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University of New Mexico
Introduction to Neutrosophic Soft Groups Tuhin Bera1 and Nirmal Kumar Mahapatra2 1
Department of Mathematics, Boror Siksha Satra High School, Bagnan, Howrah - 711312, WB, India. E-mail: tuhin78bera@gmail.com 2 Department of Mathematics, Panskura Banamali College, Panskura RS-721152, WB, India. E-mail: nirmal_hridoy@yahoo.co.in
Abstract. The notion of neutrosophic soft group is introduced, together with several related properties. Its structural characteristics are investigated with suitable examples.
The Cartesian product on neutrosophic soft groups and on neutrosophic soft subgroup is defined and illustrated by examples. Related theorems are established.
Keywords: neutrosophic soft set, neutrosophic soft groups, neutrosophic soft subgroup, Cartesian product.
1 Introduction The concept of Neutrosophic Set (NS), firstly introduced by Smarandache [1], is a generalisation of classical sets, fuzzy set [2], intuitionistic fuzzy set [3] etc. Researchers in economics, sociology, medical science and many other several fields deal daily with the complexities of modelling uncertain data. Classical methods are not always successful because the uncertainty appearing in these domains may be of various types. While probability theory, theory of fuzzy set, intuitionistic fuzzy set and other mathematical tools are well known and often useful approaches to describe uncertainty, each of these theories has its different difficulties, as pointed out by Molodtsov [4]. In 1999, Molodtsov [4] introduced a new concept of soft set theory, which is free from the parameterization inadequacy syndrome of different theories dealing with uncertainty. This makes the theory very convenient and easy to apply in practice. The classical group theory was extended over fuzzy set, intuitionistic fuzzy set and soft set by Rosenfeld [5], Sharma [6], Aktas et.al. [7], and many others. Consequently, several authors applied the theory of fuzzy soft sets, intuitionistic fuzzy soft sets to different algebraic structures, e.g. Maji et. al. [8, 9, 10], Dinda and Samanta [11], Ghosh et. al. [12], Mondal [13], Chetia and Das [14], Basu et. al. [15], Augunoglu and Aygun [16], Yaqoob et. al [17], Varol et. al. [18], Zhang [19]. Later, Maji [20] has introduced a combined concept, the Neutrosophic Soft Set (NSS). Using this concept, several mathematicians have produced their research works in different mathematical structures, e.g. Sahin et. al [21], Broumi [22], Bera and Mahapatra [23], Maji [24], Broumi and Smarandache [25]. Later, the concept has been redefined by Deli and Broumi [26]. This paper presents the notion of neutrosophic soft groups along with an investigation of some related properties and theorems. Section 2 gives some useful definitions. In Section 3, neutrosophic soft group is defined, along with some properties. Section 4 deals with the Cartesian product
of neutrosophic soft groups. Finally, the concept of neutrosophic soft subgroup is studied, with suitable examples, in Section 5. 2 Preliminaries We recall basic definitions related to fuzzy set, soft set, and neutrosophic soft. 2.1 Definition: [27] A binary operation Γ’ˆ— : [0, 1] Δ— [0, 1] Γ’†’ [0, 1] is continuous t - norm if it satisfies the following conditions: (i) Γ’ˆ— is commutative and associative. (ii) Γ’ˆ— is continuous, (iii) Δ?‘Ž Γ’ˆ— 1 = 1 Γ’ˆ— Δ?‘Ž = Δ?‘Ž, Γ’ˆ€Δ?‘Ž Γ’ˆˆ [0, 1], (iv) Δ?‘Ž Γ’ˆ— Δ?‘? Γ’‰Β€ Δ?‘? Γ’ˆ— Δ?‘‘ if Δ?‘Ž Γ’‰Β€ Δ?‘?, Δ?‘? Γ’‰Β€ Δ?‘‘ , with Δ?‘Ž, Δ?‘?, Δ?‘?, Δ?‘‘ Γ’ˆˆ [0, 1]. A few examples of continuous t-norm are Δ?‘Ž Γ’ˆ— Δ?‘? = Δ?‘ŽΔ?‘?, Δ?‘Ž Γ’ˆ— Δ?‘? = Δ?‘šΔ?‘–Δ?‘› (Δ?‘Ž, Δ?‘?), Δ?‘Ž Γ’ˆ— Δ?‘? = Δ?‘šΔ?‘ŽΔ?‘Δ½ (Δ?‘Ž + Δ?‘? Γ’€“ 1, 0). 2.2 Definition: [27] A binary operation Γ’‹„ Γ’ˆΕ [0, 1]Δ— [0, 1] Γ’†’ [0, 1] is continuous t - conorm (s - norm) if it satisfies the following conditions: (i) Γ’‹„ is commutative and associative, (ii) Γ’‹„ is continuous, (iii) Δ?‘Ž Γ’‹„ 0 = 0 Γ’‹„ Δ?‘Ž = Δ?‘Ž, Γ’ˆ€Δ?‘Ž Γ’ˆˆ [0, 1], (iv) Δ?‘Ž Γ’‹„ Δ?‘? Γ’‰Β€ Δ?‘? Γ’‹„ Δ?‘‘ if Δ?‘Ž Γ’‰Β€ Δ?‘?, Δ?‘? Γ’‰Β€ Δ?‘‘, with Δ?‘Ž, Δ?‘?, Δ?‘?, Δ?‘‘ Γ’ˆˆ [0, 1]. A few examples of continuous s-norm are Δ?‘Ž Γ’‹„ Δ?‘? = Δ?‘Ž + Δ?‘? Γ’€“ Δ?‘ŽΔ?‘?, Δ?‘Ž Γ’‹„ Δ?‘? = max(Δ?‘Ž, Δ?‘?) , Δ?‘Ž Γ’‹„ Δ?‘? = min(Δ?‘Ž + Δ?‘?, 1). Γ’ˆ€ Δ?‘Ž Γ’ˆˆ [0, 1], if Δ?‘Ž Γ’ˆ— Δ?‘Ž = Δ?‘Ž and Δ?‘Ž Γ’‹„ Δ?‘Ž = Δ?‘Ž, then Γ’ˆ— is called an idempotent t-norm and Γ’‹„ is called an idempotent s-norm. 2.3 Definition: [1] A neutrosophic set (NS) on the universe of discourse Δ?‘ˆ is defined as : Δ??Β΄ = {Δ?‘Δ½, Δ?‘‡Δ??Β΄ (Δ?‘Δ½), Δ??ΕΊΔ??Β΄ (Δ?‘Δ½), Δ??Ε‘Δ??Β΄ (Δ?‘Δ½) > : Δ?‘Δ½ Γ’ˆˆ Δ?‘ˆ} ,
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where Δ?‘‡, Δ??ΕΊ, Δ??Ε‘ Γ’ˆΕ Δ?‘ˆ Γ’&#x;Ε ] Γ’€“0,1+ [ and Γ’€“ 0 Γ’‰Β€ Δ?‘‡Δ??Β΄ (Δ?‘Δ½) + Δ??ΕΊΔ??Β΄ (Δ?‘Δ½) + Δ??Ε‘Δ??Β΄ (Δ?‘Δ½) Γ’‰Β€ 3+ . From a philosophical point of view, the neutrosophic set (NS) takes its values from real standard or nonstandard subsets of ] Γ’€“0,1+ [ . But in real life application, in scientific and engineering problems, it is difficult to use NS with values from real standard or nonstandard subset of ] Γ’€“0,1+ [. Hence, we consider the NS which takes the values from the subset of [0,1].
The tabular representation of the NSS Δ?‘ is as: Δ?‘“Δ?‘ (Δ?‘’2 )
Δ?‘“Δ?‘ (Δ?‘’3 )
Γ’„Ž1
(0.5,0.6,0.3) (0.6,0.3,0.5)
Δ?‘“Δ?‘ (Δ?‘’1 )
(0.7,0.4,0.3)
Γ’„Ž2
(0.4,0.7,0.6) (0.7,0.4,0.3)
(0.6,0.7,0.2)
Γ’„Ž3
(0.6,0.2,0.3) (0.8,0.1,0.2)
(0.7,0.2,0.5)
Table 1: Tabular form of NSS N.
