More on “ n -valued refined neutrosophic soft set theory”

1. Introduction and Preliminaries

The original concept of soft set was firstly introduced by Molodtsov [1]. His pioneer paper has undergone tremendous growth and applications in the last few years. Maji et al. [2] defined some operations on soft sets and showed that the distributive laws of soft sets are varied. Ali et al. [3] pointed out that the distributive laws of soft sets are not true in general. Later, Maji [4] introduced the concept of neutrosophic soft set and presented some definitions and operations on neutrosophic soft set. In 2013, Smarandache [5] defined the symbolic and numerical refined neutrosophic logic and set. Recently, Alkhazaleh [6] presented the concept of time-neutrosophic soft set as a generalization of neutrosophic soft set and he studied some of its properties and gave hypothetical application of this concept in decision making problems. Khalil et al. [7] corrected some errors in [6] and introduced some new notions describing ‘subset’ and ‘equal’ of time-neutrosophic soft sets, and gave some examples and related propositions. As a generalization of neutrosophic soft set Alkhazaleh [8] proposed the notion of the n-valued refined neutrosophic soft set (n-VRNSs for short) and defined some operations (namely subset, complement, union, intersection, AND and OR operations) on n-valued refined neutrosophic soft sets. Since quite a few readers (at least the readers we know) care about the notion of neutrosophic set and the like, it is necessary to present the theory in a mathematical (at least errorless) way. In this paper, we first show, by a counterexample, that assertions (3) and (4) of Proposition 3.6 proposed by Alkhazaleh [8] are incorrect. Then we introduce two new notions describing ‘subset’ of n-VRNSs and ‘equal’ of n-VRNSs, and give some examples and related propositions. Based on these results we remedy the flaw of the assertions and improve the work of Alkhazaleh.

First, we present the key notion neutrosophic set (which was defined for the finite set in the references) in a clear, rigorous, and non-burdensome manner (this will also be beneficial to the subsequent citation).

Definition 1.1. An infinitesimal [10] is a hyper-real number θ which satisfies | θ | < 1 n for all positive integers n. Let Θ be the set of all infinitesimals, and ^*  [0, 1] = {θ ∣ θ ∈ Θ, θ > 0} ∪ {a + θ ∣ a ∈ (0, 1) , θ ∈ Θ} ∪ {1 + θ ∣ θ ∈ Θ, θ < 0}. The set of all mappings (called also neutrosophic sets on U [9] if U is a finite set) from U to ^*  [0, 1] ³ is denoted by (^*  [0, 1] ³)  ^U . Customarily, for each A ∈ ^*  [0, 1] ^3U and each x ∈ U, p₁ ∘ A (x) (written as also T _A  (x)) is called truth-membership, p₂ ∘ A (x) (written as also I _A  (x)) is called indeterminacy membership, and p₃ ∘ A (x) (written as also F _A  (x)) is called falsity-membership of x, where p _j is j - th projection (j = 1, 2, 3). Therefore, one can also write A = {〈x ; T _A  (x) , I _A  (x) , F _A  (x) 〉 ; x ∈ U} , or (if U = {x, y, ⋯} is countable) A = { x 〈 T A ( x ) , I A ( x ) , F A ( x ) 〉 , y 〈 T A ( y ) , I A ( y ) , F A ( y ) 〉 , ⋯ } .

The following notions are also generalizations of their old forms (which were defined just for the finite set case).

Definition 1.2. (cf. [1]) A mapping F : E ⟶ P (U) (the set of all subsets of U) is also called a soft set over the initial universe set U indexed by the parameter set E (this soft set is customarily written as (F, A)).

Definition 1.3. (cf. [4]) A mapping F : E ⟶   (^*  [0, 1] ³)  ^U is also called a neutrosophic soft set (NSS for short) over the initial universe set U indexed by the parameter set E (this soft set is customarily written also (F, A)).

Definition 1.4. (cf. [8]) A mapping F _n  : E ⟶   (^*  [0, 1] ³)  ^Un is also called n-valued refined neutrosophic soft set (n-VRNSs for short) over the initial universe set U indexed by the parameter set E (this soft set is customarily written also (F _n , A)). The class of all n-VRNSs over a fixed initial universe set U indexed by a fixed parameter set E is denoted by n- V ℝ ℕ S ( U , E ) .

