- ment operation c if and only if c(t(a, b)) = s(c(a), c(b)) and c(s(a, b)) = t(c(a), c(b)), for all a, b ∈ [0, 1]. 2. Multi-fuzzy Sets Throughout this paper, we will use the following notations. X, N, I and I stand for; a nonempty set called a universal set, the set of all natural numbers, the unit interval [0, 1] and the set of all functions from X to I respectively. I stands for I × I ×...× I (k-times), for any positive integer k. In this section we introduce the concept of multi- fuzzy set based on multi-membership functions. Fuzzy Inf. Eng. (2011) 1: 35-43 37 Definition 6 Let X be a nonempty set, N the set of all natural numbers and {L : i ∈ N} a family of complete lattices. A multi-fuzzy set A in X is a set of ordered sequences A = {x,μ (x),μ (x),...,μ (x),... : x ∈ X}, (1) 1 2 i whereμ ∈ L , for i ∈ N. Remark 1 The function μ = μ ,μ ,... is called a multi-membership function of A 1 2 multi-fuzzy set A. If the sequences of the membership functions have only k-terms (finite number of terms), k is called the dimension of A. Let L = [0, 1] (for i = 1, 2,..., k), then the set of all multi-fuzzy sets in X of dimension k is denoted by M FS(X). Example 1 Let L = [0, 1], for i ∈ N. In the following way, a fuzzy set can be represented as a multi-fuzzy set A = μ ,μ  of dimension 2. Let μ ,μ be linearly 1 2 1 2 dependent with the relation μ (x) + μ (x) = 1,∀x ∈ X. Then the multi-fuzzy set 1 2 represents an ordinary fuzzy set with membership value μ (x) and nonmembership value μ (x). If μ (x) + μ (x) ≤ 1,∀x ∈ X, then the multi-fuzzy set represents an 2 1 2 Atanassov intuitionistic fuzzy set withπ(x) = 1−μ (x)−μ (x) as the measure of non- 1 2 specificity of x. The multi-fuzzy set of dimension 3 withμ (x)+μ (x)+μ (x) = 1 also 1 2 3 represents the same intuitionistic fuzzy set. Uncertainty (non-specificity) depends on many noticed and unnoticed factors and each factor has a membership function. For this we define multi-fuzzy membership functions with linearly dependent infinite co- ordinates. That is, A = {x,μ (x),μ (x),... : x ∈ X, μ (x) = 1}, (2) 1 2 i i=1 where μ ∈ I (for i = 1, 2,...). Here μ (x) and μ (x) are the membership and non i 1 2 membership values of x in A and π(x) = μ (x),μ (x),... is the membership value 3 4 of non-specificity or non-determinacy (or uncertainty) of x. Clearly π(x) itself is a multi-fuzzy membership function of infinite dimension. Example 2 Using 3-dimensional multi-fuzzy membership functions, we can charac- terize the colour of a pixel in a colour image. Suppose an image is approximated by an m× n matrix of pixels with multi-fuzzy membership function (μ ,μ ,μ ). The r g b membership values μ (x),μ (x),μ (x) being the normalized red value, green value r g b and blue value of the pixel x ∈ X respectively, where X is the set consisting of the mn pixels. So the colour image can be