2.4 Definition: [4] Let Δ?‘ˆ be an initial universe set and Δ??ΒΈ be a set of parameters. Let Δ?‘ƒ(Δ?‘ˆ) denote the power set of Δ?‘ˆ. Then for Δ??Β΄ Γ’Š† Δ??ΒΈ, a pair (Δ??Ε‘, Δ??Β΄) is called a soft set over Δ?‘ˆ , where Δ??Ε‘: Δ??Β΄ Γ’†’ Δ?‘ƒ(Δ?‘ˆ) is a mapping. 2.5 Definition: [20] Let Δ?‘ˆ be an initial universe set and Δ??ΒΈ be a set of parameters. Let Δ?‘ƒ(Δ?‘ˆ) denote the set of all NSs of Δ?‘ˆ. Then for Δ??Β΄ Γ’Š† Δ??ΒΈ, a pair (Δ??Ε‘, Δ??Β΄) is called an NSS over Δ?‘ˆ, where Δ??Ε‘: Δ??Β΄ Γ’†’ Δ?‘ƒ(Δ?‘ˆ) is a mapping. This concept has been modified by Deli and Broumi [26] as given below. 2.6 Definition: [26] Let Δ?‘ˆ be an initial universe set and Δ??ΒΈ be a set of parameters. Let Δ?‘ƒ(Δ?‘ˆ) denote the set of all NSs of Δ?‘ˆ. Then, a neutrosophic soft set Δ?‘ over Δ?‘ˆ is a set defined by a set valued function Δ?‘“Δ?‘ representing a mapping Δ?‘“Δ?‘ : Δ??ΒΈ Γ’†’ Δ?‘ƒ(Δ?‘ˆ) where Δ?‘“Δ?‘ is called approximate function of the neutrosophic soft set Δ?‘ . In other words, the neutrosophic soft set is a parameterized family of some elements of the set Δ?‘ƒ(Δ?‘ˆ) and therefore it can be written as a set of ordered pairs, Δ?‘ = {(Δ?‘’, {< Δ?‘Δ½ , Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) , Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) , Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) > Γ’ˆΕ Δ?‘Δ½ Γ’ˆˆ Δ?‘ˆ }) Γ’ˆΕ Δ?‘’ Γ’ˆˆ Δ??ΒΈ} where Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) , Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’ˆˆ [0,1] , respectively, called the truth-membership, indeterminacy-membership, falsity-membership function of Δ?‘“Δ?‘ (Δ?‘’) . Since supremum of each Δ?‘‡, Δ??ΕΊ, Δ??Ε‘ is 1 so the inequality 0 Γ’‰Β€ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) + Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) + Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‰Β€ 3 is obvious. 2.6.1 Example Let, Δ?‘ˆ = {Γ’„Ž1 , Γ’„Ž2 , Γ’„Ž3 } be a set of houses and Δ??ΒΈ = {Δ?‘’1 (Δ?‘?Δ?‘’Δ?‘ŽΔ?‘ΛΔ?‘ΔΔ?‘–Δ?‘“Δ?‘ΛΔ?‘™), Δ?‘’2 (Δ?‘Β€Δ?‘œΔ?‘œΔ?‘‘Δ?‘’Δ?‘›), Δ?‘’3 (Δ?‘?Δ?‘œΔ?‘ Δ?‘ΔΔ?‘™Δ?‘Ε) } be a set of parameters with respect to which the nature of houses is described. Let Δ?‘“Δ?‘ (Δ?‘’1 ) = {<Γ’„Ž1 , (0.5, 0.6, 0.3)>, <Γ’„Ž2 , (0.4, 0.7, 0.6)>, <Γ’„Ž3 , (0.6, 0.2, 0.3)>}; Δ?‘“Δ?‘ (Δ?‘’2 )= {<Γ’„Ž1 , (0.6, 0.3, 0.5) >, < Γ’„Ž2 , (0.7, 0.4, 0.3) >,< Γ’„Ž3 , (0.8, 0.1, 0.2) >}; Δ?‘“Δ?‘ (Δ?‘’3 ) = {<Γ’„Ž1 , (0.7, 0.4, 0.3) >, < Γ’„Ž2 , (0.6, 0.7, 0.2) >,< Γ’„Ž3 , (0.7, 0.2, 0.5) >}; Then, Δ?‘ = {[Δ?‘’1 , Δ?‘“Δ?‘ (Δ?‘’1 )], [Δ?‘’2 , Δ?‘“Δ?‘ (Δ?‘’2 )], [Δ?‘’3 , Δ?‘“Δ?‘ (Δ?‘’3 )]} is an NSS over (Δ?‘ˆ, Δ??ΒΈ).
2.6.2 Definition: [26] The complement of a neutrosophic soft set Δ?‘ is denoted by Δ?‘ Δ?‘? and is defined as: Δ?‘ Δ?‘? = {(Δ?‘’, {< Δ?‘Δ½, Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½), 1 Γ’ˆ’ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½), Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) >Γ’ˆΕ Δ?‘Δ½ Γ’ˆˆ Δ?‘ˆ}) Γ’ˆΕ Δ?‘’ Γ’ˆˆ Δ??ΒΈ} 2.6.3 Definition: [26] Let Δ?‘ 1 and Δ?‘ 2 be two NSSs over the common universe (Δ?‘ˆ, Δ??ΒΈ). Then Δ?‘ 1 is said to be the neutrosophic soft subset of Δ?‘ 2 if Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‰Β€ Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‰Δ½ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‰Δ½ Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½); Γ’ˆ€Δ?‘’ Γ’ˆˆ Δ??ΒΈ and Δ?‘Δ½ Γ’ˆˆ Δ?‘ˆ. We write Δ?‘ 1 Γ’Š† Δ?‘ 2 and then Δ?‘ 2 is the neutrosophic soft superset of Δ?‘ 1 . 2.6.4 Definition: [26] 1. Let Δ?‘ 1 and Δ?‘ 2 be two NSSs over the common universe (Δ?‘ˆ, Δ??ΒΈ). Then their union is denoted by Δ?‘ 1 Γ’ˆΕ Δ?‘ 2 = Δ?‘ 3 and is defined as : Δ?‘ 3 = {(Δ?‘’, {< Δ?‘Δ½, Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½) >Γ’ˆΕ Δ?‘Δ½ Γ’ˆˆ Δ?‘ˆ }) Γ’ˆΕ Δ?‘’ Γ’ˆˆ Δ??ΒΈ }, where Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½); 2. Let Δ?‘ 1 and Δ?‘ 2 be two NSSs over the common universe (Δ?‘ˆ, Δ??ΒΈ). Then their intersection is denoted by Δ?‘ 1 Γ’ˆΕ Δ?‘ 2 = Δ?‘ 4 and it is defined as: Δ?‘ 4 = {(Δ?‘’, {< Δ?‘Δ½, Δ?‘‡Δ?‘“Δ?‘ 4 (Δ?‘’) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 4 (Δ?‘’) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 4 (Δ?‘’) (Δ?‘Δ½) >Γ’ˆΕ Δ?‘Δ½ Γ’ˆˆ Δ?‘ˆ }) Γ’ˆΕ Δ?‘’ Γ’ˆˆ Δ??ΒΈ } where Δ?‘‡Δ?‘“Δ?‘ 4 (Δ?‘’) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 4 (Δ?‘’) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 4 (Δ?‘’) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½); 2.7 Definition: [8] Let (Δ??Ε‘, Δ??Β΄) be a soft set over the group Δ??Ε. Then (Δ??Ε‘, Δ??Β΄) is called a soft group over Δ??Ε if Δ??Ε‘(Δ?‘Ž) is a subgroup of Δ??Ε, Γ’ˆ€Δ?‘Ž Γ’ˆˆ Δ??Β΄.
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3 Neutrosophic soft groups In this section, we define the neutrosophic soft group and some basic properties related to it. Unless otherwise stated, Δ??ΒΈ is treated as the parametric set throughout this paper and Δ?‘’ Γ’ˆˆ Δ??ΒΈ, an arbitrary parameter. 3.1 Definition:
Δ??Ε‘Δ?‘“Δ?‘€ (Δ?‘›) (Δ?‘Δ½) = {
3.2 Proposition:
3.1.1 Example:
Proof:
Δ?‘“Δ?‘ (Δ?›ΕΊ) Δ?‘“Δ?‘ (Δ?›ΕΎ)
1 Δ?‘›
if Δ?‘Δ½ is odd
0 if Δ?‘Δ½ is even Corresponding t-norm (Γ’ˆ—) and s-norm (Γ’‹„) are defined as Δ?‘Ž Γ’ˆ— Δ?‘? = Δ?‘šΔ?‘–Δ?‘› (Δ?‘Ž, Δ?‘?), Δ?‘Ž Γ’‹„ Δ?‘? = Δ?‘šΔ?‘ŽΔ?‘Δ½ (Δ?‘Ž, Δ?‘?). Then, Δ?‘€ forms a neutrosophic soft set as well as neutrosophic soft group over [(Γ’„Β€, +), Γ’„•].