Definition 1.5. (cf. [8]) Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. (F _n , A) is said to be n-valued refined neutrosophic soft subset of (G _n , B) or (G _n , B) is said to be n-valued refined neutrosophic soft super set of (F _n , A) (write as (F _n , A) ⊆ (G _n , B) or (G _n , B) ⊇ (F _n , A)) if A ⊆ B, and T F j ( e ) ( x ) ≤ T G j ( e ) ( x ) , j ∈ { 1 , 2 , . . . , p } ; I F k ( e ) ( x ) ≤ I G k ( e ) ( x ) , k ∈ { 1 , 2 , . . . , r } ; and F F l ( e ) ( x ) ≥ F G l ( e ) ( x ) , l ∈ { 1 , 2 , . . . , s } hold for each (e, x) ∈ A × U.

Definition 1.6. (cf. [8]) Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. Then the union of (F _n , A) and (G _n , B) (denoted by (F _n , A) ∪ (G _n , B)), is a n-VRNSs (H, C) defined by C = A ∪ B and T H j ( e ) ( m ) = { T F j ( e ) ( m ) , e ∈ A - B , T G j ( e ) ( m ) , e ∈ B - A , max { T F j ( e ) ( m ) , T G j ( e ) ( m ) } , e ∈ A ∩ B , I H k ( e ) ( m ) = { I F k ( e ) ( m ) , e ∈ A - B , I G k ( e ) ( m ) , e ∈ B - A , 1 2 [ I F k ( e ) ( m ) + I G k ( e ) ( m ) ] , e ∈ A ∩ B , F H l ( e ) ( m ) = { F F l ( e ) ( m ) , e ∈ A - B , F G l ( e ) ( m ) , e ∈ B - A , min { F F l ( e ) ( m ) , F G l ( e ) ( m ) } , e ∈ A ∩ B , where j ∈ {1, 2,. . . , p}, k ∈ {1, 2,. . . , r}, and l∈ {1, 2,. . . , s}.

Definition 1.7. (cf. [8]) Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. Then the intersection of (F _n , A) and (G _n , B) (denoted by (F _n , A) ∩ (G _n , B)), is a n-VRNSs (H, C) defined by (e ∈ C = A ∩ B) T H j ( e ) ( m ) = min { T F j ( e ) ( m ) , T G j ( e ) ( m ) } ( ∀ j ∈ { 1 , 2 , . . . , p } ) , I H k ( e ) ( m ) = 1 2 [ I F k ( e ) ( m ) + I G k ( e ) ( m ) ] ( ∀ k ∈ { 1 , 2 , . . . , r } ) , F H l ( e ) ( m ) = max { F F l ( e ) ( m ) , F G l ( e ) ( m ) } ( ∀ l ∈ { 1 , 2 , . . . , s } ) .

Definition 1.8. (cf. [8]) Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. (F _n , A) AND (G _n , B) is a n-VRNSs (F _n , A × B) (denoted also by (F _n , A) ∧ (G _n , B)) over U defined by (∀ (α, β) ∈ A × B) T H j ( α , β ) ( m ) = min { T F j ( α ) ( m ) , T G j ( β ) ( m ) } ( ∀ j ∈ { 1 , 2 , . . . , p } ) I H k ( α , β ) ( m ) = 1 2 [ I F k ( α ) ( m ) + I G k ( β ) ( m ) ] ( ∀ k ∈ { 1 , 2 , . . . , r } ) F H l ( α , β ) ( m ) = max { F F l ( α ) ( m ) , F G l ( β ) ( m ) } ( ∀ l ∈ { 1 , 2 , . . . , s } ) .

Definition 1.9. (cf. [8]) Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. (F _n , A) OR (G _n , B) is a n-VRNSs (F _n , A × B) (denoted also by (F _n , A) ∨ (G _n , B)) over U defined by (∀ (α, β) ∈ A × B) T H j ( α , β ) ( m ) = max { T F j ( α ) ( m ) , T G j ( β ) ( m ) } ( ∀ j ∈ { 1 , 2 , . . . , p } ) I H k ( α , β ) ( m ) = 1 2 [ I F k ( α ) ( m ) + I G k ( β ) ( m ) ] ( ∀ k ∈ { 1 , 2 , . . . , r } ) F H l ( α , β ) ( m ) = min { F F l ( α ) ( m ) , F G l ( β ) ( m ) } ( ∀ l ∈ { 1 , 2 , . . . , s } ) .