approximated by the collection of pixels with the multi-membership function (μ ,μ ,μ ) and it can be represented as a multi-fuzzy set r g b A = {x,μ (x),μ (x),μ (x) : x ∈ X}. r g b Construct 3 different gray images of the colour image A as follows. Let x be an arbitrary pixel of the image, for i, j = 1, 2, 3; a ≥ 0 and a + a + a = 1. (i, j) (i,1) (i,2) (i,3) Define gray values h (x), h (x) and h (x)of x as 1 2 3 h (x) = a μ (x)+ a μ (x)+ a μ (x). i (i,1) r (i,2) g (i,3) b 38 Sabu Sebastian · T V Ramakrishnan(2011) The fuzzy sets H = {x, h (x) : x ∈ X}; (for i = 1, 2, 3) are the gray images (with i i different gray tones) of the colour image A. If the coefficient matrix of h (x), for i = 1, 2, 3 is invertible, then using matrix inversion we can reconstruct the original colour image from the 3 gray images. Let PA(x) = H(x) be the matrix representation of the pixel x ∈ X of the 3 gray images, where ⎡ ⎤ ⎢ ⎥ a a a ⎢ (1,1) (1,2) (1,3)⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ P = a a a , ⎢ (2,1) (2,2) (2,3)⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ a a a (3,1) (3,2) (3,3) A(x) = (μ (x),μ (x),μ (x)) , r g b and H(x) = (h (x), h (x), h (x)) . 1 2 3 −1 If P is invertible, then we have A(x) = P H(x). Hence we can reconstruct the colour membership value μ (x) = (μ (x),μ (x),μ (x)) of the pixel x from 3 linearly A r g b independent gray values. We hope that this technique is useful in image processing. Definition 7 Let{L : i ∈ N} be a family of complete lattices; A = {x,μ (x),...,μ (x),... : x ∈ X,μ ∈ L , i ∈ N}, 1 i i B = {x,ν (x),...,ν (x),... : x ∈ X,ν ∈ L , i ∈ N} 1 i i be multi-fuzzy sets in a nonempty set X. We define the following relations and opera- tions: (a) A B if and only if μ (x) ≤ ν (x),∀x ∈ X and for all i ∈ N; i i (b) A = B if and only if μ (x) = ν (x),∀x ∈ X and for all i ∈ N; i i (c) A B = {x,μ (x)∨ν (x),μ (x)∨ν (x),...,μ (x)∨ν (x),... : x ∈ X}; 1 1 2 2 i i (d) A B = {x,μ (x)∧ν (x),μ (x)∧ν (x),...,μ (x)∧ν (x),... : x ∈ X}. 1 1 2 2 i i X X k Remark 2 Let L = L, for i = 1,..., k. L = (L ) . The set of all multi-fuzzy membership functions of multi-fuzzy sets in X of dimension k does not form a vector space over the lattice L with respect to the operations and∧, where l∧ A = {x, l∧ μ (x), l ∧ μ (x),..., l ∧ μ (x),... : x ∈ X}, for any l ∈ L. So the name vector valued 1 2 i membership is not appropriate for multi-membership function. 3.T ,S andM Operations In this section we introduce the operations T , S and M on multi-fuzzy sets. From now onwards, A, B and C will denote the following multi-fuzzy sets in X of dimension k (where k is a fixed positive integer) and L = [0, 1], for i = 1,..., k. A={x,μ (x),μ (x),...,μ (x) : x ∈ X}, (3) 1 2 k B={x,ν (x),ν (x),...,ν (x) : x ∈ X}, (4) 1 2 k C={x,γ (x),γ (x),...,γ (x) : x ∈ X}. (5) 1 2 k Fuzzy Inf. Eng. (2011) 1: 35-43 39 Definition 8 