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An. S¸t. Univ. Ovidius Constan¸ta P-union and P-intersection of neutrosophic cubic sets Young Bae Jun, Florentin Smarandache and Chang Su Kim Abstract ConditionsfortheP-intersectionandP-intersectionoffalsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic sets to be an falsity-external (resp. indeterminacy-external and truth- external) neutrosophic cubic set are provided. Conditions for the P- unionandtheP-intersectionoftwotruth-external(resp. indeterminacy- external and falsity-external) neutrosophic cubic sets to be a truth- internal(resp. indeterminacy-internalandfalsity-internal)neutrosophic cubic set are discussed. 1 Introduction The concept of neutrosophic set (NS) developed by Smarandache ([3, 4]) is a more general platform which extends the concepts of the classic set and fuzzy set, intuitionistic fuzzy set and interval valued intuitionistic fuzzy set. Neutrosophic set theory is applied to various part (refer to the site http://fs.gallup.unm.edu/neutrosophy.htm). Jun et al. [2] extended the concept of cubic sets to the neutrosophic sets. They introduced the notions of truth-internal (indeterminacy-internal, falsity-internal) neutrosophic cubic setsandtruth-external(indeterminacy-external,falsity-external)neutrosophic cubic sets, and investigate related properties. Generally, the P-intersection Key Words: Truth-internal (indeterminacy-internal, falsity-internal) neutrosophic cu- bicset, truth-external(indeterminacy-external, falsity-external)neutrosophiccubicset, P- union,P-intersection. 2010MathematicsSubjectClassification: Primary03E72;Secondary,03B60. Received: November,2015. Revised: mm,201y. Accepted: mm,201y. 0 P-UNION AND P-INTERSECTION OF NEUTROSOPHIC CUBIC SETS 1 of falsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic sets may not be a falsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic set (see [2]). As a continuation of the paper [2], we provide a condition for the P-intersection of falsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic sets to be a falsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic set. We provide examples to show that the P-union of falsity-external (resp. indeterminacy-external and truth- external) neutrosophic cubic sets may not be a falsity-external (resp. indeterminacy-external and truth-external) neutrosophic cubic set. We consider a condition for the P-union of truth-external (resp. indeterminacy-external and falsity-external) neutrosophic cubic sets to be a truth-external (resp. indeterminacy-external and falsity-external) neutrosophic cubic set. We also giveaconditionfortheP-intersectionoftwoneutrosophiccubicsetstobeboth a truth-internal (resp. indeterminacy-internal and falsity-internal) neutro- sophiccubicsetandatruth-external(resp. indeterminacy-externalandfalsity- external)neutrosophiccubicset. Generally,theP-unionoftwotruth-external (resp. indeterminacy-external and falsity-external) neutrosophic cubic sets maynotbeatruth-internal(resp. indeterminacy-internalandfalsity-internal) neutrosophic cubic set. We provide conditions for the P-union and the P- intersection of two truth-external (resp. indeterminacy-external and falsity- external) neutrosophic cubic sets to be a truth-internal (resp. indeterminacy- internal and falsity-internal) neutrosophic cubic set. 2 Preliminaries Jun et al. [1] have defined the cubic set as follows: Let X be a non-empty set. A cubic set in X is a structure of the form: C={(x,A(x),λ(x))|x∈X} where A is an interval-valued fuzzy set in X and λ is a fuzzy set in X. Let X be a non-empty set. A neutrosophic set (NS) in X (see [3]) is a structure of the form: Λ:={(cid:104)x;λ (x),λ (x),λ (x)(cid:105)|x∈X} T I F where λ : X → [0,1] is a truth membership function, λ : X → [0,1] is an T I indeterminatemembershipfunction,andλ :X →[0,1]isafalsemembership F function. 