Extension of Crisp Functions on Neutrosophic Sets

Sabu Sebastian, Florentin Smarandache

Sabu Sebastian, Florentin Smarandache (2017). Extension of Crisp Functions on Neutrosophic Sets. Neutrosophic Sets and Systems 17, 88-92

Abstract. In this paper, we generalize the definition of Neutrosophic sets and present a method for extending

Keywords: Neutrosophic set, Multi-fuzzy set, Bridge function..

1 Introduction

L-fuzzy sets constitute a generalization of the notion of Zadeh's [26] fuzzy sets and were introduced by Goguen [8] in 1967, later Atanassov introduced the notion of the intuitionistic fuzzy sets [1] Gau and Buehrer [7] defined vague sets. Bustince and Burillo [2] showed that the notion of vague sets is the same as that of intuitionistic fuzzy sets. Deschrijver and Kerre [5] established the interrelationship between the theories of fuzzy sets, L-fuzzy sets, interval valued fuzzy sets, intuitionistic fuzzy sets, intuitionistic Lfuzzy sets, interval valued intuitionistic fuzzy sets, vague sets and gray sets [4].

The neutrosophic set (NS) was introduced by F. Smarandache [22] who introduced the degree of indeterminacy (i) as independent component in his manuscripts that was published in 1998.

Multi-fuzzy sets [12, 13, 16] was proposed in 2009 by Sabu Sebastian as an extension of fuzzy sets [8, 26] in terms of multi membership functions. In this paper we generalize the definition of neutrosophic sets and introduce extension of crisp functions on neutrosophic sets.

2 Preliminaries

Definition 2.1. [26] Let X be a nonempty set. A fuzzy set A of X is a mapping A : X → [0, 1], that is, A = {(x, µA(x)) : µA(x) is the grade of membership of x in A, x ∈ X}. The set of all the fuzzy sets on X is denoted by F(X). Definition 2.2. [8] 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 LX consisting of all the mappings from X to L. Definition 2.3. [1] An Intuitionistic Fuzzy Set on X is a set

A = { x, µA(x), νA(x) : x ∈ X}, where µA(x) ∈ [0, 1] denotes the membership degree and νA(x) ∈ [0, 1] denotes the nonmembership degree of x in A and µA(x) + νA(x) ≤ 1, ∀x ∈ X.

crisp functions on Neutrosophic sets and study some properties of such extended functions.

Definition 2.4. [22]A Neutrosophic Set on X is a set

A = { x, TA(x), IA(x), FA(x) : x ∈ X}, where TA(x) ∈ [0, 1] denotes the truth membership degree, IA(x) ∈ [0, 1] denotes the indetermi-nancy membership degree and FA(x) ∈ [0, 1] denotes the falsity membership degree of x in A respectively and 0 ≤ TA(x) + IA(x) + FA(x) ≤ 3, ∀x ∈ X.

For single valued neutrosophic logic (T, I, F ), the sum of the components is: 0 ≤ T +I+F ≤ 3 when all three components are independent; 0 ≤ T + I + F ≤ 2 when two components are dependent, while the third one is independent from them; 0 ≤ T + I + F ≤ 1 when all three components are dependent.

Definition 2.5. [12, 13, 16]Let X be a nonempty set, J be an indexing set and {Lj : j ∈ J} a family of partially ordered sets. A multi-fuzzy set A in X is a set :

A = { x, (µj (x))j∈J : x ∈ X, µj ∈ Lj X , j ∈ J}.

= h(h−1(B(y))) ≤ B(y). Hence f (f −1(B)) ⊆ B .

Proposition 4.2. If an order homomorphism h : I3 → I3 is the bridge function for the extension of a crisp function f : X → Y , then for any k ∈ K neutrosophic sets Ak in X and B in Y : 1. f (0X ) = 0Y ;

2. f (∪Ak) = ∪f (Ak); and

3. (f −1(B)) = f −1(B ), that is, the extension map f is an order homomorphism.

Acknowledgement

The authors are very grateful to referees for their constructive comments and suggestions.

