Annals of Fuzzy Mathematics and Informatics Volume x, No. x, (Month 201y), pp. 1–xx ISSN: 2093–9310 (print version) ISSN: 2287–6235 (electronic version) http://www.afmi.or.kr
@FMI c Research Institute for Basic Science, Wonkwang University http://ribs.wonkwang.ac.kr
Single valued neutrosophic relations J. H. Kim, P. K. Lim, J. G. Lee, K. Hur Received 6 March 2018; Revised 5 April 2018; Accepted 24 June 2018
Abstract. We introduce the concept of a single valued neutrosophic reflexive, symmetric and transitive relation. And we study single valued neutrosophic analogues of many results concerning relationships between ordinary reflexive, symmetric and transitive relations. Next, we define the concepts of a single valued neutrosophic equivalence class and a single valued neutrosophic partition, and we prove that the set of all single valued neutrosophic equivalence classes is a single valued neutrosophic partition and the single valued neutrosophic equivalence relation is induced by a single valued neutrosophic partition. Finally, we define an α-cut of a single valued neutrosophic relation and investigate some relationships between single valued neutrosophic relations and their α-cuts.
2010 AMS Classification: 04A72 Keywords: Single valued neutrosophic relation, Single valued neutrosophic reflexive [respec., symmetric and transitive]relation, Single valued neutrosophic equivalence relation, Single valued neutrosophic transitive closure, Single valued neutrosophic value. Corresponding Author: J. H. Kim (junhikim@wku.ac.kr)
1. Introduction In 1965, Zadeh [28] had introduced the concept of a fuzzy set as the generalization of a crisp set. In 1971, he [27] defined the notions of similarity relations and fuzzy orderings as the generalizations of crisp equivalence relations and partial orderings playing basic roles in many fields of pure and applied science. After that time, many researchers [5, 6, 7, 8, 9, 10, 13, 14, 18] studied fuzzy relations. In particular, Chakraborty et al. [5, 6, 7, 8] defined a fuzzy relation over a fuzzy set and obtained many properties. Furthermore, Dib and Youssef [9] defined the fuzzy Cartesian product of two ordinary sets X and Y as the collection of all L-fuzzy sets of X × Y ,
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where L = I × I and I denotes the unit closed interval. In 2009, Lee [14] obtained many results by using the notion of fuzzy relations introduced by Dib and Youssef. In 1968, Atanassov [1] defined an intutionistic fuzzy set as a generalization of a fuzzy set. After then, Atanassov and Gargov [2, 3] introduced the concept of an interval-valued intuitionistic fuzzy set an dealt with intuitionistic fuzzy logics. Moreover, Hur et al. [11] studied the category of intuitionistic H-fuzzy relation in the sense of a topological universe. Recently, Liu et al. [15, 16, 17] applied the concepts of an intuitionistic fuzzy set and an interval-valued intuitionistic fuzzy set to multi-attribute group decision making and group decision making, respectively. In 1998, Smarandache [23] defined the concept of a neutrusophic set as the generalization of an intuitionistic fuzzy set. Also he introduced neutrosophic logics, neutrosophic sets, neutrosophic probabilities, neutrosophic statistics and its applications in [21, 22] . Furthermore, Salama et al. [19, 20] introduced the concept of a neutrusophic relation and studied its some properties. Recently, Bhowmik and Pal [4] introduced the concept of a neutrosophic relation and studied some of its properties. In particular, Wang et al. [24] introduced the notion of a single valued neutrosophic set. Moreover, Yang et al. [25] defined a single valued neutrosophic relation and investigated some of its properties. In this paper, first, we introduce a single valued neutrosophic relation from a set X to Y and the composition of two single valued neutrosophic relations. Also we introduce some operations between single valued neutrosophic relations and obtain some of their properties. Second, we introduce the concept of a single valued neutrosophic reflexive, symmetric and transitive relation. And we study single valued neutrosophic analogues of many results concerning relationships between ordinary reflexive, symmetric and transitive relations. Third, we define the concepts of a single valued neutrosophic equivalence class and a single valued neutrosophic partition, and we prove that the set of all single valued neutrosophic equivalence classes is a single valued neutrosophic partition and the single valued neutrosophic equivalence relation is induced by a single valued neutrosophic partition. Finally, we define an α-cut of a single valued neutrosophic relation and investigate some relationships between single valued neutrosophic relations and their α-cuts. 2. Preliminaries In this section, we introduce the concept of single valued neutrosophic set, the complement of a single valued neutrosophic set, the inclusion between two single valued neutrosophic sets, the union and the intersection of two single valued neutrosophic sets. Definition 2.1 ([22]). Let X be a non-empty set. Then A is called a neutrosophic set (in sort, NS) in X, if A has the form A = (TA , IA , FA ), where TA : X →]− 0, 1+ [, IA : X →]− 0, 1+ [, FA : X →]− 0, 1+ [. Since there is no restriction on the sum of TA (x), IA (x) and FA (x), for each x ∈ X, −
0 ≤ TA (x) + IA (x) + FA (x) ≤ 3+ .
Moreover, for each x ∈ X, TA (x) [resp.,IA (x) and FA (x)] represent the degree of membership [resp.,indeterminacy and non-membership] of x to A. 2
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From Example 2.1.1 in [19], we can see that every IFS (intutionistic fuzzy set) A in a non-empty set X is an NS in X having the form A = (TA , 1 − (TA + FA ), FA ), where (1 − (TA + FA ))(x) = 1 − (TA (x) + FA (x)). Definition 2.2 ([22]). Let A and B be two NSs in X. Then we called A is contained in B, denoted by A ⊂ B, if for each x ∈ X, inf TA (x) ≤ inf TB (x), sup TA (x) ≤ sup TB (x), inf IA (x) ≥ inf IB (x), sup IA (x) ≥ sup IB (x), inf FA (x) ≥ inf FB (x) and sup FA (x) ≥ sup FB (x). Definition 2.3 ([24]). Let X be a space of points (objects) with a generic element in X denoted by x. Then A is called a single valued neutrosophic set (in sort, SVNS) in X, if A has the form A = (TA , IA , FA ), where TA , IA , FA : X → [0, 1]. In this case, TA , IA , FA are called truth-membership function, indeterminacymembership function, falsity-membership function, respectively and we will denote the set of all SVNSs in X as SV N S(X). Furthermore, we will denote the empty SVNS [resp. the whole SVNS] in X as 0N [resp. 1N ] and define by 0N (x) = (0, 1, 1) [resp. 1N = (1, 0, 0)], for each x ∈ X. Definition 2.4 ([24]). Let A ∈ SV N S(X). Then the complement of A, denoted by Ac , is a SVNS in X defined as follows: for each x ∈ X, TAc (x) = FA (x), IAc (x) = 1 − IA (x) and FAc (x) = TA (x). Definition 2.5 ([26]). Let A, B ∈ SV N S(X). Then (i) A is said to be contained in B, denoted by A ⊂ B, if for each x ∈ X, TA (x) ≤ TB (x), IA (x) ≥ IB (x) and FA (x) ≥ FB (x), (ii) A is said to be equal to B, denoted by A = B, if A ⊂ B and B ⊂ A. Definition 2.6 ([25]). Let A, B ∈ SV N S(X). Then (i) the intersection of A and B, denoted by A ∩ B, is a SVNS in X defined as: A ∩ B = (TA ∧ TB , IA ∨ IB , FA ∨ FB ), where (TA ∧ TB )(x) = TA (x) ∨ TB (x), (FA ∨ FB ) = FA (x) ∨ FB (x), for each x ∈ X, (ii) the union of A and B, denoted by A ∪ B, is an SVNS in X defined as: A ∪ B = (TA ∨ TB , IA ∧ IB , FA ∧ FB ). Result 2.7 ([25], Proposition 2.1). Let A, B ∈ N S(X). Then (1) A ⊂ A ∪ B and B ⊂ A ∪ B, (2) A ∩ B ⊂ A and A ∩ B ⊂ B, (3) (Ac )c = A, (4) (A ∪ B)c = Ac ∩ B c , (A ∩ B)c = Ac ∪ B c . 3. Single valued neutrosophic relations In this section, we introduce the concepts of single valued neutrosophic relation, the composition of two single valued neutrosophic relations and the inverse of a single valued neutrosophic relation, and study some properties of each concept. Let X, Y, Z be ordinary non-empty sets. 3
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Definition 3.1. R is called a single valued neutrosophic relation (in short, SVNR) from X to Y , if it is a SVNS in X × Y having the form: R = (TR , IR , FR ), where TR , IR , FR : X × Y → [0, 1] denote the truth-membership function, indeterminacy membership function, falsity-membership function, respectively. For each (x, y) ∈ X ×Y , TR (x, y) [resp.,IR (x, y) and FR (x, y)] represent the degree of membership [resp., indeterminacy and non-membership] of (x, y) to R. In particular, a SVNR from from X to X is called a SVNR in X (See [25]). The empty SVNR[resp. the whole SVNR] in X is denoted by φN [resp. XN ] and defined as follows: for each (x, y) ∈ X × X, φN (x, y) = (0, 1, 1) [resp. XN (x, y) = (1, 0, 0)]. We will denote the set of all SVNRs in X [resp. from X to Y ] as SV N R(X) [resp. SV N R(X × Y )] . Let X = {x1 , x2 , ..., xm } and let Y = {y1 , y2 , ..., yn }. Then R = (TR , IR , FR ) ∈ SV N R(X × Y ) can be expressed by m × n matrix. This kind of matrix expressing a SVNR will be called a single valued neutrosophic matrix. Definition 3.2 (See [25]). Let R ∈ SV N R(X × Y ). Then (i) the inverse of R, denoted by R−1 , is a SVNR from Y to X defined as follows: for each (y, x) ∈ Y × X, R−1 (x, y) = R(y, x), i.e., −1 TR−1 (y, x) = TR (x, y), IR (y, x) = IR (x, y), FR−1 (y, x) = FR (x, y).