2. Counterexample

Let us recall assertions (3) and (4) in [8, Proposition 3.6].

Proposition 2.1. ([8, Proposition 3.6, p.4318]) If (F _n , A), (G _n , B) and (H _n , C) are three n-VRNSs over the common universe U. Then the following equalities hold:

(3)   (F _n , A) ∨  [(G _n , B) ∧ (H _n , C) ] =

 [(F _n , A) ∨ (G _n , B) ] ∧  [(F _n , A) ∨ (H _n , C) ] ,

(4)   (F _n , A) ∧  [(G _n , B) ∨ (H _n , C) ] =

 [(F _n , A) ∧ (G _n , B) ] ∨  [(F _n , A) ∧ (H _n , C) ] .

However, the following example shows that neither assertion (3) nor assertion (4) of Proposition 2.1 is true.

Example 2.2. Let U = {u₁, u₂} be a set of universe and E = {e₁, e₂, e₃, e₄, e₅, e₆} a six-element set of parameters. Consider two-element subsets A = {e₁, e₂}, B = {e₃, e₄}, and C = {e₅} of E . Then, the following case of 4-VRNSs (F₄, A), (G₄, B) and (H₄, C) over U defined by F 4 ( e 1 ) = { u 1 〈 0.2 , ( 0.5 , 0.1 ) , 0.6 〉 , u 2 〈 0.1 , ( 0.4 , 0.2 ) , 0.8 〉 } , F 4 ( e 2 ) = { u 1 〈 0.3 , ( 0.4 , 0.3 ) , 0.7 〉 , u 2 〈 0.2 , ( 0.5 , 0.4 ) , 0.6 〉 } , G 4 ( e 3 ) = { u 1 〈 0.7 , ( 0.6 , 0.2 ) , 0.2 〉 , u 2 〈 0.1 , ( 0.8 , 0.1 ) , 0.9 〉 } , G 4 ( e 4 ) = { u 1 〈 0.5 , ( 0.2 , 0.4 ) , 0.7 〉 , u 2 〈 0.3 , ( 0.2 , 0.1 ) , 0.9 〉 } , H 4 ( e 5 ) = { u 1 〈 0.2 , ( 0.2 , 0.2 ) , 0.1 〉 , u 2 〈 0.9 , ( 0.4 , 0.4 ) , 0.1 〉 } . By Definitions 1.8 and 1.9, the parameter set A × (B × C) of the n-VRNSs (F _n , A) ∨  [(G _n , B) ∧ (H _n , C) ] is a four-element set, and the parameter set (A × B) × (A × C) of the n-VRNSs  [(F _n , A) ∨ (G _n , B) ] ∧  [(F _n , A) ∨ (H _n , C) ] is a eight-element set. Thus neither of (3) and (4) is true.

3. Main results

In the this section we first propose some new notions describing ‘subset’ of n-VRNSs and ‘equal’ of n-VRNSs, and give some examples and related propositions. Then we use these results to remedy the flaw in the two assertions above.

Definition 3.1. Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. (F _n , A) is said to be a strong n-valued refined neutrosophic soft subset (Sn-VRNS subset for short) of (G _n , B) or (G _n , B) is said to be a strong n-valued refined neutrosophic soft supset (Sn-VRNS supset for short) of (F _n , A) (denoted by (F _n , A)   ⊆  _s   (G _n , B) or (G _n , B)   ⊇  _s   (F _n , A)) if, for each α ∈ A, there exists a β ∈ B such that F _n  (α) = G _n  (β). (F _n , A) is said to be a n-valued refined neutrosophic soft subset (n-VRNS subset for short) of (G _n , B) or (G _n , B) is said to be a n-valued refined neutrosophic soft supset (n-VRNS supset for short) of (F _n , A) (denoted by ( F n , A ) ⊆ ˜ ( G n , B ) or ( G n , B ) ⊇ ˜ ( F n , A ) ) if, for each α ∈ A, there exists a β ∈ B such that {F _n  (α)∣ α ∈ A} = {G _n  (β) ∣ β ∈ B}.