A neutrosophic set Δ??Β΄ = {< Δ?‘Δ½, Δ?‘‡Δ??Β΄ (Δ?‘Δ½), Δ??ΕΊΔ??Β΄ (Δ?‘Δ½), Δ??Ε‘Δ??Β΄ (Δ?‘Δ½) >Γ’ˆΕ Δ?‘Δ½ Γ’ˆˆ Δ??Ε} over a group (Δ??Ε,Γ’ˆ˜) is called a neutrosophic subgroup of (Δ??Ε,Γ’ˆ˜) if (Δ?‘–) Δ?‘‡Δ??Β΄ (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε) Γ’‰Δ½ Δ?‘‡Δ??Β΄ (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ??Β΄ (Δ?‘Ε), Δ??ΕΊΔ??Β΄ (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε) Γ’‰Β€ Δ??ΕΊΔ??Β΄ (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ??Β΄ (Δ?‘Ε), Δ??Ε‘Δ??Β΄ (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε) Γ’‰Β€ Δ??Ε‘Δ??Β΄ (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ??Β΄ (Δ?‘Ε); for Δ?‘Δ½, Δ?‘Ε Γ’ˆˆ Δ??Ε. (Δ?‘–Δ?‘–) Δ?‘‡Δ??Β΄ (Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ??Β΄ (Δ?‘Δ½), Δ??ΕΊΔ??Β΄ (Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Β€ Δ??ΕΊΔ??Β΄ (Δ?‘Δ½), Δ??Ε‘Δ??Β΄ (Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Β€ Δ??Ε‘Δ??Β΄ (Δ?‘Δ½), for Δ?‘Δ½ Γ’ˆˆ Δ??Ε. An NSS Δ?‘ over a group (Δ??Ε,Γ’ˆ˜) is called a neutrosophic soft group if Δ?‘“Δ?‘ (Δ?‘’) is a neutrosophic subgroup of (Δ??Ε,Γ’ˆ˜) for each Δ?‘’ Γ’ˆˆ Δ??ΒΈ. 1. Let us consider the Klein's -4 group Δ?‘‰ = {Δ?‘’, Δ?‘Ž, Δ?‘?, Δ?‘?} and Δ??ΒΈ = {Δ?›ΕΊ, Δ?›Λ, Δ?›ΕΎ, Δ?›ΕΌ} be the set of parameters. We define Δ?‘“Δ?‘ (Δ?›ΕΊ), Δ?‘“Δ?‘ (Δ?›Λ), Δ?‘“Δ?‘ (Δ?›ΕΎ), Δ?‘“Δ?‘ (Δ?›ΕΌ) as given by the following table:
1Γ’ˆ’
An NSS Δ?‘ over the group (Δ??Ε,Γ’ˆ˜) is called a neutrosophic soft group iff followings hold on the assumption that truth membership (Δ?‘‡), indeterministic membership (Δ??ΕΊ) and falsity membership (Δ??Ε‘) functions of an NSS obey the idempotent t-norm and idempotent s-norm disciplines. Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε), Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε Γ’ˆ’1 ) Γ’‰Β€ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε), Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε Γ’ˆ’1 ) Γ’‰Β€ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε); Γ’ˆ€ Δ?‘Δ½, Δ?‘Ε Γ’ˆˆ Δ??Ε; Γ’ˆ€Δ?‘’ Γ’ˆˆ Δ??ΒΈ; Firstly, suppose Δ?‘ is an NSS group over (Δ??Ε,Γ’ˆ˜). Then, Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε),
Δ?‘“Δ?‘ (Δ?›Λ) Δ?‘“Δ?‘ (Δ?›ΕΌ)
Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε Γ’ˆ’1 ) Γ’‰Β€ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε Γ’ˆ’1 ) Γ’‰Β€ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε),
Δ?‘’
(0.65, 0.34, 0.14) (0.72, 0.21, 0.16)
(0.88, 0.12,0.72) (0.69, 0.31, 0.32)
Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε Γ’ˆ’1 ) Γ’‰Β€ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε Γ’ˆ’1 )
Δ?‘Ž
(0.71, 0.22, 0.78) (0.84, 0.16, 0.25)
(0.71, 0.19, 0.44) (0.62, 0.32, 0.42)
Conversely, for the identity element Δ?‘’Δ??Ε in Δ??Ε;
Δ?‘?
(0.75, 0.25, 0.52) (0.69, 0.31, 0.39)
(0.83, 0.11, 0.28) (0.58, 0.41, 0.66)
Δ?‘?
(0.67, 0.32, 0.29) (0.79, 0.19, 0.41)
(0.75, 0.21, 0.19) (0.71, 0.27, 0.53)
Table 2: Tabular form of neutrosophic soft group N.
Corresponding t-norm (Γ’ˆ—) and s-norm (Γ’‹„) are defined as Δ?‘Ž Γ’ˆ— Δ?‘? = Δ?‘šΔ?‘ŽΔ?‘Δ½ ( Δ?‘Ž + Δ?‘? Γ’ˆ’ 1, 0) , Δ?‘Ž Γ’‹„ Δ?‘? = Δ?‘šΔ?‘–Δ?‘› (Δ?‘Ž + Δ?‘?, 1). Then, Δ?‘ forms a neutrosophic soft group over (Δ?‘‰, Δ??ΒΈ). 2. Let Δ??ΒΈ = Γ’„• ( the set of natural no.), be the parametric set and Δ??Ε = (Γ’„Β€, +) be the group of all integers. Define a mapping Δ?‘“Δ?‘€ : Γ’„• Γ’†’ Δ?‘ Δ?‘†(Γ’„Β€) where, for any Δ?‘› Γ’ˆˆ Γ’„• and Δ?‘Δ½ Γ’ˆˆ Γ’„Β€, 0 if Δ?‘Δ½ is odd Δ?‘‡Δ?‘“Δ?‘€ (Δ?‘›) (Δ?‘Δ½) = {1 if Δ?‘Δ½ is even 1
Δ?‘›
if Δ?‘Δ½ is odd Δ??ΕΊΔ?‘“Δ?‘€ (Δ?‘›) (Δ?‘Δ½) = {2Δ?‘› 0 if Δ?‘Δ½ is even
Γ’‰Β€ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε); Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘’Δ??Ε ) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘’Δ??Ε ) = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Β€ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘’Δ??Ε ) = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Β€ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½); Now, Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ’1 ) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘’Δ??Ε Γ’ˆ˜ Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘’Δ??Ε ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ’1 ) = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘’Δ??Ε Γ’ˆ˜ Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Β€ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘’Δ??Ε ) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Β€ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½),
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πΉππ(π) (π₯ β1 ) = πΉππ(π) (ππΊ β π₯ β1 ) β€ πΉππ(π) (ππΊ ) β πΌππ(π) (π₯ β1 ) β€ πΉππ(π) (π₯) β πΉππ(π) (π₯) = πΉππ(π) (π₯); Next, πππ(π) (π₯ β π¦) = πππ(π) (π₯ β (π¦ β1 )β1 ) β₯ πππ(π) (π₯) β πππ(π) (π¦ β1 ) β₯ πππ(π) (π₯) β πππ(π) (π¦), πΌππ(π) (π₯ β π¦) = πΌππ(π) (π₯ β (π¦ β1 )β1 ) β€ πΌππ(π) (π₯) β πΌππ(π) (π¦ β1 ) β€ πΌππ(π) (π₯) β πΌππ(π) (π¦), πΉππ(π) (π₯ β π¦) = πΉππ(π) (π₯ β (π¦ β1 )β1 ) β€ πΉππ(π) (π₯) β πΉππ(π) (π¦ β1 ) β€ πΉππ(π) (π₯) β πΉππ(π) (π¦). This completes the proof. 3.2.1 Proposition: Let π be a neutrosophic soft group over the group (πΊ,β). Then for π₯ β πΊ, followings hold. (π) πππ(π) (π₯ β1 ) = πππ(π) (π₯) , πΌππ(π) (π₯ β1 ) = πΌππ(π) (π₯), πΉππ(π) (π₯ β1 ) = πΉππ(π) (π₯) ; (ππ) πππ(π) (ππΊ ) β₯ πππ(π) (π₯) , πΌππ(π) (ππΊ ) β€ πΌππ(π) (π₯) , πΉππ(π) (ππΊ ) β€ πΉππ(π) (π₯) ; if π follows the idempotent t-norm and πΌ, πΉ follow the idempotent s-norm disciplines, respectively. (ππΊ being the identity element of πΊ.)
Proof: Let π1 β π2 = π3 ; Now for π₯, π¦ β πΊ; π ππ3 (π) (π₯ β π¦) = π ππ1 (π) (π₯ β π¦) β π ππ2 (π) (π₯ β π¦) β₯ [π ππ1 (π) (π₯) β π ππ1 (π) (π¦)] β [π ππ2 (π) (π₯) β π ππ2 (π) (π¦)] = [π ππ1 (π) (π₯) β π ππ1 (π) (π¦)] β [π ππ2 (π) (π¦) β π ππ2 (π) (π₯)] (as β is commutative) = π ππ1 (π) (π₯) β [π ππ1 (π) (π¦) β π ππ2 (π) (π¦)] β π ππ2 (π) (π₯) (as β is associative) = π ππ1 (π) (π₯) β π ππ3 (π) (π¦) β π ππ2 (π) (π₯) = π ππ1 (π) (π₯) β π ππ2 (π) (π₯) β π ππ3 (π) (π¦) (as β is commutative) = π ππ3 (π) (π₯) β π ππ3 (π) (π¦) Also, πππ3 (π) (π₯ β1 ) = πππ1 (π) (π₯ β1 ) β πππ2 (π) (π₯ β1 ) β₯ π ππ1 (π) (π₯) β π ππ2 (π) (π₯) = π ππ3 (π) (π₯); Next, πΌ ππ3 (π) (π₯ β π¦) = πΌ ππ1 (π) (π₯ β π¦) β πΌ ππ2 (π) (π₯ β π¦) β€ [πΌ ππ1 (π) (π₯) β πΌ ππ1 (π) (π¦)] β
Proof: (π) πππ(π) (π₯) = πππ(π) (π₯ β1 )β1 β₯ πππ(π) (π₯ β1 ) πΌππ(π) (π₯) = πΌππ(π) (π₯ β1 )β1 β€ πΌππ(π) (π₯ β1 ) πΉππ(π) (π₯) = πΉππ(π) (π₯ β1 )β1 β€ πΉππ(π) (π₯ β1 ) Now, from definition of neutrosophic soft group, the result follows. (ππ) For the identity element ππΊ in πΊ, πππ(π) (ππΊ ) = πππ(π) (π₯ β π₯ β1 ) β₯ πππ(π) (π₯) β πππ(π) (π₯) = πππ(π) (π₯) , πΌππ(π) (ππΊ ) = πΌππ(π) (π₯ β π₯ β1 ) β€ πΌππ(π) (π₯) β πΌππ(π) (π₯) = πΌππ(π) (π₯) , πΉππ(π) (ππΊ ) = πΉππ(π) (π₯ β π₯ β1 ) β€ πΉππ(π) (π₯) β πΉππ(π) (π₯) = πΉππ(π) (π₯) ; Hence, the proposition is proved. 3.3 Theorem: Let π1 and π2 be two neutrosophic soft groups over the group (πΊ,β). Then, π1 β© π2 is also a neutrosophic soft group over (πΊ,β).