(ii) the complement of R, denoted by Rc , is a SVNR from X to Y defined as follows: for each (x, y) ∈ X × Y , c TRc (x, y) = FR (x, y), IR (x, y) = 1 − IR (x, y), FRc (x, y) = TR (x, y).
Example 3.3. Let X = {a, b, c} and let R be a SVNR in X given by the single valued neutrosophic matrix: (0.2, 0.4, 0.3) (1, 0.2, 0) (0.4, 1, 0.7) (0.6, 0.2, 0.1) (0.3, 0.2, 0.6) . R = (0, 0, 0) (0, 0, 0) (0, 0, 0) (0.2, 0.4, 0.1) Then the inverse and the complement of R are given as below: (0.2, 0.4, 0.3) (0, 0, 0) (0, 0, 0) (0.6, 0.2, 0.1) (0, 0, 0) , R−1 = (1, 0.2, 0) (0.4, 1, 0.7) (0.3, 0.2, 0.6) (0.2, 0.4, 0.1)
(0.3, 0.6, 0.2) Rc = (0, 1, 0) (0, 1, 0)
(0, 0.8, 1) (0.1, 0.8, 0.6) (0, 1, 0)
(0.7, 0, 0.4) (0.6, 0.8, 0.3) . (0.1, 0.6, 0.2)
Remark 3.4. For each R ∈ SV N R(X), R ∩ Rc = φN and R ∪ Rc = XN do not hold, in general. 4
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Consider the SVNR R in Example 3.3. Then (0.2, 0.6, 0.3) (0, 0.8, 1) (0.4, 1, 0.7) c (0.1, 0.8, 0.6) (0.3, 0.8, 0.6) 6= φN , R ∩ R = (0, 1, 0) (0, 1, 0) (0, 1, 0) (0.1, 0.6, 0.2) (0.3, 0.4, 0.2) (1, 0.2, 0) (0.7, 0, 0.4) (0.6, 0.2, 0.1) (0.6, 0.2, 0.3) 6= XN . R ∪ Rc = (0, 0, 0) (0, 0, 0) (0, 0, 0) (0.2, 0.4, 0.1) Definition 3.5 (See [25]). Let R, S ∈ SV N R(X × Y ). Then (i) R is said to be contained in S, denoted by R ⊂ S, if TR (x, y) ≤ TS (x, y), IR (x, y) ≥ IS (x, y) and FR (x, y) ≥ FS (x, y), for each (x, y) ∈ X × Y , (ii) R is said to equal to S, denoted by R = S, if R ⊂ S and S ⊂ R, (iii) the intersection of R and S, denoted by R ∩ S, is a SVNR from X to Y defined as: A ∩ B = (TA ∧ TB , IA ∨ IB , FA ∨ FB ), where (TA ∧ TB )(x, y) = TA (x, y) ∧ TB (x, y), (FA ∨ FB )(x, y) = FA (x, y) ∨ FB (x, y), for each (x, y) ∈ X × Y , (iv) the union of R and S, denoted by R ∪ S, is a SVNR in X to Y defined as: A ∪ B = (TA ∨ TB , IA ∧ IB , FA ∧ FB ). Proposition 3.6 (See [25], Theorem 3.1). Let R, S, P ∈ SV N R(X × Y ). Then (1) (Rc )−1 = (R−1 )c , (2) (R−1 )−1 = R, (Rc )c = R, (3) R ⊂ R ∪ S and S ⊂ R ∪ S, (4) R ∩ S ⊂ R and R ∩ S ⊂ S, (5) if R ⊂ S, then R−1 ⊂ S −1 , (6) if R ⊂ P and S ⊂ P , then R ∪ S ⊂ P , (7) if P ⊂ R and P ⊂ S, then P ⊂ R ∩ S, (8) if R ⊂ S, then R ∪ S = S and R ∩ S = R, (9) (R ∪ S)−1 = R−1 ∪ S −1 , (R ∩ S)−1 = R−1 ∩ S −1 , (10) (R ∪ S)c = Rc ∩ S c , (R ∩ S)c = Rc ∪ S c . Proof. The proofs are similar to Theorem 3.1 in [25].