Example 3.2. Consider the two n-VRNSs (F _n , A) and (G _n , B) over U = {u₁, u₂} defined below (where A = {e₁, e₃, e₆, e₉}, B = {e₂, e₄, e₅, e₇, e₈}, and n = 4): F 4 ( e 1 ) = G 4 ( e 4 ) = { u 1 〈 0.4 , ( 0.5 , 0.3 ) , 0.6 〉 , u 2 〈 0.8 , ( 0.1 , 0.4 ) , 0.4 〉 } , F 4 ( e 3 ) = G 4 ( e 2 ) = { u 1 〈 0.3 , ( 0.3 , 0.4 ) , 0.4 〉 , u 2 〈 0.9 , ( 0.1 , 0.6 ) , 0.2 〉 } , F 4 ( e 6 ) = G 4 ( e 5 ) = { u 1 〈 0.1 , ( 0.2 , 0.6 ) , 0.9 〉 , u 2 〈 0.1 , ( 0.5 , 0.2 ) , 0.7 〉 } , F 4 ( e 9 ) = G 4 ( e 7 ) = { u 1 〈 0.7 , ( 0.4 , 0.3 ) , 0.1 〉 , u 2 〈 0.1 , ( 0.5 , 0.2 ) , 0.7 〉 } , G 4 ( e 9 ) = { u 1 〈 0.7 , ( 0.4 , 0.1 ) , 0.1 〉 , u 2 〈 0.1 , ( 0.5 , 0.2 ) , 0.7 〉 } . Then (F _n , A)   ⊆  _s   (G _n , B).

The strong n-valued refined neutrosophic soft equal and n-valued refined neutrosophic soft equal are defined as follows:

Definition 3.3. Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. (F _n , A) is said to be a strong n-valued refined neutrosophic soft equal to (Sn-VRNS equal for short) of (G _n , B) (denoted by (F _n , A) =  _s  (G _n , B) if (F _n , A) ⊆  _s (G _n , B) and (G _n , B) ⊆  _s  (F _n , A) hold. (F _n , A) is said to be n-valued refined neutrosophic soft equal to (n-VRNS equal to for short) (G _n , B) (denoted by (F _n , A) = ˜ ( G n , B ) ) if ( F n , A ) ⊆ ˜ ( G n , B ) and (G _n , B) ⊆ ˜ ( F n , A ) hold.

Based on the Definitions 3.1 and 3.3, respectively, we propose the following proposition.

Proposition 3.4. Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. Then

(1) (F _n , A)   ⊆  _s  (G _n , B) implies (F _n , A) ⊆ ˜ ( G n , B ) and (F _n , A) =  _s  (G _n , B) implies (F _n , A) = ˜ ( G n , B ) .

(2) ⊆ _s , and ⊆ ˜ are generalized relations on the class n- V ℝ ℕ S ( U , E ) which are reflexive and transitive (but not antisymmetric). Moreover, if (F _n , A)   ⊆  _s   (G _n , B) (resp., ( F n , A ) ⊆ ˜ ( G n , B ) ) and (G _n , B)   ⊆  _s   (F _n , A) (resp., ( G n , B ) ⊆ ˜ (F _n , A)), then (G _n , B)   =  _s   (F _n , A) (resp., ( G n , B ) = ˜ ( F n , A ) ).

(3) =  _s , and = ˜ are generalized relations on the class n- V ℝ ℕ S ( U , E ) which are reflexive, symmetric, and transitive.

(4) (F _n , A)   =  _s   (G _n , B) if and only if the set {H _n  (α) ∣ α ∈ A} of sequences is equal to the set {G _n  (β) ∣ β ∈ B} of sequences.

(5) ( F n , A ) = ˜ ( G n , B ) if and only if the set {H _n  (α) ∣α ∈ A} of families is equal to the set {G _n  (β) ∣β ∈ B} of families.