References

[1] K.T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96. [2] H. Bustince and P. Burillo, Vague sets are intuitionistic fuzzy sets, Fuzzy Sets and Systems 79 (1996) 403-405. [3] D. Coker, Fuzzy rough sets are intuitionistic L-fuzzy sets, Fuzzy Sets and Systems 96 (1998) 381-383. [4] J.L. Deng, Introduction to grey system theory, J. rey Systems 1 (1989) 1-24. [5] G. Deschrijver and E.E. Kerre, On the relationship between some extensions of fuzzy set theory, Fuzzy Sets and Systems 133 (2003) 227-235. [6] C.A. Drossos, Foundations of fuzzy sets: a nonstandard approch, Fuzzy Sets and Systems 37 (1990) 287-307. [7] W.L. Gau and D.J. Buehrer, Vague sets, IEEE Trans. Syst., Man, Cybern. 23 (1993) 610-614. [8] J.A. Goguen, L-fuzzy sets, J. Math. Anal. Appl. 18 (1967) 145-174. [9] G.J. Klir and B. Yuan, Fuzzy Sets and Fuzzy Logic: Theory and Applications, Prentice-Hall, New Delhi (1995). [10] K. Menger, Statistical metrics, Proc. the National Academy of Sciences of the United States of America 28 (1942) 535-537. [11] T. V. Ramakrishnan and S. Sebastian, A study on multi-fuzzy sets, Int. J. Appl. Math. 23 (2010) 713–721. [12] S. Sebastian and T. V. Ramakrishnan, Multi-fuzzy sets, Int. Math. Forum 50 (2010) 2471–2476. [13] S. Sebastian and T. V. Ramakrishnan, Multi-fuzzy sets: an extension of fuzzy sets, Fuzzy Inf. Eng. 1 (2011) 35–43. [14] S. Sebastian and T. V. Ramakrishnan, Multi-fuzzy extensions of functions, Advance in Adaptive Data Analysis 3 (2011) 339 – 350. [15] S. Sebastian and T. V. Ramakrishnan, Multi-fuzzy extension of crisp functions using bridge functions, Ann. Fuzzy Math. Inform. 2 (2011) 1–8. [16] S. Sebastian and T. V. Ramakrishnan, Multi-fuzzy topology, Int. J. Appl. Math. 24 (2011) 117–129. [17] S. Sebastian and T. V. Ramakrishnan, Multi-fuzzy subgroups, Int. J. Contemp. Math. Sci. 6 (2011) 365–372. [18] S. Sebastian and T. V. Ramakrishnan, Atanassov intuitionistic fuzzy sets generating maps, J. Intell. Fuzzy Systems 25 (2013) 859–862. [19] S. Sebastian, A Study on Multi-Fuzziness, Ph.D Thesis 2011. [20] S. Sebastian, Multi-Fuzzy Sets, Lap Lambert Academic Publishing, Germany 2013. [21] S. Sebastian and R. John, Multi-fuzzy sets and their correspondence to other sets, Ann. Fuzzy Math. Inform. 11(2) (2016) 341–348. [22] F. Smarandache, Neutrosophy. Neutrosophic Probability, Set, and Logic, Amer. Res. Press, Rehoboth, USA, 105 p., 1998, 2000, 2002, 2005, 2006. [23] G.J. Wang, Order-homomorphism on fuzzes, Fuzzy Sets and Systems, 12 (1984) 281-288. [24] H. Wei, Generalized Zadeh function, Fuzzy Sets and Systems 97 (1998) 381-386. [25] L. Ying-Ming and L. Mao-Kang, Fuzzy Topology, World Scientific, Singapore 1997. [26] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965) 338-353. [27] F. Smarandache, M. Ali, Neutrosophic Triplet as eñtension of Matter Plasma, Unmatter Plasma, and Antimatter Plasma, 69th Annual Gaseous Electronics Conference, Bochum, Germany, Ruhr-Universitat, October 10-14, 2016. [28] F. Smarandache, Neutrosophic Perspectives: Triplets, Duplets, Multisets, Hybrid Operators, Modal Logic, Hedge Algebras. And Applications. Pons Editions, 325 p., 2017.