Let A and B be multi-fuzzy sets in X. The T operations, S operations andM operations can be defined as: T (A, B)={x, t (μ (x),ν (x)), t (μ (x),ν (x)),..., t (μ (x),ν (x)),... : x ∈ X}, 1 1 1 2 2 2 n n n S(A, B)={x, s (μ (x),ν (x)), s (μ (x),ν (x)),..., s (μ (x),ν (x)),... : x ∈ X}, 1 1 1 2 2 2 n n n M(A, B)={x, m (μ (x),ν (x)), m (μ (x),ν (x)),..., m (μ (x),ν (x)),... : x ∈ X}, 1 1 1 2 2 2 n n n where n is a positive integer; t ,s and m are different t-norm, s-norm and aggrega- i j l tion operation [6] from I × I into I respectively, for 1 ≤ i, j, l ≤ k. Example 3 Let A and B be multi-fuzzy sets in X of dimension k. (a) A• B = {x,μ (x)·ν (x),μ (x)·ν (x),...,μ (x)·ν (x) : x ∈ X}. 1 1 2 2 k k (b) A+ B = {x,μ (x)+ν (x)−μ (x)·ν (x),...,μ (x)+ν (x)−μ (x)·ν (x) : x ∈ X}. 1 1 1 1 k k k k (c) A⊕B = {x, min(1,μ (x)+ν (x)),..., min(1,μ (x)+ν (x)) : x ∈ X}. 1 1 k k Example 4 Let A and B be multi-fuzzy sets in X of dimension k. Hamacher T operations andS operations of A and B are: T (A, B) = {x, t (μ (x),ν (x)),..., t (μ (x),ν (x)) : x ∈ X}, (6) r 1 1 r k k 1 k ab where t (a, b) = , r > 0; r+(1−r)(a+b−ab) S (A, B) = {x, s (μ (x),ν (x)),..., s (μ (x),ν (x)) : x ∈ X}, (7) r 1 1 r k k 1 k a+b+(r−2)ab where s (a, b) = , r > 0. 1+(r−1)ab Theorem 1 Let A, B and C be multi-fuzzy sets in X of dimension k. (a) T (A, 1) = A, where 1 = {x, 1, 1,..., 1 : x ∈ X}. (b) T (A, B) = T (B, A). (c) T (A,T (B, C)) = T (T (A, B), C). (d) B C impliesT (A, B) T (A, C). Proof (a) T (A, 1)={x, t (μ (x), 1), t (μ (x), 1),..., t (μ (x), 1) : x ∈ X} 1 1 2 2 k k ={x,μ (x),μ (x),...,μ (x) : x ∈ X} 1 2 k = A. Proofs of (b), (c) and (d) are similar to (a). Remark 3 Similarly S operations are commutative, associative and non-decreasing operations withS(A, 0) = A, where 0 = {x, 0, 0,..., 0 : x ∈ X}. In the remaining part, we will use t and s as different t-norm and s-norm re- i j spectively, for 1 ≤ i, j ≤ k. Here we introduce the notion of p-conjugates of T and S Operations, which are very useful links between multi-fuzzy sets and Atanassov intuitionistic fuzzy sets. 40 Sabu Sebastian · T V Ramakrishnan(2011) Definition 9 Let A and B be multi-fuzzy sets in X of dimension k and let s-norm s be the dual of t-norm t with respect to the fuzzy complement operation c. For a fixed p ∈{1, 2,..., k}, the p-conjugate of T (A, B) with respect to c is the multi-fuzzy set T (A, B) obtained from T (A, B) by replacing t (μ (x),ν (x)) as s (μ (x),ν (x)) and p p p p p p the remaining co-ordinates kept to be the same as inT (A, B). That is, T (A, B) = {x, t (μ (x),ν (x)),..., t (μ (x),ν (x)), s (μ (x),ν (x)), 1 1 1 p−1 p−1 p−1 p p p t (μ (x),ν (x)),..., t (μ (x),ν (x)) : x ∈ X}. p+1 p+1 p+1 k k k Definition 10 Let A, B be multi-fuzzy sets in X of dimension k and let t-norm t be the dual of s-norm s with respect to the fuzzy complement operation c. For a fixed p ∈{1, 