Proof. We only show (4). Notice that (F _n , A)   ⊆  _s   (G _n , B) ⇔ {F _n  (α) ∣ α ∈ A} ⊆ {G _n  (β) ∣ β ∈ B}, and that (G _n , B) ⊆  _s  (F _n , A) ⇔ {F _n  (α) ∣ α ∈ A} ⊇ {G _n  (β) ∣ β ∈ B}. Thus we have (F _n , A) =  _s  (G _n , B) ⇔ (F _n , A) {⊆}  _s  (G _n , B) and (G _n , B) {⊆}  _s    (F _n , A) ⇔ {F _n  (α) ∣ α ∈ A} ⊆ {G _n  (β) ∣ β ∈ B} and {F _n  (α) ∣ α ∈ A} ⊇ {G _n  (β) ∣ β ∈ B} ⇔ {F _n  (α)∣ α ∈ A} = {G _n  (β) ∣ β ∈ B}.

The following Proposition 3.5 is the corrected version of assertions (3) and (4) of Proposition 2.1, originally written as Proposition 5.2 of Alkhazaleh [8].

Proposition 3.5. Let (F _n , A), (G _n , B) and (H _n , C) are three n-VRNSs over the common universe U. Then

(1) (F _n , A) ∨  [(G _n , B) ∧ (H _n , C) ] ⊆  _s  [(F _n , A) ∨ (G _n , B) ] ∧  [(F _n , A) ∨ (H _n , C) ]  ,

(2) (F _n , A) ∧  [(G _n , B) ∨ (H _n , C) ] ⊆  _s  [(F _n , A) ∧ (G _n , B) ] ∨  [(F _n , A) ∧ (H _n , C) ] .

Proof. We only show (1). Let (M _n , B × C) stand for (G _n , B) ∧ (H _n , C) , where M n ( β , γ ) = G n ( β ) ∩ H n ( γ ) ( ∀ ( β , γ ) ∈ B × C ) . Again let ( F n , A ) ∨ ( M n , B × C ) = ( L n , A × ( B × C ) ) , where L n ( α , ( β , γ ) ) = F n ( α ) ∪ M n ( β , γ ) = F n ( α ) ∪ [ G n ( β ) ∩ H n ( γ ) ] ( ∀ ( α , ( β , γ ) ) ∈ A × ( B × C ) ) . Similarly, let R n ( ( α , β ) , ( α , γ ) ) = [ F n ( α ) ∪ G n ( β ) ] ∩ [ F n ( α ) ∪ H n ( γ ) ] ( ∀ ( ( α , β ) , ( α , γ ) ) ∈ ( ( A × B ) × ( A × C ) ) ) . Hence for each (α, (β, γ)) ∈ A × (B × C) , there exists a ((α, β) , (α, γ)) ∈ (A × B) × (A × C) such that L n ( α , ( β , γ ) ) = F n ( α ) ∪ [ G n ( β ) ∩ H n ( γ ) ] = [ F n ( α ) ∪ G n ( β ) ] ∩ [ F n ( α ) ∪ H n ( γ ) ] = R n ( ( α , β ) , ( α , γ ) ) . By Definition 3.1, we obtain ( F n , A ) ∨ [ ( G n , B ) ∧ ( H n , C ) ] ⊆ s [ ( F n , A ) ∨ ( G n , B ) ] ∧ [ ( F n , A ) ∨ ( H n , C ) ] .

However, ‘equal’ of n-VRNSs in Proposition 3.5 is impossible in general as illustrated by the following Example 3.6 (we will put forward Propositions 3.7, 3.8 and 3.9 to change this situation).