From Definitions 3.2 and 3.5, we can easily obtain the following results. Proposition 3.7. Let R, S, P ∈ SV N R(X × Y ). Then (1) (Idempotent laws): R ∪ R = R, R ∩ R = R, (2) (Commutative laws): R ∪ S = S ∪ R, R ∩ S = S ∩ R, (3) (Associative laws): R ∪ (S ∪ P ) = (R ∪ S) ∪ P , R ∩ (S ∩ P ) = (R ∩ S) ∩ P , (4) (Distributive laws): R ∪ (S ∩ P ) = (R ∪ S) ∩ (R ∪ P ), R ∩ (S ∪ P ) = (R ∩ S) ∪ (R ∩ P ), (5) (Absorption laws): R ∪ (R ∩ S) = R, R ∩ (R ∪ S) = R. Definition 3.8. Let (Rj )j∈J ⊂ SV N R(X × Y ). Then 5
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T T (i) the the intersection of (Rj )j∈J , denoted by j∈J Rj (simply, Rj ), is a SVNR from X to Y defined as: \ ^ _ _ Rj = ( TRj , IRj , FRj ), S S (ii) the the union of (Rj )j∈J , denoted by j∈J Rj (simply, Rj ), is a SVNR from X to Y defined as: [ _ ^ ^ Rj = ( TRj , IRj , FRj ). The followings are the immediate result of Definitions 3.2, 3.5 and 3.8 Proposition 3.9.SLet RS ∈ SV N R(X ) and let (Rj )j∈J ⊂ SV N R(X ×Y ). Then T T ×Y c c c (1) ( RjS )c = RS , ( R ) = R , j j j T T (2) R ∩ ( Rj ) = (R ∩ Rj ), R ∪ ( Rj ) = (R ∪ Rj ). Definition 3.10. Let R ∈ SV N R(X × Y ) and let S ∈ SV N R(Y × Z). Then the composition of R and S, denoted by S ◦ R, is a SVNR from X to Z defined as: S ◦ R = (TS◦R , IS◦R , FS◦R ), where for each (x, z) ∈ X × Z,W TS◦R (x, z) = y∈Y (TR (x, y) ∧ TS (y, z)), V IS◦R (x, z) = y∈Y (IR (x, y) ∨ IS (y, z)), V FS◦R (x, z) = y∈Y (FR (x, y) ∨ FS (y, x)). Proposition 3.11. (1) P ◦ (S ◦ R) = (P ◦ S) ◦ R), where R ∈ SV N R(X × Y ), S ∈ SV N R(Y × Z) and P ∈ SV N R(Z × W ). (2) P ◦ (R ∪ S) = (P ◦ R) ∪ (P ◦ S), where R, S ∈ SV N R(X × Y ) and P ∈ SV N R(Y × Z). (3) If R ⊂ S, then P ◦ R ⊂ P ◦ S, where R, S ∈ SV N R(X × Y ) and P ∈ SV N R(Y × Z). (4) (S ◦ R)−1 = R−1 ◦ S −1 , where R ∈ SV N R(X × Y ) and S ∈ SV N R(Y × Z). Proof. (1) Let R ∈ SV N R(X × Y ), S ∈ SV N R(Y × Z) and P ∈ SV N R(Z × W ) and let (x, w) ∈ (X × Z). Then W TP ◦(S◦R) (x, w) = Wz∈Z (TW S◦R (x, z) ∧ TP (z, w)) = z∈Z ([ y∈Y (TR (x, y) ∧ TS (y, z)] ∧ TP (z, w)) W W = y∈Y (TR (x, y) ∧ [ z∈Z (TS (y, z) ∧ TP (z, w))]) W = y∈Y (TR (x, y) ∧ TP ◦S (y, w)) = T(P ◦S)◦R) (x, w). Similarly, we can prove that IP ◦(S◦R) (x, w) = I(P ◦S)◦R) (x, w) and FP ◦(S◦R) (x, w) = F(P ◦S)◦R) (x, w). Thus the result holds. (2) Let R, S ∈ SV N R(X × Y ) and P ∈ SV N R(Y × Z) and let (x, z) ∈ X × Z. Then W TP ◦(R∪S) (x, z) = y∈Y (TR∪S (x, y) ∧ TP (y, z)) W = y∈Y ([TR (x, y) ∨ TS (x, y)] ∧ TP (y, z)) W W = [ y∈Y (TR (x, y) ∧ TP (y, z)] ∨ [ y∈Y (TS (x, y) ∧ TP (y, z)] = TP ◦R (x, z) ∨ TP ◦S (x, z) 6
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= T(P ◦R)∪(P ◦S) (x, z). Similarly, we can see that IP ◦(R∪S) (x, z) = I(P ◦R)∪(P ◦S) (x, z) and FP ◦(R∪S) (x, z) = F(P ◦R)∪(P ◦S) (x, z). Thus the result holds. (3) Let R, S ∈ SV N R(X × Y ) and P ∈ SV N R(Y × Z). Suppose R ⊂ S and let (x, z) ∈ X × Z. Then W TP ◦R (x, z) = y∈Y (TR (x, y) ∧ TP (y, z)) W ≤ y∈Y (TS (x, y) ∧ TP (y, z)) [Since R ⊂ S, TR (x, y) ≤ TS (x, y)] = TP ◦S (x, z). Similarly, we can prove that IP ◦R (x, z) ≥ IP ◦S (x, z) and FP ◦R (x, z) ≥ FP ◦S (x, z). Thus the result holds. (4) Let R ∈ SV N R(X × Y ) and S ∈ SV N R(Y × Z) and let (x, z) ∈ X × Z. Then T(S◦R)−1 (z, x) = T W(S◦R) (x, z) = y∈Y (TR (x, y) ∧ TS (y, z)) W = y∈Y (TS −1 (z, y) ∧ TR−1 (y, x)) = TR−1 ◦S −1 (z, x). Similarly we can see that I(S◦R)−1 (z, x) = IR−1 ◦S −1 (z, x) and F(S◦R)−1 (z, x) = FR−1 ◦S −1 (z, x). Thus the result holds. Remark 3.12. (1) For any SVNRs R and S, S ◦ R 6= R ◦ S, in general. (2) For any R, S ∈ SV N R(X × Y ) and P ∈ SV N R(Y × Z), P ◦ (R ∩ S) 6= (P ◦ R) ∩ (P ◦ S), in general. Example 3.13. Let X = Y = {a, b}, Z = {x, y}. Consider two SVNRs R and S in X, and an SVNR P from X to Z given by following single valued neutrosophic matrices: (0.6, 0.3, 0.4) (0.7, 0.2, 0.1) R= , (0.4, 0.6, 0.3) (0.6, 0.4, 0.2) (0.7, 0.4, 0.2) (0.4, 0.6, 0.4) S= (0.5, 0.2, 0.6) (0.3, 0.6, 0.5) and (0.7, 0.2, 0.3) (0.4, 0.6, 0.4) P = . (0.4, 0.6, 0.2) (0.8, 0.2, 0.3) Then TP ◦(R∩S) (a, x) = 0.6 6= 0.4 = T(P ◦R)∩(P ◦S) (a, x). Thus P ◦ (R ∩ S) 6= (P ◦ R) ∩ (P ◦ S). 4. Single valued neutrosophic reflexve, smmetric and ansitive relations In this section, we introduce single valued neutrosophic reflexve, smmetric and ansitive relations and obtain some properties related to them. Definition 4.1 ([25]). The single valued neutrosophic identity relation in X, denoted by IX (simply, I), is a SVNR in X defined as: for each (x, y) ∈ X × X, 1 if x = y 0 if x = y 0 if x = y TIX (x, y) = I (x, y) = F (x, y) = 0 if x 6= y, IX 1 if x 6= y, IX 1 if x 6= y. It is clear that I = I −1 and I c = (I c )−1 . 7
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Definition 4.2 ([25]). R ∈ SV N R(X) is said to be: (i) reflexive, if for each x ∈ X, TR (x, x)1, IR (x, x) = FR (x, x) = 0, (ii) anti-reflexive, if for each x ∈ X, TR (x, x) = 0, IR (x, x) = FR (x, x) = 1. From Definitions 4.1 and 4.2, it is obvious that R is neutrosophic reflexive if and only if I ⊂ R. The followings are the immediate results of the above definition. Proposition 4.3 (See [19], Theorem 2.5.2). Let R ∈ SV N R(X). (1) R is reflexive if and only if R−1 is reflexive. (2) If R is reflexive, then R ∪ S is reflexive, for each S ∈ SV N R(X). (3) If R is reflexive, then R ∩ S is reflexive if and only if S ∈ SV N R(X) is reflexive. The followings are the immediate result of Definitions 3.2, 3.5 and 4.2. Proposition 4.4. Let R ∈ SV N R(X). (1) R is anti-reflexive if and only R−1 is anti-reflexive. (2) If R is anti-reflexive, then R ∪ S is anti-reflexive if and only if S ∈ SV N R(X) is anti-reflexive. (3) If R is anti-reflexive, then R ∩ S is anti-reflexive, for each S ∈ SV N R(X). Proposition 4.5. Let R, S ∈ SV N R(X). If R and S are reflexive, then S ◦ R is reflexive. Proof. Let x ∈ X. Since R and S are reflexive, TR (x, x) = 1, IR (x, x) = FR (x, x) = 0 and TS (x, x) = 1, IR (x, x) = FS (x, x) = 0. Thus W TS◦R = y∈X (TR (x, y) ∧ TS (y, x)) W = [ x6=y∈X (TR (x, y) ∧ TS (y, x))] ∨ (TR (x, x) ∧ TS (x, x)) W = [ x6=y∈X (TR (x, y) ∧ TS (y, x))] ∨ (1 ∧ 1) = 1. On the other V hand, IS◦R = y∈X (IR (x, y) ∨ IS (y, x)) V = [ x6=y∈X (IR (x, y) ∨ IS (y, x))] ∧ (IR (x, x) ∨ IS (x, x)) V = [ x6=y∈X (IR (x, y) ∨ IS (y, x))] ∧ (0 ∨ 0) = 0. Similarly, FS◦R = 0. So S ◦ R is reflexive. Definition 4.6. Let R = (TR , IR , FR ) ∈ SV N R(X). Then (i)[19, 25] R is said to be symmetric, if for each x, y ∈ X, TR (x, y) = TR (y, x), IR (x, y) = IR (y, x), FR (x, y) = FR (y, x), (ii)[19] R is said to be anti-symmetric, if for each (x, y) ∈ X × X with x 6= y, TR (x, y) 6= TR (y, x), IR (x, y) 6= IR (y, x), FR (x, y) 6= FR (y, x), 8