2,..., k}, the p-conjugate of S(A, B) with respect to c is the multi-fuzzy set S (A, B) obtained from S(A, B) by replacing s (μ (x),ν (x)) as t (μ (x),ν (x)) and p p p p p p the remaining co-ordinates kept to be the same as inS(A, B). That is, S (A, B) = {x, s (μ (x),ν (x)),..., s (μ (x),ν (x)), t (μ (x),ν (x)), 1 1 1 p−1 p−1 p−1 p p p s (μ (x),ν (x)),..., s (μ (x),ν (x)) : x ∈ X}. p+1 p+1 p+1 k k k Example 5 Let A and B be multi-fuzzy sets in X of dimension 2, c be the classical fuzzy complement. Let S(A, B) = {x,μ (x)+ν (x)−μ (x)·ν (x),μ (x)+ν (x)−μ (x)·ν (x) : x ∈ X}. 1 1 1 1 2 2 2 2 Then S (A, B) = {x,μ (x)+ν (x)−μ (x)·ν (x),μ (x)·ν (x) : x ∈ X}. 1 1 1 1 2 2 Remark 4 q-conjugate of T (A, B) with respect to the fuzzy complement operation (c,d) d is denoted byT (A, B), for every p, q∈{1, 2,..., k}. That is (p,q) (c,d) d c T (A, B) = T (T (A, B)). q p (p,q) Theorem 2 Let A, B be multi-fuzzy sets in X of dimension k and let T , S be fixed operators. If p , p and p are distinct integers in{1, 2,..., k}, then 1 2 3 (c,c) (a) T (A, B) = T (A, B); (p,p) (c ,c ) (c ,c ) 1 2 2 1 (b) T (A, B)=T (A, B); (p ,p ) (p ,p ) 1 2 2 1 ((c ,c ),c ) (c ,(c ,c )) 1 2 3 1 2 3 (c) T (A, B)=T (A, B); ((p ,p ),p ) (p ,(p ,p )) 1 2 3 1 2 3 where c , c , c and c are fuzzy complement operations. 1 2 3 Proof (a)T (A, B) is obtained fromT (A, B) by replacing t (μ (x),ν (x)) as s (μ (x), p p p p p (c,c) ν (x)) and remaining coordinates are unchanged. Similarly T (A, B) is obtained (p,p) fromT (A, B) by replacing s (μ (x),ν (x)) as t (μ (x),ν (x)) and remaining coordi- p p p p p p (c,c) nates are unchanged. HenceT (A, B) = T (A, B). (p,p) Fuzzy Inf. Eng. (2011) 1: 35-43 41 1 th (b) The operation T makes a change in the p co-ordinate only. Similar change may be obtained in the case of T also. Since p  p , the sequential order of the 1 2 operatorsT s will not effect their compositions. (c) Similar to the proofs of (a) and (b). Remark 5 • Similar results are obtained when replaceT byS. (c ,...,c ) (c ,...,c ) 1 m 1 m−1 • In generalT (A, B)bethe p conjugates ofT (A, B) with respect (p ,...,p ) (p ,...,p ) 1 m 1 m−1 to the complement operation c , for p ,..., p ∈{1, 2,..., k} and for any positive m 1 m integer m ≥ 2. (c ,...,c ) (c ,...,c ) 1 m 1 m−1 • SimilarlyS (A, B) be the p conjugates ofS (A, B) with respect (p ,...,p ) (p ,...,p ) 1 m 1 m−1 to the complement operation c , for p ,..., p ∈{1, 2,..., k} and for any positive m 1 m integer m ≥ 2. • For simplicity we avoid the superscript of the p-conjugate operators. Theorem 3 Let A and B be multi-fuzzy sets in X of dimension k. (a) T (A, B) = S(A, B). (1,2,...,k) (b) S (A, B) = T (A, B). (1,2,...,k) Proof (a) Let T (A, B) = {x, t (μ (x),ν (x)), t (μ (x),ν (x)),..., t (μ (x),ν (x)) : x ∈ X} 1 1 1 2 2 2 k k k and s (μ (x),ν (x)) be the dual of t (μ (x),ν (x)), for p = 1, 2,..., k. Therefore, 1 to p p p p p p k conjugates ofT (A, B)is {x, s (μ (x),ν (x)), s (μ (x),ν (x)),..., s (μ (x),ν (x)) : x ∈ X} = S(A, B). 