Example 3.6. Consider n-VRNSs (F _n , A), (G _n , B) and (H _n , C) over U = {u₁, u₂} defined below (where A = {e₁, e₂}, B = {e₃}, C = {e₄}, and n = 4): F 4 ( e 1 ) = { u 1 〈 0.3 , ( 0.5 , 0.3 ) , 0.6 〉 , u 2 〈 0.8 , ( 0.4 , 0.1 ) , 0.2 〉 } , F 4 ( e 2 ) = { u 1 〈 0.1 , ( 0.5 , 0.2 ) , 0.7 〉 , u 2 〈 0.4 , ( 0.5 , 0.5 ) , 0.3 〉 } , G 4 ( e 3 ) = { u 1 〈 0.6 , ( 0.2 , 0.4 ) , 0.3 〉 , u 2 〈 0.5 , ( 0.7 , 0.3 ) , 0.3 〉 } , H 4 ( e 4 ) = { u 1 〈 0.4 , ( 0.3 , 0.2 ) , 0.6 〉 , u 2 〈 0.3 , ( 0.3 , 0.7 ) , 0.6 〉 } . Let M = A × (B × C), N = (A × B) × (A × C), (F _n , A) ∨ [(G _n , B) ∧ (H _n , C)] = (I _n , M), and [(F _n , A)  ∨  (G _n , B)]  ∧  [(F _n , A)  ∨ (H _n , C)]  = (J _n , N). Then by Definitions 1.8 and 1.9, we have I 4 ( e 1 , e 3 , e 4 ) = { u 1 〈 0.4 , ( 0.375 , 0.3 ) , 0.6 〉 , u 2 〈 0.8 , ( 0.45 , 0.3 ) , 0.2 〉 } , I 4 ( e 2 , e 3 , e 4 ) = { u 1 〈 0.4 , ( 0.375 , 0.25 ) , 0.6 〉 , u 2 〈 0.4 , ( 0.5 , 0.5 ) , 0.3 〉 } , J 4 ( e 1 , e 3 , e 1 , e 4 ) = { u 1 〈 0.4 , ( 0.375 , 0.3 ) , 0.6 〉 , u 2 〈 0.8 , ( 0.45 , 0.3 ) , 0.2 〉 } , J 4 ( e 1 , e 3 , e 2 , e 4 ) = { u 1 〈 0.4 , ( 0.375 , 0.3 ) , 0.6 〉 , u 2 〈 0.4 , ( 0.475 , 0.4 ) , 0.3 〉 } , J 4 ( e 2 , e 3 , e 1 , e 4 ) = { u 1 〈 0.4 , ( 0.375 , 0.275 ) , 0.6 〉 , u 2 〈 0.5 , ( 0.475 , 0.4 ) , 0.3 〉 } , J 4 ( e 2 , e 3 , e 2 , e 4 ) = { u 1 〈 0.4 , ( 0.375 , 0.25 ) , 0.6 〉 , u 2 〈 0.4 , ( 0.5 , 0.5 ) , 0.3 〉 } . Clearly, [ ( F n , A ) ∨ ( G n , B ) ] ∧ [ ( F n , A ) ∨ ( H n , C ) ] ⊆ s ( F n , A ) ∨ [ ( G n , B ) ∧ ( H n , C ) ] does not hold. Thus ( F n , A ) ∨ [ ( G n , B ) ∧ ( H n , C ) ] = s [ ( F n , A ) ∨ ( G n , B ) ] ∧ [ ( F n , A ) ∨ ( H n , C ) ] does not hold. Similarly, ( F n , A ) ∧ [ ( G n , B ) ∨ ( H n , C ) ] = s [ ( F n , A ) ∧ ( G n , B ) ] ∨ [ ( F n , A ) ∧ ( H n , C ) ] does not hold.

In order to give a deeper insight on associativity, intersection and union of ‘equal’ of n-VRNSs, we propose the following propositions 3.7, 3.8 and 3.9 along with their corresponding proofs.

Proposition 3.7. Let (F _n , A), (G _n , B) and (H _n , C) are three n-VRNSs over the common universe U. Then

(1) (F _n , A) ∧  [(G _n , B) ∨ (H _n , C) ] ⊆  _s  [(F _n , A) ∨ (G _n , B) ] ∧  [(F _n , A) ∨ (H _n , C) ] ,

(2) (F _n , A) ∨  [(G _n , B) ∧ (H _n , C) ] ⊆  _s  [(F _n , A) ∧ (G _n , B) ] ∨  [(F _n , A) ∧ (H _n , C) ] .

Proof. Similar to the proof of Proposition 3.5 (1) and (2) respectively.