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From Definitions 4.2 and 4.6, it is obvious that φN is a symmetric and antireflexive SVNR, XN and I are symmetric and reflexive SVNRs and I c is an antireflexive SVNR. The following is the immediate result of Definitions 3.5 and 4.6. Result 4.7 ([25], Theorem 3.1). Let R ∈ SV N R(X). Then R is symmetric iff R = R−1 . Proposition 4.8. Let R ∈ SV N R(X). If R is symmetric, then R−1 is symmetric. Proposition 4.9. Let R, S ∈ SV N R(X). If R and S are symmetric, then R ∪ S and R ∩ S are symmetric. Proof. Let (x, y) ∈ X × X. Since R and S and are symmetric, TR (x, y) = TR (y, x), IR (x, y) = IR (y, x), FR (x, y) = FR (y, x) and TS (x, y) = TS (y, x), IS (x, y) = IS (y, x), FS (x, y) = FS (y, x). Thus TR∪S (x, y) = TR (x, y) ∨ SR (x, y) = TR (y, x) ∨ SR (y, x) = TR∪S (y, x). Similarly, we can see that IR∪S (x, y) = IR∪S (y, x) and FR∪S (x, y) = FR∪S (y, x). So R ∪ S is symmetric. Similarly, we can prove that R ∩ S is symmetric. Remark 4.10. R and S are nsymmetric, but S ◦ R is not symmetric, in general. Example 4.11. Let X = {a, b, c} and consider two NRs R and S in X given by the following single valued neutrosophic matrices: (0.2, 0.4, 0.3) (1, 0.2, 0) (0.4, 1, 0.7) (0.6, 0.2, 0.1) (0.3, 0.2, 0.6) R = (1, 0.2, 0) (0.4, 1, 0.7) (0.3, 0.2, 0.6) (0.2, 0.4, 0.1) and
(0.2, 0.4, 0.3) (0, 0.2, 0.6) (0.2, 0.6, 0.3) S = (0, 0.2, 0.6) (0.6, 0.2, 0.1) (0.3, 0.2, 0.6) . (0.2, 0.6, 0.3) (0.3, 0.2, 0.6) (0.2, 0.4, 0.1) Then clearly, R and S are symmetric. But TS◦R (a, b) = 0.6 6= 0.2 = TS◦R (b, a). Thus S ◦ R is not nsymmetric. The following gives the condition for its being symmetric. Proposition 4.12. Let R, S ∈ SV N R(X). Let R and S be symmetric. Then S ◦ R is symmetric if and only if S ◦ R = R ◦ S. Proof. Suppose S ◦ R is symmetric. Since R and S and are symmetric, by Result 4.7, R = R−1 and S = S −1 . Thus S ◦ R = (S ◦ R)−1 [By the hypothesis and Result 4.7] = R−1 ◦ S −1 [By Proposition 3.11] = R ◦ S. Conversely, suppose S ◦ R = R ◦ S. Then (S ◦ R)−1 = R−1 ◦ S −1 [By Proposition 3.11] 9
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=R◦S [Since R and S and are symmetric, R = R−1 and S = S −1 ] = S ◦ R. [By the hypothesis] This completes the proof.
The following is the immediate result of Proposition 4.12. Corollary 4.13. If R is symmetric, then Rn is symmetric, for all positive integer n, where Rn = R ◦ R ◦ ... n times. Definition 4.14. (See [25]) R ∈ SV N R(X) is said to be transitive, if R ◦ R ⊂ R, i.e., R2 ⊂ R. Proposition 4.15. Let R ∈ SV N R(X). If R is transitive, then R−1 is so. Proof. Let (x, y) ∈ X × X. Then TR (y, x) ≥ TR◦R (y, x) TR−1 (x, y) = W = Wz∈X (TR (y, z) ∧ TR (z, x)) = Wz∈X (TR−1 (z, y) ∧ TR−1 (x, z)) = z∈X (TR−1 (x, z) ∧ TR−1 (z, y)) = TR−1 ◦R−1 (x, y). Similarly, we can prove that IR−1 (x, y) ≤ IR−1 ◦R−1 (x, y) and FR−1 (x, y) ≤ FR−1 ◦R−1 (x, y). Thus the result holds.
Proposition 4.16. Let R ∈ SV N R(X). If R is transitive, then so is R2 . Proof. Let (x, y) ∈W X × X. Then TR2 (x, y) = Wz∈X (TR (x, z) ∧ TR (z, y)) ≥ z∈X (TR2 (x, z) ∧ TR2 (z, y)) = TR2 ◦R2 (x, y). Similarly, we can see that IR2 (x, y) ≤ IR2 ◦R2 (x, y) and FR2 (x, y) ≤ FR2 ◦R2 (x, y). Thus the result holds. Proposition 4.17. Let R, S ∈ SV N R(X). If R and S are transitive, then R ∩ S is transitive. Proof. Let (x, y) ∈ X × X.WThen T(R∩S)◦(R∩S) (x, y) = Wz∈X (TR∩S (y, z) ∧ TR∩S (z, x)) = Wz∈X ([TR (x, z) ∧ TS (x, z)] ∧ [TR (z, y) ∧ TS (z, y)]) = Wz∈X ([TR (x, z) ∧ TR (z, y)] ∧ [TW S (x, z) ∧ TS (z, y)]) = ( z∈X [TR (x, z) ∧ TR (z, y)]) ∧ ( z∈X [TS (x, z) ∧ TS (z, y)]) = TR◦R (x, y) ∧ TS◦S (x, y) ≤ TR (x, y) ∧ TS (x, y) [Since R and S are transitive] = TR∩S (x, y). Similarly, we can prove that I(R∩S)◦(R∩S) (x, y) ≥ IR∩S (x, y) and F(R∩S)◦(R∩S) (x, y) ≥ FR∩S (x, y). Thus the result holds. Remark 4.18. For two single valued neutrosophic transitive relation R and S in X, R ∪ S is not transitive, in general. 10
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Example 4.19. Let X = {a, b} and consider two SVNRs R and S in X given by following single valed neutrosophic matrices: (0.8, 0.5, 0.4) (0.6, 0.4, 0.5) R= (0.7, 0.6, 0.2) (0.7, 0.6, 0.3) and
(0.7, 0.4, 0.2) (0.4, 0.6, 0.4) S= . (0.5, 0.4, 0.3) (0.5, 0.4, 0.4) Then we can easily see that R and S are transitive. On the other hand, (0.8, 0.4, 0.2) (0.6, 0.4, 0.4) R∪S = . (0.7, 0.4, 0.2) (0.7, 0.4, 0.3) Then T(R∪S)◦(R∪S) (a, b) = 0.7 ≥ 0.6 = TR∪S (a, b). Thus R ∪ S is not transitive. 5. Single valurd neutrosophic transitive closure In this section, we define the concept of the single valued neutrosophic transitive closure of an SVNR and study some of its properties. Definition 5.1. Let R ∈ SV N R(X). Then the single valued neutrosophic transitive closure of R, denoted by R̂, is defined as: R̂ = R ∪ R2 ∪ .... The following is the immediate result of Definition 5.1. Proposition 5.2. Let R ∈ SV N R(X). Then (1) R̂ is transitive. (2 ) R is transitive iff R = R̂. Proposition 5.3. Let R, S ∈ SV N R(X). If R ⊂ S, then R̂ ⊂ Ŝ. Proof. By Definition 5.1, R̂ = R ∪ R2 ∪ ... and Ŝ = S ∪ S 2 ∪ .... Since R ⊂ S, by Proposition 3.11, R ◦ R ⊂ S ◦ R ⊂ S ◦ S. Then R2 ⊂ S 2 . Thus R3 ⊂ S 3 and so on. So R̂ ⊂ Ŝ. Proposition 5.4. Let R, S ∈ SV N R(X). If R is symmetric, then R̂ is symmetric. Proof. By Corollary 4.13, R2 , R3 , ..., are symmetric. Then by Proposition 4.9, R̂ is symmetric. Proposition 5.5. Let R ∈ SV N R(X). Then (R̂)−1 = Rˆ−1 . Proof. (Rn )−1 = (R ◦ R ◦ ... ◦ R)−1 ntimes = R−1 ◦ R−1 ◦ ... ◦ R−1 = (R−1 )n = (R−1 )n . Then (R̂)−1 = (R ∪ R2 ∪ ...)−1 = R−1 ∪ (R2 )−1 ∪ ... = R−1 ∪ (R−1 )2 ∪ ... = Rˆ−1 .