1 1 1 2 2 2 k k k (b) Similar to the proof of (a). 4. Relation Between Intuitionistic Fuzzy Operations andT, S, M Operations In this section we study the multi-fuzzy sets of dimension 2 (that is k = 2). Through- out this section conjugate means 2-conjugate with respect to classical fuzzy comple- ment operation and intuitionistic fuzzy set means an Atanassov intuitionistic fuzzy set. Here triangular norms t ≡ t ≡ t and s ≡ s ≡ s. 1 2 1 2 Remark 6 A multi-fuzzy set A = {x,μ (x),μ (x) : x ∈ X}, is called an intuitionistic 1 2 fuzzy set, if μ (x) + μ (x) ≤ 1 and if μ (x) and μ (x) represent the membership 1 2 1 2 and nonmembership values of x in A. Therefore every intuitionistic fuzzy set is a multi-fuzzy set, but the converse need not be true. For example, the multi-fuzzy set A = {x, 1, 1 : x ∈ X} is not an intuitionistic fuzzy set. Remark 7 If A and B are multi-fuzzy sets in X of dimension 2, then T (A, B) = S (A, B) andS (A, B) = T (A, B). 2-conjugates ofT andS operations on Atanassov 2 1 2 intuitionistic fuzzy sets are intuitionistic operations. 42 Sabu Sebastian · T V Ramakrishnan(2011) Theorem 4 Let A = {x,μ (x),μ (x) : x ∈ X} and B = {x,ν (x),ν (x) : x ∈ 1 2 1 2 X} be multi-fuzzy sets in X of dimension 2 and let s, t be dual with respect to the classical fuzzy complement operation. If μ (x) + μ (x) ≤ 1, ν (x) + ν (x) ≤ 1, then 1 2 1 2 the conjugates of T (A, B)={x, t(μ (x),ν (x)), t(μ (x),ν (x)) : x ∈ X}, (8) 1 1 2 2 S(A, B)={x, s(μ (x),ν (x)), s(μ (x),ν (x)) : x ∈ X} (9) 1 1 2 2 are intuitionistic fuzzy operations. Proof Let T (A, B) = {x, t(μ (x),ν (x)), s(μ (x),ν (x)) : x ∈ X} 2 1 1 2 2 and S (A, B) = {x, s(μ (x),ν (x)), t(μ (x),ν (x)) : x ∈ X} 2 1 1 2 2 be the conjugates of T (A, B) andS(A, B), respectively. Since μ (x) ≤ 1−μ (x), 2 1 ν (x) ≤ 1−ν (x), 2 1 t(μ (x),ν (x)) = 1− s(1−μ (x), 1−ν (x)) 1 1 1 1 and monotonicity of s,wehave t(μ (x),ν (x))+ s(μ (x),ν (x))≤ t(μ (x),ν (x))+ s(1−μ (x), 1−ν (x)) 1 1 2 2 1 1 1 1 = t(μ (x),ν (x))+ (1− t(μ (x),ν (x))) 1 1 1 1 = 1. Since t-norms and s-norms are non-negative functions, we have 0 ≤ t(μ (x),ν (x))+ s(μ (x),ν (x)) ≤ 1. 1 1 2 2 Similarly 0 ≤ s(μ (x),ν (x))+ t(μ (x),ν (x)) ≤ 1. 1 1 2 2 Therefore,T (A, B) andS (A, B) are intuitionistic fuzzy operations. 2 2 Example 6 Let A and B be multi-fuzzy sets in X of dimension 2 (that is A, B ∈ M FS(X)) withμ (x)+μ (x) ≤ 1,ν (x)+ν (x) ≤ 1. Then the intuitionistic fuzzy op- 1 2 1 2 erations ‘ ’,‘ ’,‘·’,‘+’, ‘ &’,‘ $’, ‘ #’ (defined by Atanassov [2]) can be expressed as the operations in M FS(X). (a) A B = {x, max(μ (x),ν (x)), min(μ (x),ν (x)) : x ∈ X} = S (A, B) 1 1 2 2 2 with s ≡ max. (b) A B = {x, min(μ (x),ν (x)), max(μ (x),ν (x)) : x ∈ X} = T (A, B) 1 1 2 2 2 with t ≡ min. Fuzzy