Proposition 3.8. Let (F _n , A) and (G _n , B) be two n-VRNSs over the common universe U. Then

(1) (F _n , A) ∧ (G _n , B) =  _s  (G _n , B) ∧ (F _n , A) ,

(2) (F _n , A) ∨ (G _n , B) =  _s  (G _n , B) ∨ (F _n , A) .

Proof. We only prove the first assertion. Let (L _n , A × B) stand for (F _n , A) ∧ (G _n , B) , where L n ( α , β ) = F n ( α ) ∩ G n ( β ) ( ∀ ( α , β ) ∈ A × B ) . Let (R _n , B × A) stand for (G _n , B) ∧ (F _n , A), where R n ( β , α ) = G n ( β ) ∩ F n ( α ) ( ∀ ( α , β ) ∈ A × B ) . For each (α, β) ∈ A × B, there exists a (β, α) ∈ B × A such that L n ( α , β ) = F n ( α ) ∩ G n ( β ) = G n ( β ) ∩ F n ( α ) = R n ( β , α ) . By Definition 3.1, (L _n , A × B)   ⊆  _s   (R _n , B × A) . Similarly, we can show that (R _n , B × A)   ⊆  _s   (L _n , A × B) . Therefore, ( F n , A ) ∧ ( G n , B ) = s ( G n , B ) ∧ ( F n , A ) .

Proposition 3.9. Let (F _n , A), (G _n , B) and (H _n , C) are three n-VRNSs over the common universe U. Then the following two equalities hold 1 :

(1) (F _n , A) ∧  [(G _n , B) ∧ (H _n , C) ] =  _s  [(F _n , A) ∧ (G _n , B) ] ∧ (H _n , C) ,

(2) (F _n , A) ∨  [(G _n , B) ∨ (H _n , C) ] =  _s  [(F _n , A) ∨ (G _n , B) ] ∨ (H _n , C) .

Proof. We only show (1). Let (N _n , B × C) stand for (G _n , B) ∧ (H _n , C) , where

N n ( β , γ ) = G n ( β ) ∩ H n ( γ ) ( ∀ ( β , γ ) ∈ B × C ) . Let (F _n , A) ∧ (N _n , B × C) = (L _n , A × (B × C)) , where L n ( α , ( β , γ ) ) = F n ( α ) ∩ N n ( β , γ ) = F n ( α ) ∩ [ G n ( β ) ∩ H n ( γ ) ] ( ∀ ( α , ( β , γ ) ) ∈ A × ( B × C ) ) . Again, let (M _n , A × B) stand for (F _n , A) ∧ (G _n , B) , where M _n  (α, β) = F _n  (α)  ∩ G _n  (β)    (∀ (α, β) ∈ A × B) .

Let (M _n , A × B) ∧ {(H _n , C) = (R _n , (A × B) × C)) , where R n ( ( α , β ) , γ ) = M n ( α , β ) ∩ H n ( γ ) = [ F n ( α ) ∩ G n ( β ) ] ∩ H n ( γ ) ( ∀ ( ( α , β ) , γ ) ∈ ( A × B ) × C ) . Since F _n  (α) ∩ [G _n  (β) ∩ H _n  (γ)] = [F _n  (α) ∩ G _n  (β)] ∩H _n  (γ) , we deduce that {L _n  (α, (β, γ)) ∣ (α, (β, γ)) ∈ A × (B × C)} and {R _n  ((α, β) , γ) ∣ ((α, β) , γ) ∈ (A × B) × C} are indeed the same set. Hence by Proposition 3.4 (4), we can conclude that (L _n , A × (B × C))   =  _s   (R _n , (A × B) × C).

4. Conclusion

Alkhazaleh [8] introduced the concept of n-valued refined neutrosophic soft set and studied some of its properties and proposed several proportions and some operations on a n-valued refined neutrosophic soft set. We pointed out that assertions in Proposition 3.6 (3) and (4) of [8] are flawed. We introduce some new notions describing ‘subset’ of n-valued refined neutrosophic soft sets (n-VRNSs for short) and ‘equal’ of n-VRNSs, and give some examples and related propositions. Then we use these results to remedy the flaw of the assertions, and to improve the work of Alkhazaleh. It should pointed that all notions and results in this note can be extended to arbitrary set case; such an idea can also be used in the study of products of L-topological spaces.