Proposition 5.6. For any R ∈ SV N R(X), Then R̂ is the intersection of all single valued neutrosophic transitive relations containing R. 11
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Proof. Let T R ∈ SV N R(X) and let R∗ = {RT : RT is a transitive relation containing R}. Then clearly, R∗ is the smallest transitive relation containing R. Since R̂ is a transitive relation containing R, R∗ ⊂ R̂. Conversely, let RT be any transitive relation containing R. Then by Proposition 5.3, R̂ ⊂ RˆT . Since RT is transitive, by Proposition 5.2, RˆT = RT . Thus R̂ ⊂ RˆT , for each RT . So R̂ ⊂ R∗ . This completes the proof. 6. Single valued neutrosophic equivalence relation In this section, we define the concept of a single valued neutrosophic equivalence class and a single valued neutrosophic partition, and we prove that the set of all single valued neutrosophic equivalence classes is a neutrosophic partition and induce the single valued neutrosophic equivalence relation from a single valued neutrosophic partition. Definition 6.1. R ∈ SV N R(X × X) is called a: (i) tolerance relation on X, if it is reflexive and symmetric, (ii) similarity (or equivalence) relation on X, if it is reflexive, symmetric and transitive. (iii) order relation on X, if it is reflexive, anti-symmetric and transitive. We will denote the set of all tolerance [resp., equivalence and order] relations on X as SV N T (X) [resp., SV N E(X) and SV N O(X)]. The following is the immediate result of Propositions 4.3, 4.9 and 4.17. Proposition 6.2. Let (Rj )j∈J ⊂ SV N T (X) [resp., SV N E(X) and SV N O(X)]. T Then Rj ∈ SV N T (X) [resp., SV N E(X) and SV N O(X)]. Proposition 6.3. Let R ∈ SV N E(X). Then R = R ◦ R. Proof. From Definition 4.14, it is clear that R ◦ R ⊂ R. Let (x, y) ∈ X ×WX. Then TR◦R (x, y) = z∈X (TR (x, z) ∧ TR (z, y)) ≥ TR (x, x) ∧ TR (x, y) = 1 ∧ TR (x, y) [Since R is reflexive] = TR (x, y) and V IR◦R (x, y) = z∈X (IR (x, z) ∨ IR (z, y)) ≤ IR (x, x) ∨ IR (x, y) = 0 ∨ IR (x, y) [Since R is reflexive] = IR (x, y). Similarly, FR◦R (x, y) ≤ FR (x, y). Thus R ◦ R ⊃ R. So R ◦ R = R. Definition 6.4. Let V V A ∈ SV N S(X). Then A is said to be normal, if 1, x∈X IA (x) = x∈X FA (x) = 0.
W
x∈X
TA (x) =
Definition 6.5. Let R ∈ SV N E(X) and let x ∈ X. Then the single valued neutrosophic equivalence class of x by R, denoted by Rx , is a SVNS in X defined as: Rx = (TRx , IRx , FRx ), 12
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where TRx , IRx , FRx : X → [0, 1] are mappings and TRx (y) = TR (x, y), IRx (y) = IR (x, y), FRx (y) = FR (x, y), for each y ∈ X. We will denote the set of all single valued neutrosophic equivalence class by R as X/R and it will be called the single valued neutrosophic quotient set of X by R. Proposition 6.6. Let R ∈ SV N E(X) and let x, y ∈ X. Then (1) Rx is normal, in fact, Rx 6= 0N , (2) Rx ∩ Ry = 0N iff R(x, y) = (0, 1, 1), (3) Rx = Ry iff R(x, y) = (1, 0, 0). Proof. (1) Since R is reflexive, TRW(x, x) = TRx (x) =V1, IR (x, x) = IRx (x) = V0 and FR (x, x) = FRx (x) = 0. Thus y∈X TRx (y) = 1, y∈X IRx (y) = 0 and y∈X FRx (y) = 0. So Rx is normal. Moreover, Rx = (1, 0, 0) 6= (0, 1, 1) = 0N (x). Hence Rx 6= 0N . (2) Suppose Rx ∩ Ry = 0N and let z ∈ X. Then 0 = TRx ∩Ry (z) = TRx (z) ∧ TRy (z) = TR (x, z) ∧ TR (y, z) [By Definition 6.5] = TR (x, z) ∧ TR (z, y) [Since R is symmetric] and 1 = IRx ∪Ry (z) = IRx (z) ∨ IRy (z) = IR (x, z) ∨ IR (y, z) [By Definition 6.5] = IR (x, z) ∨ F IR(z, y). [Since R is symmetric] Thus W 0 = z∈X (TR (x, z) ∧ TR (z, y)) = TR◦R (x, y) = TR (x, y) [By Proposition 6.3] and V 1 = z∈X (IR (x, z) ∨ IR (z, y)) = IR◦R (x, y) = IR (x, y) [By Proposition 6.3]. Similarly, FR (x, y) = 1. So R(x, y) = (0, 1, 1). The sufficient condition is easily proved. (3) Suppose Rx = Ry and let z ∈ X. Then R(x, z) = R(y, z). In particular, R(x, y) = R(y, y). Since R is reflexive, R(x, y) = (1, 0, 0). Conversely, suppose R(x, y) = (1, 0, 0) and let z ∈ X. Since R is transitive, R ◦ R ⊂ R. Then TR (x, y) ∧ TR (y, z) ≤ TR (x, z), IR (x, y) ∨ IR (y, z) ≥ IR (x, z), FR (x, y) ∨ FR (y, z) ≥ FR (x, z). Since R(x, y) = (1, 0, 0), TR (x, y) = 1 and IR (x, y) = FR (x, y) = 0. Thus TR (y, z) ≤ TR (x, z), IR (y, z) ≥ IR (x, z), FR (y, z) ≥ FR (x, z). So TRy (z) ≤ TRx (z), IRy (z) ≥ IRx (z), FRy (z) ≥ FRx (z). Hence Ry ⊂ Rx . Similarly, we can see that Rx ⊂ Ry . Therefore Rx = Ry . 13
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Definition 6.7. Let Σ = (Aj )j∈J ⊂ SV N S(X). Then Σ is called a single valued neutrosophic partition of X, if it satisfies the followings: (i) Aj is normal, for each j ∈ J, (ii) either Aj = Ak or Aj 6= Ak , for any j, k ∈ J, S (iii) j∈J Aj = 1N . The following is the immediate result of Proposition 6.6 and Definition 6.7. Corollary 6.8. Let R ∈ SV N E(X). Then X/R is a single valued neutrosophic partition of X. Proposition 6.9. Let Σ be a single valued neutrosophic partition of X. We define R(Σ) = (TR(Σ) , IR(Σ) , FR(Σ) ) as: for each (x, y) ∈ X × X, _ TR(Σ) (x, y) = [TA (x) ∧ TA (y)], A∈Σ
IR(Σ) (x, y) =
^
[IA (x) ∨ IA (y)],
A∈Σ
FR(Σ) (x, y) =
^
[FA (x) ∨ FA (y)],
A∈Σ
where TR(Σ) , IR(Σ) , FR(Σ) : X × X → [0, 1] are mappings. Then R(Σ) ∈ SV N E(X). Proof. Let x ∈ X. Then by Definition 6.7 (iii), _ _ TR(Σ) (x, x) = (TA (x) ∧ TA (x) = (TA (x) = 1 A∈Σ
A∈Σ
and IR(Σ) (x, y) =
^
(IA (x) ∨ IA (y) =
A∈Σ
^
(IA (x) = 0 = FR(Σ) (x, y).