[πΌ ππ2 (π) (π₯) β πΌ ππ2 (π) (π¦)] = [πΌ ππ1 (π) (π₯) β πΌ ππ1 (π) (π¦)] β [πΌ ππ2 (π) (π¦) β πΌ ππ2 (π) (π₯)] (as β is commutative) = πΌ ππ1 (π) (π₯) β [πΌ ππ1 (π) (π¦) β πΌ ππ2 (π) (π¦)] β πΌ ππ2 (π) (π₯) (as β is associative) = πΌ ππ1 (π) (π₯) β πΌ ππ3 (π) (π¦) β πΌ ππ2 (π) (π₯) = πΌ ππ1 (π) (π₯) β πΌ ππ2 (π) (π₯) β πΌ ππ3 (π) (π¦) (as β is commutative) = πΌ ππ3 (π) (π₯) β πΌ ππ3 (π) (π¦) Also, πΌ ππ3 (π) (π₯ β1 ) = πΌ ππ1 (π) (π₯ β1 ) β πΌ ππ2 (π) (π₯ β1 ) β€ πΌ ππ1 (π) (π₯) β πΌ ππ2 (π) (π₯) = πΌ ππ3 (π) (π₯); Similarly, πΉ ππ3 (π) (π₯ β π¦) β€ πΉ ππ3 (π) (π₯) β πΉ ππ3 (π) (π¦), πΉ ππ (π) (π₯ β1 ) β€ πΉ ππ (π) (π₯); 3
3
This ends the theorem. The theorem is also true for a family of neutrosophic soft groups over a group.
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3.3.1 Remark: For two neutrosophic soft groups Δ?‘ 1 and Δ?‘ 2 over the group Δ??Ε, Δ?‘ 1 Γ’‹ƒΔ?‘ 2 is not generally a neutrosophic soft group over Δ??Ε. It is possible if anyone is contained in other. For example, let, Δ??Ε = (Γ’„Β€, +), Δ??ΒΈ = 2Γ’„Β€. Consider two neutrosophic soft groups Δ?‘ 1 and Δ?‘ 2 over Δ??Ε as following. For Δ?‘Δ½, Δ?‘› Γ’ˆˆ Γ’„Β€, 1 if Δ?‘Δ½ = 4kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€ Δ?‘‡Δ?‘“Δ?‘ 1 (2Δ?‘›) (Δ?‘Δ½) = {2 0 others 0 if Δ?‘Δ½ = 4kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€ Δ??ΕΊΔ?‘“Δ?‘ 1 (2Δ?‘›) (Δ?‘Δ½) = { 1 others 4
0 if Δ?‘Δ½ = 4kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€ Δ??Ε‘Δ?‘“Δ?‘ 1 (2Δ?‘›) (Δ?‘Δ½) = { 1 others 10 and 2 if Δ?‘Δ½ = 6kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€ Δ?‘‡Δ?‘“2(2Δ?‘›) (Δ?‘Δ½) = {3 0 others 0 if Δ?‘Δ½ = 6kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€ Δ??ΕΊΔ?‘“Δ?‘ 2 (2Δ?‘›) (Δ?‘Δ½) = { 1 others
Then, it can be easily verified that Δ?‘ 2 Γ’Š† Δ?‘ 1 and Δ?‘ 1 Γ’‹ƒΔ?‘ 2 is a neutrosophic soft group over Δ??Ε. 3.4 Definition: 1. Let Δ?‘ 1 and Δ?‘ 2 be two NSSs over the common universe (Δ?‘ˆ, Δ??ΒΈ). Then their Γ’€˜Δ??Β΄Δ?‘ Δ??ΛΓ’€Λ operation is denoted by Δ?‘ 1 Γ’‹€ Δ?‘ 2 = Δ?‘ 3 and is defined as: Δ?‘ 3 = {[(Δ?‘Ž, Δ?‘?), {< Δ?‘Δ½, Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) >Γ’ˆΕ Δ?‘Δ½ Γ’ˆˆ Δ?‘ˆ} ]: (Δ?‘Ž, Δ?‘?) Γ’ˆˆ Δ??ΒΈΔ—Δ??ΒΈ} where Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½); 2. Let Δ?‘ 1 and Δ?‘ 2 be two NSSs over the common universe (Δ?‘ˆ, Δ??ΒΈ). Then their Γ’€˜Δ?‘‚Δ?‘…Γ’€Λ operation is denoted by Δ?‘ 1 Γ’‹ Δ?‘ 2 = Δ?‘ 4 and is defined as: Δ?‘ 4 = {[(Δ?‘Ž, Δ?‘?), {< Δ?‘Δ½, Δ?‘‡Δ?‘“Δ?‘ 4 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 4 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 4 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) >Γ’ˆΕ Δ?‘Δ½ Γ’ˆˆ Δ?‘ˆ} ]: (Δ?‘Ž, Δ?‘?) Γ’ˆˆ Δ??ΒΈΔ—Δ??ΒΈ} where Δ?‘‡Δ?‘“Δ?‘ 4 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 4 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 4 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½);
5
1
if Δ?‘Δ½ = 6kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€ Δ??Ε‘Δ?‘“Δ?‘ 2 (2Δ?‘›) (Δ?‘Δ½) = {6 1 others Corresponding t-norm (Γ’ˆ—) and s-norm (Γ’‹„) are defined as Δ?‘Ž Γ’ˆ— Δ?‘? = Δ?‘šΔ?‘–Δ?‘› (Δ?‘Ž, Δ?‘?), Δ?‘Ž Γ’‹„ Δ?‘? = Δ?‘šΔ?‘ŽΔ?‘Δ½ (Δ?‘Ž, Δ?‘?). Let Δ?‘ 1 Γ’‹ƒ Δ?‘ 2 = Δ?‘ 3 ; Then for Δ?‘› = 3, Δ?‘Δ½ = 12, Δ?‘Ε = 18 we have, Δ?‘‡Δ?‘“Δ?‘ 3 (6) (12 Γ’ˆ’ 18) = Δ?‘‡Δ?‘“Δ?‘ 1 (6) ( Γ’ˆ’6) Γ’‹„ Δ?‘‡Δ?‘“Δ?‘ 2 (6) ( Γ’ˆ’6) = max (0, 0) = 0 and Δ?‘‡Δ?‘“Δ?‘ 3 (6) (12) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 3 (6) (18) = { Δ?‘‡Δ?‘“Δ?‘ 1 (6) (12) Γ’‹„ Δ?‘‡Δ?‘“Δ?‘ 2 (6) (12)} Γ’ˆ— {Δ?‘‡Δ?‘“Δ?‘ 1 (6) (18) Γ’‹„ Δ?‘‡Δ?‘“Δ?‘ 2 (6) (18) } 1 2 = Δ?‘šΔ?‘–Δ?‘› {Δ?‘šΔ?‘ŽΔ?‘Δ½ ( , 0) , Δ?‘šΔ?‘ŽΔ?‘Δ½ (0, )} 2 3 1 2 1 = Δ?‘šΔ?‘–Δ?‘› ( , ) = 2 3 2 Hence, Δ?‘‡Δ?‘“Δ?‘ (6) (12 Γ’ˆ’ 18) < Δ?‘‡Δ?‘“Δ?‘ (6) (12) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (6) (18); 3
3
3
i,e Δ?‘ 1 Γ’‹ƒ Δ?‘ 2 is not a neutrosophic soft group, here. Now, if we define Δ?‘ 2 over Δ??Ε as following: 1
Δ?‘‡Δ?‘“Δ?‘ 2 (2Δ?‘›) (Δ?‘Δ½) = { Δ??ΕΊΔ?‘“Δ?‘ 2 (2Δ?‘›) (Δ?‘Δ½) = {
8
if Δ?‘Δ½ = 8kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€
0
others
0
if Δ?‘Δ½ = 8kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€ others
2 5 1
Δ??Ε‘Δ?‘“Δ?‘ 2 (2Δ?‘›) (Δ?‘Δ½) = {61 2
if Δ?‘Δ½ = 8kn, Γ’ˆƒk Γ’ˆˆ Γ’„Β€ others