Inf. Eng. (2011) 1: 35-43 43 (c) A· B = {x,μ (x)·ν (x),μ (x)+ν (x)−μ (x)·ν (x) : x ∈ X} = T (A, B) 1 1 2 2 2 2 2 with t as the algebraic product. (d) A+ B = {x,μ (x)+ν (x)−μ (x)·ν (x),μ (x)·ν (x) : x ∈ X} = S (A, B) 1 1 1 1 2 2 2 with s as the algebraic sum. μ (x)+ν (x) μ (x)+ν (x) 1 1 2 2 (e) A&B = {x, ,  : x ∈ X} = M(A, B) 2 2 with m ≡ m as the algebraic mean. 1 2 (f) A$B = {x, μ (x)ν (x), μ (x)ν (x) : x ∈ X} = M(A, B) 1 1 2 2 with m ≡ m as the geometric mean. 1 2 2μ (x)ν (x) 2μ (x)ν (x) 1 1 2 2 (g) A#B = {x, ,  : x ∈ X} = M(A, B) μ (x)+ν (x) μ (x)+ν (x) 1 1 2 2 μ (x)ν (x) i i with m ≡ m as the harmonic mean, for which = 0, 1 2 μ (x)+ν (x) i i if μ (x) = ν (x) = 0, i = 1, 2. i i 5. Concluding Remarks If k=1 and L = [0, 1], then the multi-fuzzy set becomes an ordinary fuzzy set. An or- dinary L-fuzzy set is a multi-fuzzy set of dimension k=1. By Remark 4 we conclude that every intuitionistic fuzzy set is a multi-fuzzy set and operations in intuitionistic fuzzy sets (see [2]) are certain operation in multi-fuzzy sets. Hence the theory of multi-fuzzy sets is an extension of fuzzy set theory, L-fuzzy set theory and intuition- istic fuzzy set theory. Acknowledgments The authors are very grateful to Prof. T. Thrivikraman and referees for their very con- structive comments. The first author is thankful to the University Grants Commission of India for awarding teacher fellowship under the Faculty Development Programme (No. F.FIP/XI Plan/KLKA 001 TF 01). References 1. Atanassov K T (1986) Intuitionistic fuzzy sets. Fuzzy Sets and Systems 20(1): 87-96 2. Atanassov K T (1999) Intuitionistic Fuzzy Sets. Heidelberg: Springer-Physica Verlag, Vol. 35 3. Deng J L (1989) Introduction to grey system theory. Journal of Grey Systems 1: 1-24 4. Gau W L, Buehrer D J (1993) Vague sets. IEEE Trans. on Systems, Man, and Cybernetics 23(2): 610-614 5. Goguen J A (1967) L-fuzzy sets. Journal of Mathematical Analysis and Applications 18(1): 145-174 6. Klir G J, Yuan B (1995) Fuzzy sets and fuzzy logic: Theory and applications. New Delhi: Prentice- Hall 7. Menger K (1942) Statistical metrics. Proceedings of the National Academy of Sciences of the United States of America 28: 535-537 8. Mizumoto M, Tanaka K (1976) Some properties of fuzzy sets of type-2. Information and Control 31: 312-340 9. Pawlak Z (1982) Rough sets. International Journal of Computer and Information Sciences 11(5): 341-356 10. Sambuc R (1975) Fonctionsφ-floues. Applications I’Aide au Diagnostic en Pathologie Thyroidienne, Ph.D Thesis, Univ. Marseille, France 11. Zadeh L A (1965) Fuzzy sets. Information and Control 8(3): 338-353

Journal

Fuzzy Information and Engineering – Taylor & Francis

Published: Mar 1, 2011

Keywords: Multi-fuzzy set; Intuitionistic fuzzy set; L-fuzzy set; Uncertainty; p-conjugat; T operation

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