A∈Σ
Thus R(Σ) is reflexive. From the definition of R(Σ), it is clear that R(Σ) is symmetric. Let (x, y) ∈ X × X. Then T WR(Σ)◦R(Σ) (x, y) = Wz∈X [T WR(Σ) (x, z) ∧ TR(Σ) (z, y)]W = Wz∈X [ WA∈Σ (TA (x) ∧ TWA (z)) ∧ B∈Σ (TB (z) ∧ TB (y))] = Wz∈X [( A∈Σ TA (z) ∧ B∈Σ TB (z)) ∧ (TA (x) ∧ TB (y))] = Wz∈X [(1 ∧ 1) ∧ (TA (x) ∧ TB (y))] [Since A and B are normal] = z∈X [TA (x) ∧ TB (y)] = TR(Σ) (x, y). Similarly, we can prove that IR(Σ)◦R(Σ) (x, y) = IR(Σ) (x, y) and FR(Σ)◦R(Σ) (x, y) = FR(Σ) (x, y). Thus R(Σ) is transitive. So R(Σ) ∈ SV N E(X). Proposition 6.10. Let R, S ∈ SV N E(X). Then R ⊂ S iff Rx ⊂ Sx , for each x ∈ X. Proof. Suppose R ⊂ S and let x ∈ X. Let y ∈ X. Then by the hypothesis, TRx (y) = TR (x, y) ≤ TS (x, y) = TSx (y), 14
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IRx (y) = IR (x, y) ≥ IS (x, y) = ISx (y), FRx (y) = FR (x, y) ≥ FS (x, y) = FSx (y). Thus Rx ⊂ Sx . The converse can be easily proved.
Proposition 6.11. Let R, S ∈ SV N E(X). Then S ◦ R ∈ N E(X) iff S ◦ R = R ◦ S. Proof. Suppose S ◦ R = R ◦ S. Since R and S are reflexive, by Proposition 4.5, S ◦ R is reflexive. Since R and S are symmetric, by the hypothesis and Proposition 4.12, S ◦ R is symmetric. Then it is sufficient to show that S ◦ R is transitive. (S ◦ R) ◦ (S ◦ R) = S ◦ (R ◦ S) ◦ R [By Proposition 3.11 (2)] = S ◦ (S ◦ R)◦) = (S ◦ S) ◦ (R ◦ R) ⊂ S ◦ R. Thus S ◦ R is transitive. So S ◦ R ∈ SV N E(X). The converse is immediate. Proposition 6.12. Let R, S ∈ SV N E(X). If R ∪ S = S ◦ R, then R ∪ S ∈ SV N E(X). Proof. Suppose R ∪ S = S ◦ R. Since R and S are reflexive, by Result 4.3 (2), R ∪ S is neutrosophic reflexive. Since R and S are symmetric, by the hypothesis and Proposition 4.8, R ∪ S is symmetric. Then by the hypothesis, S ◦ R is symmetric. Thus by Proposition 4.12, S ◦R = R◦S. So by Proposition 6.11, S ◦R ∈ SV N E(X). Hence R ∪ S ∈ SV N E(X). 7. Relationships between a neutrosophic relation and its α-cut For Tα , Iα , Fα ∈ [0, 1], α = (Tα , Iα , Fα ) will be called a single valued neutrosophic value. For two single valued neutrosophic values α and β, (i) α ≤ β iff Tα ≤ Tβ , Iα ≥ Iβ and Fα ≥ Fβ . (ii) α < β iff Tα < Tβ , Iα > Iβ and Fα > Fβ . In particular, the form α = (α, 1 − α, 1 − α) is called a single valued neutrosophic constant and denoted by α∗ . We will denote that set of all single valued neutrosophic values [resp. constant] as SVNV [resp. SVNC]. Definition 7.1. Let R ∈ SV N R(X × Y ) and let α ∈ SVNV. (i) The strong α-level subset or strong α-cut of R, denoted by [R]ᾱ , is an ordinary relation from X to Y defined as: [R]ᾱ = {(x, y) ∈ X × Y : TR (x, y) > Tα , IR (x, y) < Iα , FR (x, y) < Fα }. (ii) The α-level subset or α-cut of R, denoted by [R]α , is an ordinary relation from X to Y defined as: [R]α = {(x, y) ∈ X × Y : TR (x, y) ≥ Tα , IR (x, y) ≤ Iα , FR (x, y) ≤ Fα }. Definition 7.2. Let R ∈ SV N R(X × Y ) and let α∗ ∈ SVNC. 15
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(i) The strong α∗ -level subset or strong α∗ -cut of R, denoted by [R]α¯∗ , is an ordinary relation from X to Y defined as: [R]α¯∗ = {(x, y) ∈ X × Y : TR (x, y) > α, IR (x, y) < 1 − α, FR (x, y) < 1 − α}. (ii) The α∗ -level subset or α∗ -cut of R, denoted by [R]α∗ , is an ordinary relation from X to Y defined as: [R]α∗ = {(x, y) ∈ X × Y : TR (x, y) ≥ α, IR (x, y) ≤ 1 − α, FR (x, y) ≤ 1 − α}. Example 7.3. In Example 3.3, [R](0.2,0.3,0.1) ¯ = {(x, y) ∈ X × X : TR (x, y) ≥ 0.2, IR (x, y) ≥ 0.3, FR (x, y) ≤ 0.1} = φ, [R](0.2,0.3,0.1) = {(c, c)}, and [R](0.2,0.3,0.1) = φ, ¯ = {(c, c)}, [R](0.2,0.3,0.8) ¯ [R](0.2,0.3,0.8) = {(a, a), (a, c), (c, c)}. [R]0.2∗ = [R](0.2,0.2,0.9) = {(a, a), (c, c)} = [R]0.2 ¯∗. Proposition 7.4. Let R, S ∈ SV N R(X × Y ) and let α, β ∈ SVNV. (1) If R ⊂ S, then [R]α ⊂ [S]α and [R]ᾱ ⊂ [S]ᾱ . (2) If α ≤ β, then [R]β ⊂ [R]α and [R]β̄ ⊂ [R]ᾱ . Proof. (1) Let (x, y) ∈ [R]α . Then TR (x, y) ≥ Tα , IR (x, y) ≤ Iα and FR (x, y) ≤ Fα . Since R ⊂ S, TR (x, y) ≤ TS (x, y), IR (x, y) ≥ IS (x, y) and FR (x, y) ≥ FS (x, y). Thus SR (x, y) ≥ Tα , IS (x, y) ≤ Iα and FS (x, y) ≤ Fα . Hence [R]α ⊂ [S]α . The proof of the second part is similar. (2) Let (x, y) ∈ [R]β . Then TR (x, y) ≥ Tβ , IR (x, y) ≤ Iβ and FR (x, y) ≤ Fβ . Since α ≤ β, Tα ≤ Tβ , Iα ≥ Iβ and Fα ≥ Fβ . Thus TR (x, y) ≥ Tα , IR (x, y) ≤ Iα and FR (x, y) ≤ Fα . So (x, y) ∈ [R]α . Hence [R]β ⊂ [R]α . The proof of the second part is similar. The following is the particular case of the above Proposition. Corollary 7.5. Let R, S ∈ SV N R(X × Y ) and let α∗ , β ∗ ∈ SVNC. (1) If R ⊂ S, then [R]α∗ ⊂ [S]α∗ and [R]α¯∗ ⊂ [S]α¯∗ . (2) If α∗ ≤ β ∗ , then [R]β ∗ ⊂ [R]α∗ and [R]β¯∗ ⊂ [R]α¯∗ . Proposition 7.6. Let R ∈ SV N R(X × Y ). (1) [R]r is an ordinary relation from X to Y , for each r ∈ SVNV. (2) [R]r̄ is an ordinary relation from X to Y , for each r ∈ SVNV, where Tr ∈ [0.1) and Ir , F Tr ∈ (0, 1]. (3) [R]r = Ss<r [R]s , for each r ∈ SVNV, where Tr ∈ (0, 1] and Ir , Fr ∈ [0, 1). (4) [R]r̄ = s>r [R]s̄ , for each r ∈ SVNV, where Tr ∈ [0, 1) and Ir , Fr ∈ (0, 1]. Proof. The proofs of (1) and (2) are clear from Definition 7.1. (3) From Proposition 7.4, it is obvious that {[R]r : r ∈ SVNV} is a descending family of ordinary relations from X toT Y . Let r ∈ SVNV such that Tr ∈ (0, 1] and Ir , Fr ∈ [0, 1). Then clearly, [R]r ⊂ s<r [R]s . Assume that (x, y) ∈ / [R]r . Then TR (x, y) < Tr or IR (x, y) > Ir or FR (x, y) > Fr . Suppose TR (x, y) < Tr . Then there exists T Ts ∈ (0, 1] such T that TR (x, y) < Ts < Tr . Thus (x, y) ∈ / [R] , i.e. , (x, y) ∈ / [R] . So s s s<r s<r [R]s ⊂ [R]r . Hence T [R]r = s<r [R]s . 16