3.5 Theorem: Let Δ?‘ 1 and Δ?‘ 2 be two neutrosophic soft groups over the group (Δ??Ε,Γ’ˆ˜). Then, Δ?‘ 1 Γ’‹€ Δ?‘ 2 is also a neutrosophic soft group over (Δ??Ε,Γ’ˆ˜). Proof: Let Δ?‘ 1 Γ’‹€ Δ?‘ 2 = Δ?‘ 3 . Then for Δ?‘Δ½, Δ?‘Ε Γ’ˆˆ Δ??Ε and (Δ?‘Ž, Δ?‘?) Γ’ˆˆ Δ??ΒΈΔ— Δ??ΒΈ, Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½ Γ’ˆ˜ Δ?‘Ε) Γ’‰Δ½ [ Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Ε)] Γ’ˆ— [ Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε)] = [ Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Ε)] Γ’ˆ— [ Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½)] (as Γ’ˆ— is commutative) (Δ?‘Δ½) (Δ?‘Ε) (Δ?‘Ε)] = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) Γ’ˆ— [ Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½) (as Γ’ˆ— is associative) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Ε) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Ε) (as Γ’ˆ— is commutative) = Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Ε) Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½ Γ’ˆ’1 ) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½ Γ’ˆ’1 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½ Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½)
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Similarly, πΌππ3 (π,π) (π₯ β π¦) β€ πΌππ3 (π,π) (π₯) β πΌππ3 (π,π) (π¦), πΌππ3 (π,π) (π₯ β1 ) β€ πΌππ3 (π,π) (π₯) ; πΉππ3 (π,π) (π₯ β π¦) β€ πΉππ3 (π,π) (π₯) β πΉππ3 (π,π) (π¦), πΉππ3 (π,π) (π₯ β1 ) β€ πΉππ3 (π,π) (π₯) ; This completes the proof. The theorem is true for a family of neutrosophic soft groups over a group. 3.6 Definition: Let π be a mapping from a set π to a set π. If π and π are two neutrosophic soft sets over π and π, respectively, then the image of π under π is defined as a neutrosophic soft set π(π) = {[π, ππ(π) (π)]: π β πΈ} over π, where πππ(π) (π) (π¦) = πππ (π) [πβ1 (π¦)], πΌππ(π)(π) (π¦) = πΌππ (π) [πβ1 (π¦)], πΉππ(π) (π) (π¦) = πΉππ (π) [πβ1 (π¦)], βπ¦ β π. The pre-image of π under π is defined as a neutrosophic soft set given by: πβ1 (π) = {[π, ππβ1(π) (π)]: π β πΈ} over π, where ππ β1 (π) (π₯) = πππ(π) [π(π₯)], πΌπ β1 (π) (π₯) = π
(π)
π
πΌππ(π) [π(π₯)], πΉπ
(π) (π₯) πβ1 (π)
(π)
= πΉππ(π) [π(π₯)],
βπ₯ β π.
Similarly, πΉππ(π) (π) (π¦1 β π¦2 ) β€ πΉππ(π) (π) (π¦1 ) β πΉππ(π) (π) (π¦2 ) Next, πππ(π) (π) (π¦1β1 ) = πππ (π) [πβ1 (π¦1β1 )] = πππ (π) [(πβ1 (π¦1 ))β1 ] = πππ (π) (π₯1β1 ) β₯ πππ (π) (π₯1 ) = πππ (π) [πβ1 (π¦1 )] = πππ(π) (π) (π¦1 ) i,e πππ(π) (π) (π¦1β1 ) β₯ πππ(π) (π) (π¦1 ); πΌππ(π) (π) (π¦1β1 ) = πΌππ (π) [πβ1 (π¦1β1 )] = πΌππ (π) [(πβ1 (π¦1 ))β1 ] = πΌππ (π) (π₯1β1 ) β€ πΌππ (π) (π₯1 ) = πΌππ (π) [πβ1 (π¦1 )] = πΌππ(π) (π) (π¦1 ) i,e πΌππ(π) (π) (π¦1β1 ) β€ πΌππ(π) (π) (π¦1 ); Similarly, πΉππ(π) (π) (π¦1β1 ) β€ πΉππ(π) (π) (π¦1 ); This proves the theorem. 3.8 Theorem: Let π βΆ π β π be an homomorphism in classical sense. If π is a neutrosophic soft group over π, then πβ1 (π) is a neutrosophic soft group over π. [Note that πβ1 (π) is the inverse image of π under the mapping π. Here πβ1 may not be a mapping.] Proof:
3.7 Theorem: Let π βΆ π β π be an isomorphism in classical sense. If π is a neutrosophic soft group over π then π(π) is a neutrosophic soft group over π. Proof: Let π₯1 , π₯2 β π; π¦1 , π¦2 β π; such that π¦1 = π(π₯1 ), π¦2 = π(π₯2 ). Now, πππ(π) (π) (π¦1 β π¦2 ) = πππ (π) [πβ1 (π¦1 β π¦2 )] = πππ (π) [πβ1 (π¦1 ) β πβ1 (π¦2 )] (as πβ1 is homomorphism) = πππ (π) (π₯1 β π₯2 ) β₯ πππ (π) (π₯1 ) β πππ (π) (π₯2 ) = πππ (π) [πβ1 (π¦1 )] β πππ (π) [πβ1 (π¦2 )] = πππ(π) (π) (π¦1 ) β πππ(π) (π) (π¦2 ) Next, πΌππ(π) (π) (π¦1 β π¦2 ) = πΌππ (π) [πβ1 (π¦1 β π¦2 )] = πΌππ (π) [πβ1 (π¦1 ) β πβ1 (π¦2 )] (as πβ1 is homomorphism) = πΌππ (π) (π₯1 β π₯2 ) β€ πΌππ (π) (π₯1 ) β πΌππ (π) (π₯2 ) = πΌππ (π) [πβ1 (π¦1 )] β πΌππ (π) [πβ1 (π¦2 )] = πΌππ(π) (π) (π¦1 ) β πΌππ(π) (π) (π¦2 )
Let π₯1 , π₯2 β π; π¦1 , π¦2 β π; such that π¦1 = π(π₯1 ), π¦2 = π(π₯2 ). Now, ππ β1 (π) (π₯1 β π₯2 ) (π)
π
= πππ(π) [π(π₯1 β π₯2 )] = πππ(π) [π(π₯1 ) β π(π₯2 )] (as π is homomorphism) = πππ(π) (π¦1 β π¦2 ) β₯ πππ(π) (π¦1 ) β πππ(π) (π¦2 ) = πππ(π) [π(π₯1 )] β πππ(π) [π(π₯2 )] = ππ β1 (π) (π₯1 ) β ππ β1 (π) (π₯2 ) (π)
π
π
Next, πΌπ
(π) πβ1 (π)
(π)
(π₯1 β π₯2 )
= πΌππ(π) [π(π₯1 β π₯2 )] = πΌππ(π) [π(π₯1 ) β π(π₯2 )] (as π is homomorphism) = πΌππ(π) (π¦1 β π¦2 ) β€ πΌππ(π) (π¦1 ) β πΌππ(π) (π¦2 ) = πΌππ(π) [π(π₯1 )] β πΌππ(π) [π(π₯2 )] = πΌπ β1 (π) (π₯1 ) β πΌπ β1 (π) (π₯2 ) π
(π)
Similarly, πΉπ
π
(π) πβ1 (π)
β€ πΉπ Next, ππ
(π) πβ1 (π)
(π) πβ1 (π)
(π)
(π₯1 β π₯2 ) (π₯1 ) β πΉπ
(π) πβ1 (π)
(π₯2 )
(π₯1β1 ) = πππ(π) [π(π₯1β1 )]
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Neutrosophic Sets and Systems, Vol. 13, 2016
Γ’ˆ’1
= Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) [(Δ?‘”(Δ?‘Δ½1 )) ] = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε1Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Ε1 ) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) [Δ?‘”(Δ?‘Δ½1 )] = Δ?‘‡Δ?‘“ Γ’ˆ’1 (Δ?‘’) (Δ?‘Δ½1 ) i,e Δ?‘”
(Δ?‘ )
Δ?‘‡Δ?‘“
(Δ?‘’) Δ?‘”Γ’ˆ’1 (Δ?‘ )
Similarly,
(Δ?‘Δ½1Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“
(Δ?‘’) Δ?‘”Γ’ˆ’1 (Δ?‘ )
Δ??ΕΊΔ?‘“ Δ??Ε‘Δ?‘“
Δ?‘”Γ’ˆ’1 (Δ?‘ )
Γ’ˆ’1 (Δ?‘’) (Δ?‘Δ½1 )
(Δ?‘’) Δ?‘”Γ’ˆ’1 (Δ?‘ )
(Δ?‘Δ½1 ); (Δ?‘Δ½1 ),
(Δ?‘Δ½1Γ’ˆ’1 ) Γ’‰Β€ Δ??Ε‘Δ?‘“
(Δ?‘Δ½1 );
Hence, the theorem is proved.
4 Cartesian product of neutrosophic soft groups 4.1 Definition:
Γ’‰Β€ Δ??ΕΊΔ?‘“
(Δ?‘’) Δ?‘”Γ’ˆ’1 (Δ?‘ )
(Δ?‘’) Δ?‘”Γ’ˆ’1 (Δ?‘ )
Δ??Ε‘Δ?‘“Δ?œ™(Δ?‘ )(Δ?‘’) (Δ?‘Ε) = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) [Δ?œ™ Γ’ˆ’1 (Δ?‘Ε)] = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‚; This completes the 2nd part.
3.9 Definition: Let Δ?‘ be a neutrosophic soft group over the group Δ??Ε and Δ?œ†, Δ?œ‡, Δ?œ‚ Γ’ˆˆ (0,1] with Δ?œ† + Δ?œ‡ + Δ?œ‚ Γ’‰Β€ 3. Then, 1. Δ?‘ is called (Δ?œ†, Δ?œ‡, Δ?œ‚) -identity neutrosophic soft group over Δ??Ε if Γ’ˆ€Δ?‘’ Γ’ˆˆ Δ??ΒΈ, Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ†, Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‡, Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‚ ; for Δ?‘Δ½ = Δ?‘’Δ??Ε , the identity element of G. Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = 0, Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = 1; otherwise. 2. Δ?‘ is called (Δ?œ†, Δ?œ‡, Δ?œ‚) -absolute neutrosophic soft group over Δ??Ε if Γ’ˆ€Δ?‘Δ½ Γ’ˆˆ Δ??Ε, Δ?‘’ Γ’ˆˆ Δ??ΒΈ, Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ†, Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‡, Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‚. 3.10 Theorem: Let Δ?œ™ Γ’ˆΕ Δ?‘‹ Γ’†’ Δ?‘Œ be an isomorphism in classical sense. 1. If Δ?‘ is a neutrosophic soft group over Δ?‘‹, then Δ?œ™(Δ?‘ ) is a ( Δ?œ†, Δ?œ‡, Δ?œ‚ ) -identity neutrosophic soft group over Δ?‘Œ if Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ†, Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‡, Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‚; when Δ?‘Δ½ Γ’ˆˆ Δ??ΕΎΔ?‘’Δ?‘&#x;Δ?œ™. Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = 0, Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = 1; otherwise, Γ’ˆ€Δ?‘Δ½ Γ’ˆˆ Δ?‘‹, Δ?‘’ Γ’ˆˆ Δ??ΒΈ. 2. If Δ?‘ is a (Δ?œ†, Δ?œ‡, Δ?œ‚) -absolute neutrosophic soft group over Δ?‘‹, then Δ?œ™(Δ?‘ ) is also so over Δ?‘Œ. Proof: 1. Clearly, Δ?œ™(Δ?‘ ) is a neutrosophic soft group over Δ?‘Œ by theorem (3.7). Now, if Δ?‘Δ½ Γ’ˆˆ Δ?‘˜Δ?‘’Δ?‘&#x;Δ?œ™ then Δ?œ™(Δ?‘Δ½) = Δ?‘’Δ?‘Œ , the identity element of Δ?‘Œ. Then, Δ?‘‡Δ?‘“Δ?œ™(Δ?‘ )(Δ?‘’) (Δ?‘’Δ?‘Œ ) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) [Δ?œ™ Γ’ˆ’1 (Δ?‘’Δ?‘Œ )] = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ† Δ??ΕΊΔ?‘“Δ?œ™(Δ?‘ ) (Δ?‘’) (Δ?‘’Δ?‘Œ ) = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) [Δ?œ™ Γ’ˆ’1 (Δ?‘’Δ?‘Œ )] = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‡ Δ??Ε‘Δ?‘“Δ?œ™(Δ?‘ )(Δ?‘’) (Δ?‘’Δ?‘Œ ) = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) [Δ?œ™ Γ’ˆ’1 (Δ?‘’Δ?‘Œ )] = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‚ Similarly, Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = 0, Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = 1, Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = 1; if Δ?‘Δ½ otherwise. This ends the 1st part. 2. Let, Δ?‘Ε = Δ?œ™(Δ?‘Δ½) for Δ?‘Δ½ Γ’ˆˆ Δ?‘‹, Δ?‘Ε Γ’ˆˆ Δ?‘Œ. Then Γ’ˆ€Δ?‘’ Γ’ˆˆ Δ??ΒΈ, Δ?‘‡Δ?‘“Δ?œ™(Δ?‘ )(Δ?‘’) (Δ?‘Ε) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) [Δ?œ™ Γ’ˆ’1 (Δ?‘Ε)] = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ†, Δ??ΕΊΔ?‘“Δ?œ™(Δ?‘ ) (Δ?‘’) (Δ?‘Ε) = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) [Δ?œ™ Γ’ˆ’1 (Δ?‘Ε)] = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?œ‡,
Let Δ?‘ 1 and Δ?‘ 2 be two neutrosophic soft groups over the groups Δ?‘‹ and Δ?‘Œ, respectively. Then their cartesian product is Δ?‘ 1 Δ— Δ?‘ 2 = Δ?‘ 3 where Δ?‘“Δ?‘ 3 (Δ?‘Ž, Δ?‘?) = Δ?‘“Δ?‘ 1 (Δ?‘Ž)Δ—Δ?‘“Δ?‘ 2 (Δ?‘?) for (Δ?‘Ž, Δ?‘?) Γ’ˆˆ Δ??ΒΈ Δ— Δ??ΒΈ. Analytically, Δ?‘“Δ?‘ 3 (Δ?‘Ž, Δ?‘?) = {< (Δ?‘Δ½, Δ?‘Ε), Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½, Δ?‘Ε) Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½, Δ?‘Ε), Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½, Δ?‘Ε) >Γ’ˆΕ (Δ?‘Δ½, Δ?‘Ε) Γ’ˆˆ Δ?‘‹Δ—Δ?‘Œ}
where Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½, Δ?‘Ε) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε), Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½, Δ?‘Ε) = Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε), Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½, Δ?‘Ε) = Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε); This definition can be extended for more than two neutrosophic soft groups. 4.2 Theorem: Let Δ?‘ 1 and Δ?‘ 2 be two neutrosophic soft groups over the groups Δ?‘‹ and Δ?‘Œ , respectively. Then their cartesian product Δ?‘ 1 Δ— Δ?‘ 2 is also a neutrosophic soft group over Δ?‘‹ Δ— Δ?‘Œ. Proof: Let Δ?‘ 1 Δ— Δ?‘ 2 = Δ?‘ 3 where Δ?‘“Δ?‘ 3 (Δ?‘Ž, Δ?‘?) = Δ?‘“Δ?‘ 1 (Δ?‘Ž)Δ—Δ?‘“Δ?‘ 2 (Δ?‘?) for (Δ?‘Ž, Δ?‘?) Γ’ˆˆ Δ??ΒΈΔ— Δ??ΒΈ. Then for (Δ?‘Δ½1 , Δ?‘Ε1 ), (Δ?‘Δ½2 , Δ?‘Ε2 ) Γ’ˆˆ Δ?‘‹ Δ— Δ?‘Œ , Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) [(Δ?‘Δ½1 , Δ?‘Ε1 ) Γ’ˆ˜ (Δ?‘Δ½2 , Δ?‘Ε2 )] = Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1 Γ’ˆ˜ Δ?‘Δ½2 , Δ?‘Ε1 Γ’ˆ˜ Δ?‘Ε2 ) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½1 Γ’ˆ˜ Δ?‘Δ½2 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) ( Δ?‘Ε1 Γ’ˆ˜ Δ?‘Ε2 ) Γ’‰Δ½ [Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½1 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½2 )] Γ’ˆ— [Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) ( Δ?‘Ε1 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε2 )] = [Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½1 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε1 )] Γ’ˆ— [Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) ( Δ?‘Δ½2 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε2 )] = Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1 , Δ?‘Ε1 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½2 , Δ?‘Ε2 ) Next, Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) [(Δ?‘Δ½1 , Δ?‘Ε1 ) Γ’ˆ˜ (Δ?‘Δ½2 , Δ?‘Ε2 )] = Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1 Γ’ˆ˜ Δ?‘Δ½2 , Δ?‘Ε1 Γ’ˆ˜ Δ?‘Ε2 ) = Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½1 Γ’ˆ˜ Δ?‘Δ½2 ) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) ( Δ?‘Ε1 Γ’ˆ˜ Δ?‘Ε2 ) Γ’‰Β€ [Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½1 )Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½2 )] Γ’‹„ [Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) ( Δ?‘Ε1 ) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε2 )] = [Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½1 ) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε1 )] Γ’‹„ [Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) ( Δ?‘Δ½2 ) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε2 )] = Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1 , Δ?‘Ε1 ) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½2 , Δ?‘Ε2 ) Similarly, Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) [(Δ?‘Δ½1 , Δ?‘Ε1 ) Γ’ˆ˜ (Δ?‘Δ½2 , Δ?‘Ε2 )] Γ’‰Β€ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1 , Δ?‘Ε1 ) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½2 , Δ?‘Ε2 ). 3
Next,
Tuhin Bera and Nirmal Kumar Mahapatra, Introduction to Neutrosophic Soft Groups
3
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Neutrosophic Sets and Systems, Vol. 13, 2016
Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) [(Δ?‘Δ½1 , Δ?‘Ε1 )Γ’ˆ’1 ] = Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1Γ’ˆ’1 , Δ?‘Ε1Γ’ˆ’1 ) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½1Γ’ˆ’1 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε1Γ’ˆ’1 ) Γ’‰Δ½ Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½1 ) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Ε1 ) = Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1 , Δ?‘Ε1 ) Similarly, Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) [(Δ?‘Δ½1 , Δ?‘Ε1 )Γ’ˆ’1 ] Γ’‰Β€ Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1 , Δ?‘Ε1 ), Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) [(Δ?‘Δ½1 , Δ?‘Ε1 )Γ’ˆ’1 ] Γ’‰Β€ Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½1 , Δ?‘Ε1 ). Hence, the theorem is proved.
b
(0.75, 0.25, 0.52) (0.69, 0.31, 0.39)
(0.83, 0.11, 0.28) (0.58, 0.41, 0.66)
c
(0.67, 0.32, 0.29) (0.79, 0.19, 0.41)
(0.75, 0.21, 0.19) (0.71, 0.27, 0.53)
Table 4: Tabular form of neutrosophic soft group N.