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Suppose IR (x, y) > Ir or FR (x, y) > Fr . Then each case can be similarly proved. (4) Also from Proposition 7.4, it is obvious that {[R]r̄ : r ∈ SVNV]} is a descending family of ordinary relations from X to S Y . Let r ∈ SVNV such that Tr , Ir ∈ [0, 1) and Fr ∈ (0, 1]. Then clearly, [R]r̄ ⊃ s>r Rs̄ . Assume that (x, y) ∈ / [R]r̄ . Then TR (x, y) ≤ Tr or IR (x, y) ≤ Ir or FR (x, y) ≥ Fr . Suppose TR (x, y) ≤ Tr . Then there exists Ts ∈ [0, 1) S S such that TR (x, y) ≤ Tr < TS / [R]s̄ , i.e., (x, y) ∈ / s>r [R]s̄ . So s>r [R]s̄ ⊂ [R]r̄ . Hence s . Thus (x, y) ∈ [R]r̄ = s>r [R]s̄ . Suppose IR (x, y) ≤ Ir or FR (x, y) ≥ Fr . Then each case can be similarly proved. The following is the particular case of the above Proposition. Corollary 7.7. Let R ∈ SV N R(X × Y ). (1) [R]r∗ is an ordinary relation from X to Y , for each r∗ ∈ SVNC. (2) [R]r¯∗ is an ordinary relation from X to Y , for each r∗ ∈ SVNC, where r ∈ [0, 1). T (3) [R]r∗ = Ss∗ <r∗ [R]s∗ , for each r∗ ∈ SVNV, where r ∈ (0, 1]. (4) [R]r¯∗ = s∗ >r∗ [R]s¯∗ , for each r∗ ∈ SVNC, where r ∈ [0, 1). Proposition 7.8. Let X, Y be non-empty sets and let {Rr : r ∈ [0, 1]} be a nonempty descending family of ordinary relations from X to Y such that R0 = X × Y . (1) We define TR , IR , FR : X × W Y → [0, 1] as follows: for each (x, y) ∈ X × Y , TR (x, y) = {r ∈ [0, 1] : (x, y) ∈ Rr }, IR (x, y) = V FR (x, y) = V{r ∈ [0, 1] : (x, y) ∈ / Rr } r = {(1 W− r) ∈ [0, 1] : (x, y) ∈ rR } = 1 − {r ∈ [0, 1] : (x, y) ∈ R }. Then R ∈ SV N R(X × Y ). T (2) For each r ∈ (0, 1], if Rr = Ss<r Rs , then [R]r∗ = Rr . (3) For each r ∈ [0, 1), if Rr = s>r Rs , then [R]r¯∗ = Rr . In the above proposition, R is called the single valued neutrosophic relation from X to Y induced by {Rr : r ∈ [0, 1]}. Proof. (1) It is obvious T from the definition of R. (2) Suppose RrW= s<r Rs , for each r ∈ (0, 1] and let (x, y) ∈ Rr . Then TR (x, y) = {r ∈ [0, 1] : (x, y) ∈ Rr } ≥ r and W IR (x, y) = FR (x, y) = 1 − {r ∈ [0, 1] : (x, y) ∈ / Rr } ≤ 1 − r. r Thus (x, y) ∈ Rr . So R ⊂ [R]r∗ , for each r ∈ (0, 1]. Now let (x, y) ∈ [R]r∗ . Then TR (x, y) ≥ r, IR (x, y) ≤ 1 − r, FR (x, y) ≤ 1 − r, say TR (x, y) ≥ r. Thus by the definition W of R, TR (x, y) = {k ∈ [0, 1] : (x, y) ∈ Rk } = s ≥ r. Let > 0. Then there exists k ∈ (0, 1] such that s − < k and (x, y) ∈ Rk . Thus r − < s − < k and (x, y) ∈ Rk . So (x, y) ∈ Rr− . Since > 0 is arbitrary, by the hypothesis, (x, y) ∈ Rr . Hence [R]r∗ ⊂ Rr . Therefore [R]r∗ = Rr , for each r ∈ (0, 1]. (3) By the similar argument of the proof of (2), it is proved. 17
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The following is the immediate result of Corollary 7.7 and Proposition 7.8 Corollary 7.9. Let X, Y be non-empty sets, let R ∈ SV N R(X × Y ) and let {[R]r∗ : r ∈ [0, 1]} be a family of all ordinary relations from X to Y . We define mappings TS , IS , FS : X × Y →]− 0, 1+ [ as follows: for each (x, y) ∈ X × Y , W ∗ TS (x, y) = {r ∈ [0, 1] : (x, Wy) ∈ [R]r }, IS (x, y) = FS (x, y) = 1 − {r ∈ [0, 1] : (x, y) ∈ [R]r∗ }. Then S ∈ SV N R(X × Y ) and R = S. From the above corollary, we have the following. Corollary 7.10. Let X, Y be non-empty sets and let , R, S ∈ SV N R(X × Y ). Then R = S iff [R]r∗ = [S]r∗ , for each r ∈ [0, 1], or alternatively, iff [R]r¯∗ = [S]r¯∗ , for each r ∈ [0, 1]. Definition 7.11. Let X, Y be non-empty sets, let R be an ordinary relation from X to Y and let RN ∈ SV N R(X × Y ). Then RN is said to be compatible with R, if R = S(RN ), where S(RN ) = {(x, y) : TRN (x, y) > 0, IRN (x, y) < 1, FRN (x, y) < 1}. Example 7.12. (1) Let X, Y be non-empty sets, let φX×Y be the ordinary empty relation from X to Y and let 0N,X×Y be the single valued neutrosophic empty relation from X to Y defined by 0N,X×Y = (0, 1, 1), for each x ∈ X. Then clearly, S(0N,X×Y ) = φX×Y . Thus 0N,X×Y is compatible with φX×Y . (2) Let X, Y be non-empty sets, let X × Y be the whole ordinary relation from X to Y and let 1N,X×Y be the single valued neutrosophic whole relation from X to Y defined by 0N,X×Y = (1, 0, 0), for each x ∈ X. Then clearly, S(1N,X×Y ) = X × Y . Thus 1N,X×Y is compatible with X × Y . (3) Let X, Y be non-empty sets, let r ∈ (0, 1) be fixed. We define the mappings TR , IR , FR : X × Y → [0, 1] as follows: for each (x, y) ∈ X × Y , TR (x, y) = r, IR (x, y) = FR (x, y) = 1 − r. T ∗ Then clearly, T R ∈ SV N R(X × Y ) and S(R) = r∗ ∈SV N C [R]r . Thus R is compatible with r∗ ∈SV N C [R]r∗ . From the following result, every ordinary relation can be consider as a single valued neutrosophic relation. Proposition 7.13. Let X, Y be non-empty sets, let R be an ordinary relation from X to Y and let r ∈ (0, 1]. Then there exists Rr∗ ∈ SV N R(X × Y ) such that Rr∗ is compatible with R and [Rr∗ ]r∗ = R. In this case, Rr∗ will be called an r∗ -th single valued neutrosophic relation from X to Y . Proof. We define the mappings TR , IR , FR : X × Y → [0, 1] as follows: for each (x, y) ∈ X × Y , r if (x, y) ∈ R TRr∗ (x, y) = 0 if (x, y) ∈ / R, 1 − r if (x, y) ∈ R IRr∗ (x, y) = FRr∗ (x, y) = 1 if (x, y) ∈ / R. Then clearly, Rr∗ ∈ SV N R(X × Y ) and [Rr∗ ]r∗ = R. Moreover, by the definition of Rr∗ , S(Rr∗ ) = R. Thus Rr∗ is compatible with R. 18
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The following is the immediate result of Definitions 3.5 and 7.1. Proposition 7.14. Let R, S ∈ SV N R(X × Y ) and let α ∈ SVNV. Then (1) [R ∪ S]α = [R]α ∪ [S]α , [R ∪ S]ᾱ = [R]ᾱ ∪ [S]ᾱ , (2) [R ∩ S]α = [R]α ∩ [S]α , [R ∩ S]ᾱ = [R]ᾱ ∩ [S]ᾱ . The following is the immediate result of Definition 7.2 and Proposition 7.14. Corollary 7.15. Let R, S ∈ SV N R(X × Y ) and let α∗ ∈ SVNC. Then (1) [R ∪ S]α∗ = [R]α∗ ∪ [S]α∗ , [R ∪ S]α¯∗ = [R]α¯∗ ∪ [S]α¯∗ , (2) [R ∩ S]α∗ = [R]α∗ ∩ [S]α∗ , [R ∩ S]α¯∗ = [R]α¯∗ ∩ [S]α¯∗ . From Definitions 4.2, 4.6 and 7.1 it is clear that R ∈ SV N R(X) is reflexive [resp. symmetric], then [R]α and [R]ᾱ are ordinary reflexive [resp. symmetric] on X, for each α ∈ SVNV. Proposition 7.16. Let R ∈ SV N R(X × Y ) and let α ∈ SVNV. If R is transitive, then [R]α and [R]ᾱ are ordinary transitive on X. Proof. Suppose R is transitive. Then R ◦ R ⊂ R, i.e., TR◦R ⊂ TR , IR◦R ⊃ IR and FR◦R ⊃ FR . Let (x, z) ∈ [R]α ◦ [R]α . Then there exists y ∈ X such that (x, z), (z, y) ∈ [R]α . Thus TR (x, z) ≥ Tα , IR (x, z) ≤ Iα , FR (x, z) ≤ Fα and TR (z, y) ≥ Tα , IR (z, y) ≤ Iα , FR (z, y) ≤ Fα . So TR (x, z) ∧ TR (z, y) ≥ Tα , IR (x, z) ∨ IR (z, y) ≤ Iα , FR (x, z) ∨ FR (z, y) ≤ Fα . Since R ◦ R ⊂ R, TR (x, y) ≥ TR (x, z) ∧ TR (z, y), IR (x, y) ≤ IR (x, z) ∨ IR (z, y), FR (x, y) ≤ FR (x, z) ∧ FR (z, y). Hence TR (x, y) ≥ Tα , IR (x, y) ≤ Iα , FR (x, y) ≤ Fα , i.e., (x, y) ∈ [R]α . Therefore [R]α is ordinary transitive. The prof of the second part is similar. From Definitions 4.2, 4.6 and 7.2 it is clear that R ∈ N R(X) is reflexive [resp. symmetric], then [R]α∗ and [R]α¯∗ are ordinary reflexive [resp. symmetric] on X, for each α∗ ∈ NCV. Moreover, we obtain the following from Proposition 7.16. Corollary 7.17. Let R ∈ N R(X × Y ) and let α∗ ∈ NCV. If R is transitive, then [R]α∗ and [R]α¯∗ are ordinary transitive on X. The followings are the immediate results of 4.2, 4.6, Proposition 7.16 and Corollary 7.17. Corollary 7.18. Let R ∈ SV N E(X) and let α ∈ SVNV. Then [R]α and [R]ᾱ are ordinary equivalence relation on X Corollary 7.19. Let R ∈ SV N E(X) and let α∗ ∈ SVNC. Then [R]α∗ and [R]α¯∗ are ordinary equivalence relation on X 19
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8. Conclusions From now on, we dealt with properties of single valued neutrosophic reflexive, symmetric, transitive relations and single valued neutrosophic equivalence relations. In particular, we defined a single valued neutrosophic equivalence class of a point in a set X modulo a single valued neutrosophic equivalence relation R and a single valued neutrosophic partition of a set X. And we proved that the set of all single valued neutrosophic equivalence classes is a single valued neutrosophic partition and induced the single valued neutrosophic equivalence relation by a single valued neutrosophic partition. However, we did not deal with the quotient of S by R, for any SVNRs R and S such that R ⊂ S and decomposition of a mapping f : X → Y by Vneutrosophic relations. Furthermore, we defined α-cut of a SVNR and investigated some relationships between SVNRs and their α-cuts. In the future, we will solve by the above two problems and deal with single valued neutrosophic relations in a fixed SVNS A.
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J. Kim (junhikim@wku.ac.kr) Department of Mathematics Education, Wonkwang University, 460, Iksan-daero, Iksan-Si, Jeonbuk 54538, Korea P. K. Lim (pklim@wku.ac.kr) Division of Mathematics and Informational Statistics, Institute of Basic Natural Science, Wonkwang University, 460, Iksan-daero, Iksan-Si, Jeonbuk 54538, Korea J. G. Lee (jukolee@wku.ac.kr) Division of Mathematics and Informational Statistics, Institute of Basic Natural Science, Wonkwang University, 460, Iksan-daero, Iksan-Si, Jeonbuk 54538, Korea K. Hur (kulhur@wku.ac.kr) Division of Mathematics and Informational Statistics, Institute of Basic Natural Science, Wonkwang University, 460, Iksan-daero, Iksan-Si, Jeonbuk 54538, Korea
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