Obviously, Δ?‘€ is the neutrosophic soft subgroup of Δ?‘ over (Δ?‘‰, Δ??ΒΈ). 5.2 Theorem:
Let Δ?‘ 1 and Δ?‘ 2 be two neutrosophic groups over the group Δ??Ε. Then Δ?‘ 1 is neutrosophic soft subgroup of Δ?‘ 2 if Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‰Β€ Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½), Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‰Δ½ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½), Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‰Δ½ Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½); Γ’ˆ€Δ?‘Δ½ Γ’ˆˆ Δ??Ε, Δ?‘’ Γ’ˆˆ Δ??ΒΈ.
Let Δ?‘ be a neutrosophic soft group over the group Δ??Ε and Δ?‘ 1 , Δ?‘ 2 be two neutrosophic soft subgroups of Δ?‘ . If Δ?‘‡, Δ??ΕΊ, Δ??Ε‘ of neutrosophic soft group Δ?‘ obey the disciplines of idempotent t-norm and idempotent s-norm, then, (i) Δ?‘ 1 Γ’‹‚ Δ?‘ 2 is a neutrosophic soft subgroup of Δ?‘ . (ii) Δ?‘ 1 Γ’‹€ Δ?‘ 2 is a neutrosophic soft subgroup of Δ?‘ Γ’‹€ Δ?‘ .
5.1.1 Example:
Proof:
We consider the Klein's -4 group Δ?‘‰ = { Δ?‘’, Δ?‘Ž, Δ?‘?, Δ?‘? } and Δ??ΒΈ = { Δ?›ΕΊ, Δ?›Λ, Δ?›ΕΎ, Δ?›ΕΌ } be a set of parameters. The two neutrosophic soft groups Δ?‘€, Δ?‘ defined over (Δ?‘‰, Δ??ΒΈ) are given by the following tables when corresponding t-norm and s-norm are defined as Δ?‘Ž Γ’ˆ— Δ?‘? = Δ?‘šΔ?‘ŽΔ?‘Δ½ (Δ?‘Ž + Δ?‘? Γ’ˆ’ 1, 0) and Δ?‘Ž Γ’‹„ Δ?‘? = Δ?‘šΔ?‘–Δ?‘› (Δ?‘Ž + Δ?‘?, 1).
The intersection(Γ’‹‚), AND (Γ’‹€) of two neutrosophic soft groups is also so by theorems (3.3) and (3.5). Now to complete this theorem, we only verify the criteria of neutrosophic soft subgroup in each case.
5 Neutrosophic soft subgroup 5.1 Definition:
(i) Let Δ?‘ 3 = Δ?‘ 1 Γ’‹‚ Δ?‘ 2 . For Δ?‘Δ½ Γ’ˆˆ Δ??Ε, Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½) Γ’‰Β€ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) ,
Δ?‘“Δ?‘€ (Δ?›ΕΊ) Δ?‘“Δ?‘€ (Δ?›ΕΎ) (0.65, 0.42, 0.54) (0.70, 0.31, 0.32)
Δ?‘“Δ?‘€ (Δ?›Λ) Δ?‘“Δ?‘€ (Δ?›ΕΌ) (0.68, 0.21, 0.76) (0.59, 0.38, 0.62)
Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½) Γ’‰Δ½ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) ,
a
(0.61, 0.44, 0.78) (0.67, 0.41, 0.39)
(0.62, 0.31, 0.79) (0.41, 0.49, 0.64)
Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘’) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½) Γ’‰Δ½ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘’) (Δ?‘Δ½) ;
b
(0.55, 0.55, 0.59) (0.60, 0.36, 0.48)
(0.59, 0.42, 0.80) (0.56, 0.43, 0.68)
c
(0.47, 0.49, 0.69) (0.48, 0.52, 0.54)
(0.67, 0.43, 0.84) (0.49, 0.50, 0.70)
e
Table 3: Tabular form of neutrosophic soft group M.
(ii) Let Δ?‘ 3 = Δ?‘ 1 Γ’‹€ Δ?‘ 2 and Δ?‘Δ½ Γ’ˆˆ Δ??Ε; Then, Δ?‘‡Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½) Γ’‰Β€ Δ?‘‡Δ?‘“Δ?‘ (Δ?‘Ž) (Δ?‘Δ½) Γ’ˆ— Δ?‘‡Δ?‘“Δ?‘ (Δ?‘?) (Δ?‘Δ½) = Δ?‘‡Δ?‘“Δ?‘ (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) , Δ??ΕΊΔ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½) Γ’‰Δ½ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘?) (Δ?‘Δ½) = Δ??ΕΊΔ?‘“Δ?‘ (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) ,
Δ?‘“Δ?‘ (Δ?›ΕΊ) Δ?‘“Δ?‘ (Δ?›ΕΎ)
Δ?‘“Δ?‘ (Δ?›Λ) Δ?‘“Δ?‘ (Δ?›ΕΌ)
Δ??Ε‘Δ?‘“Δ?‘ 3 (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ 1 (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ 2 (Δ?‘?) (Δ?‘Δ½) Γ’‰Δ½ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘Ž) (Δ?‘Δ½) Γ’‹„ Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘?) (Δ?‘Δ½) = Δ??Ε‘Δ?‘“Δ?‘ (Δ?‘Ž,Δ?‘?) (Δ?‘Δ½) ;
e
(0.65, 0.34, 0.14) (0.72, 0.21, 0.16)
(0.88, 0.12, 0.72) (0.69, 0.31, 0.32)
The theorems are also true for a family of neutrosophic soft subgroups of Δ?‘ .
a
(0.71, 0.22, 0.78) (0.84, 0.16, 0.25)
(0.71, 0.19, 0.44) (0.62, 0.32, 0.42)
5.3 Example: We consider the group (S,.), cube root of unity where S= {1, Δ?œ”, Δ?œ”2 } and let E= { Δ?›ΕΊ, Δ?›Λ, Δ?›ΕΎ } be a set of parameters.
Tuhin Bera and Nirmal Kumar Mahapatra, Introduction to Neutrosophic Soft Groups
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The t-norm and s-norm are defined as: Δ?‘Ž Γ’ˆ— Δ?‘? = Δ?‘ŽΔ?‘? and Δ?‘Ž Γ’‹„ Δ?‘? = Δ?‘Ž + Δ?‘? Γ’€“ Δ?‘ŽΔ?‘?. The neutrosophic soft group Δ?‘ and itΓ’€™s two subgroups Δ?‘ 1 , Δ?‘ 2 defined over (S,.) are given by the following tables. Δ?‘“Δ?‘ (Δ?›ΕΊ) (0.7,0.3,0.2) (0.6,0.5,0.6)
1 Δ?œ” Δ?œ”
2
Δ?‘“Δ?‘ (Δ?›Λ)
Δ?‘“Δ?‘ (Δ?›ΕΎ) (0.6,0.3,0.5)
1
(0.7,0.2,0.4) (0.5,0.5,0.7)
(0.7,0.3,0.5)
Δ?œ”
(0.6,0.3,0.3) (0.4,0.4,0.6)
(0.5,0.4,0.6)
Δ?œ”2
Table 5: Tabular form of neutrosophic soft group N.
Δ?‘“Δ?‘ 1 (Δ?›ΕΊ) (0.4,0.4,0.9) (0.5,0.6,0.6)
1 Δ?œ” Δ?œ”2
Δ?‘“Δ?‘ 1 (Δ?›Λ)
Δ?‘“Δ?‘ 1 (Δ?›ΕΎ) (0.6,0.6,0.6)
(0.6,0.4,0.7) (0.4,0.5,0.7)
(0.5,0.8,0.5)
(0.3,0.5,0.8) (0.4,0.8,0.7)
(0.5,0.6,0.7)
1 Δ?œ” Δ?œ”2
Δ?‘“Δ?‘ 2 (Δ?›Λ)
Δ?‘“Δ?‘ 2 (Δ?›ΕΎ) (0.6,0.4,0.6)
(0.7,0.3,0.4) (0.4,0.5,0.8)
(0.6,0.4,0.5)
(0.6,0.4,0.3) (0.3,0.6,0.7)
(0.5,0.5,0.7)
Table 7: Tabular form of neutrosophic soft subgroup N2.
1 Δ?œ” Δ?œ”2
Δ?‘“Δ?‘€ (Δ?›ΕΊ) (0.24,0.70,0.92) (0.25,0.80,0.88)
Δ?‘“Δ?‘€ (Δ?›Λ)
Δ?‘“Δ?‘€ (Δ?›ΕΎ) (0.36,0.76,0.84)
(0.42,0.58,0.82) (0.16,0.75,0.94)
(0.30,0.88,0.75)
(0.18,0.70,0.86) (0.12,0.92,0.91)
(0.25,0.80,0.91)
Table 8: Tabular form of neutrosophic soft subgroup Δ?‘€ = Δ?‘ 1 Γ’‹‚Δ?‘ 2 .
Δ?‘“Δ?‘ƒ (Δ?›Λ, Δ?›ΕΊ) Δ?‘“Δ?‘ƒ (Δ?›Λ, Δ?›Λ) Δ?‘“Δ?‘ƒ (Δ?›Λ, Δ?›ΕΎ)
Δ?‘“Δ?‘ƒ (Δ?›ΕΎ, Δ?›ΕΊ) Δ?‘“Δ?‘ƒ (Δ?›ΕΎ, Δ?›Λ) Δ?‘“Δ?‘ƒ (Δ?›ΕΎ, Δ?›ΕΎ) (0.36,0.80,0.68) (0.36,0.76,0.84) (0.30,0.80,0.88)
(0.42,0.58,0.82) (0.28,0.65,0.82) (0.36,0.64,0.85) (0.24,0.70,0.85) (0.24,0.70,0.94) (0.16,0.75,0.94)
(0.35,0.86,0.70)
(0.18,0.70,0.86) (0.24,0.88,0.79) (0.15,0.75,0.94) (0.20,0.90,0.91) (0.09,0,80,0.94) (0.12,0.92,0.91)
(0.30,0.76,0.79)
(0.30,0.88,0.75) (0.20,0.90,0.90)
(0.25,0.80,0.91) (0.15,0.84,0.91)
Table 9: Tabular form of neutrosophic soft subgroup Δ?‘ƒ = Δ?‘ 1 Γ’‹€Δ?‘ 2 .
Table 6: Tabular form of neutrosophic soft subgroup N1.
Δ?‘“Δ?‘ 2 (Δ?›ΕΊ) (0.6,0.5,0.2) (0.5,0.5,0.7)
Δ?‘“Δ?‘ƒ (Δ?›ΕΊ, Δ?›ΕΊ) Δ?‘“Δ?‘ƒ (Δ?›ΕΊ, Δ?›Λ) Δ?‘“Δ?‘ƒ (Δ?›ΕΊ, Δ?›ΕΎ) (0.24,0.70,0.92) (0.30,0.80,0.68) (0.24,0.64,0.96) (0.30,0.76,0.84) (0.20,0.70,0.97) (0.25,0.80,0.88)
1 Δ?œ”
Δ?œ”2
Δ?‘“Δ?‘ƒ (Δ?›ΕΊ, Δ?›ΕΊ) Δ?‘“Δ?‘ƒ (Δ?›ΕΊ, Δ?›Λ) Δ?‘“Δ?‘ƒ (Δ?›ΕΊ, Δ?›ΕΎ) (0.49,0.51,0.36) (0.42,0.65,0.68) (0.42,0.51,0.60) (0.36,0.65,0.80) (0.42,0.65,0.68) (0.36,0.75,0.84)
Δ?‘“Δ?‘ƒ (Δ?›Λ, Δ?›ΕΊ) Δ?‘“Δ?‘ƒ (Δ?›Λ, Δ?›Λ) Δ?‘“Δ?‘ƒ (Δ?›Λ, Δ?›ΕΎ)
Δ?‘“Δ?‘ƒ (Δ?›ΕΎ, Δ?›ΕΊ) Δ?‘“Δ?‘ƒ (Δ?›ΕΎ, Δ?›Λ) Δ?‘“Δ?‘ƒ (Δ?›ΕΎ, Δ?›ΕΎ) (0.42,0.51,0.60) (0.36,0.51,0.75) (0.36,0.65,0.80)
(0.49,0.36,0.64) (0.35,0.60,0.82) (0.49,0.44,0.70) (0.35,0.65,0.85) (0.35,0.60,0.82) (0.25,0.75,0.91)
(0.49,0.44,0.70)
(0.36,0.51,0.51) (0.24,0.58,0.72) (0.30,0.58,0.72) (0.20,0.64,0.84) (0.24,0.58,0.72) (0.16,0.64,0.84)
(0.30,0.58,0.72)
(0.49,0.51,0.75) (0.35,0.65,0.85)
(0.25,0.64,0.84) (0.20,0.64,0.84)
Table 10: Tabular form of neutrosophic soft subgroup Δ?‘ƒ = Δ?‘ Γ’‹€Δ?‘ .
Tuhin Bera and Nirmal Kumar Mahapatra, Introduction to Neutrosophic Soft Groups
Neutrosophic Sets and Systems, Vol. 13, 2016
Tables 5 & 8 show the 1st result and Tables 9 & 10 show the 2nd result in theorem (5.2). 5.4 Theorem: Let Δ?‘ 1 and Δ?‘ 2 be two neutrosophic soft groups over the group Δ?‘‹ such that Δ?‘ 1 is the neutrosophic soft subgroup of Δ?‘ 2 . Let Δ?‘” Γ’ˆΕ Δ?‘‹ Γ’†’ Δ?‘Œ be an isomorphism in classical sense. Then Δ?‘”(Δ?‘ 1 ) and Δ?‘”(Δ?‘ 2 ) are two neutrosophic soft groups over Δ?‘Œ. Moreover Δ?‘”(Δ?‘ 1 ) is the neutrosophic soft subgroup of Δ?‘”(Δ?‘ 2 ). Proof: The 1st part is already proved in theorem (3.7). Let Δ?‘Δ½ Γ’ˆˆ Δ?‘‹, Δ?‘Ε Γ’ˆˆ Δ?‘Œ so that Δ?‘Ε = Δ?‘”(Δ?‘Δ½). Then, Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘’) (Δ?‘Δ½) Γ’‰Β€ Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘’) (Δ?‘Δ½) Γ’&#x;Ε‘ Δ?‘‡Δ?‘“Δ?‘ 1 (Δ?‘’) [Δ?‘”Γ’ˆ’1 (Δ?‘Ε)] Γ’‰Β€ Δ?‘‡Δ?‘“Δ?‘ 2 (Δ?‘’) [Δ?‘”Γ’ˆ’1 (Δ?‘Ε)] Γ’&#x;Ε‘ Δ?‘‡Δ?‘“Δ?‘”(Δ?‘ ) (Δ?‘’) (Δ?‘Ε) Γ’‰Β€ Δ?‘‡Δ?‘“Δ?‘”(Δ?‘ ) (Δ?‘’) (Δ?‘Ε) 1
2
Similarly, Δ??ΕΊΔ?‘“Δ?‘”(Δ?‘ ) (Δ?‘’) (Δ?‘Ε) Γ’‰Δ½ Δ??ΕΊΔ?‘“Δ?‘”(Δ?‘ ) (Δ?‘’) (Δ?‘Ε) , 1 2 (Δ?‘Ε) Δ??Ε‘Δ?‘“Δ?‘”(Δ?‘ ) (Δ?‘’) Γ’‰Δ½ Δ??Ε‘Δ?‘“Δ?‘”(Δ?‘ ) (Δ?‘’) (Δ?‘Ε); 1 2 Hence, the theorem is proved. Conclusion In the present paper, the theoretical point of view of neutrosophic soft group has been discussed with suitable examples. Here, we also have defined the Cartesian product on neutrosophic soft groups and neutrosophic soft subgroup. Some theorems have been established. We extended the concept of group in NSS theory context. This concept will bring a new opportunity in research and development of NSS theory. References [1] F. Smarandache, Neutrosophic set, a generalisation of the intuitionistic fuzzy sets, Inter.J.Pure Appl.Math., 24, 287-297, (2005). [2] L. A. Zadeh, Fuzzy sets, Information and control, 8, 338-353, (1965). [3] K. Atanassov, Intuitionistic fuzzy sets, Fuzzy sets and systems, 20(1), 87-96,(1986). [4] D. Molodtsov, Soft set theory- First results, Computer and Mathematics with Applications, 37(4-5), 19-31, (1999). [5] A. Rosenfeld, Fuzzy groups, Journal of mathematical analysis and applications, 35, 512-517, (1971). [6] P. K. Sharma, Intuitionistic fuzzy groups, IFRSA International journal of data warehousing and mining, 1(1), 86-94, (2011). [7] H. Aktas and N. Cagman, Soft sets and soft groups, Information sciences,177, 2726-2735, (2007).
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Received: November 25, 2016. Accepted: December 15, 2016
Tuhin Bera and Nirmal Kumar Mahapatra, Introduction to Neutrosophic Soft Groups