2 Young Bae Jun, Florentin Smarandacheand Chang Su Kim Let X be a non-empty set. An interval neutrosophic set (INS) in X (see [5]) is a structure of the form: A:={(cid:104)x;A (x),A (x),A (x)(cid:105)|x∈X} T I F where A , A and A are interval-valued fuzzy sets in X, which are called T I F anintervaltruthmembershipfunction,anintervalindeterminacymembership function and an interval falsity membership function, respectively. Jun et al. [2] considered the notion of neutrosophic cubic sets as an exten- sion of cubic sets. Let X be a non-empty set. A neutrosophic cubic set (NCS) in X is a pair A = (A,Λ) where A := {(cid:104)x;A (x),A (x),A (x)(cid:105) | x ∈ X} is an inter- T I F val neutrosophic set in X and Λ := {(cid:104)x;λ (x),λ (x),λ (x)(cid:105) | x ∈ X} is a T I F neutrosophic set in X. Definition 2.1 ([2]). Let X be a non-empty set. A neutrosophic cubic set A =(A,Λ) in X is said to be • truth-internal (briefly, T-internal) if the following inequality is valid (cid:161) (cid:162) (∀x∈X) A−(x)≤λ (x)≤A+(x) , (2.1) T T T • indeterminacy-internal (briefly, I-internal) if the following inequality is valid (cid:161) (cid:162) (∀x∈X) A−(x)≤λ (x)≤A+(x) , (2.2) I I I • falsity-internal (briefly, F-internal) if the following inequality is valid (cid:161) (cid:162) (∀x∈X) A−(x)≤λ (x)≤A+(x) . (2.3) F F F Definition 2.2 ([2]). Let X be a non-empty set. A neutrosophic cubic set A =(A,Λ) in X is said to be • truth-external (briefly, T-external) if the following inequality is valid (cid:161) (cid:162) (∀x∈X) λ (x)∈/ (A−(x),A+(x)) , (2.4) T T T • indeterminacy-external (briefly, I-external) if the following inequality is valid (cid:161) (cid:162) (∀x∈X) λ (x)∈/ (A−(x),A+(x)) , (2.5) I I I • falsity-external (briefly, F-external) if the following inequality is valid (cid:161) (cid:162) (∀x∈X) λ (x)∈/ (A−(x),A+(x)) . (2.6) F F F P-UNION AND P-INTERSECTION OF NEUTROSOPHIC CUBIC SETS 3 3 P-union and P-intersection of neutrosophic cubic sets Note that P-intersection of F-external (resp. I-external and T-external) neu- trosophiccubicsetsmaynotbeanF-external(resp. I-externalandT-external) neutrosophic cubic set (see [2]). We provide a condition for the P-intersection of F-external (resp. I-external and T-external) neutrosophic cubic sets to be an F-external (resp. I-external and T-external) neutrosophic cubic set. Theorem 3.1. Let A = (A,Λ) and B = (B,Ψ) be T-external neutrosophic cubic sets in X such that (cid:169) (cid:170) max min{A+(x),B−(x)},min{A−(x),B+(x)} <(λ ∧ψ )(x) (cid:169) T T T T (cid:170) T T (3.1) ≤min max{A+(x),B−(x)},max{A−(x),B+(x)} T T T T for all x ∈ X. Then the P-intersection A ∩ B = (A∩B,Λ∧Ψ) is a T- P external neutrosophic cubic set in X. Proof. For any x∈X, let (cid:169) (cid:170) a :=min max{A+(x),B−(x)},max{A−(x),B+(x)} x T T T T and (cid:169) (cid:170) b :=max min{A+(x),B−(x)},min{A−(x),B+(x)} . x T T T T Then a =A−(x), a =B−(x), a =A+(x), or a =B+(x). It is possible to x T x T x T x T consider the cases a = A−(x) and a = A+(x) only because the remaining x T x T cases are similar to these cases. If a =A−(x), then x T B−(x)≤B+(x)≤A−(x)≤A+(x). T T T T Thus b =B+(x), and so x T B−(x)=(A ∩B )−(x)≤(A ∩B )+(x) T T T T T =B+(x)=b <(λ ∧ψ )(x). T x T T Hence (λ ∧ψ )(x)∈/ ((A ∩B )−(x),(A ∩B )+(x)). If a =A+(x), then T T T T T T x T B−(x) ≤ A+(x) ≤ B+(x) and thus b = max{A−(x),B−(x)}. Suppose that T T T x T T b =A−(x). Then x T B−(x)≤A−(x)<(λ ∧ψ )(x)≤A+(x)≤B+(x). (3.2) T T T T T T It follows that B−(x)≤A−(x)<(λ ∧ψ )(x)<A+(x)≤B+(x) (3.3) T T T T T T 4 Young Bae Jun, Florentin Smarandacheand Chang Su Kim or B−(x)≤A−(x)<(λ ∧ψ )(x)=A+(x)≤B+(x). (3.4) T T T T T T The case (3.3) induces a contradiction. The case (3.4) implies that (cid:161) (cid:162) (λ ∧ψ )(x)∈/ (A ∩B )−(x),(A ∩B )+(x) T T T T T T since (λ ∧ψ )(x)=A+(x)=(A ∩B )+(x). Now, if b =B−(x), then T T T T T x T A−(x)≤B−(x)<(λ ∧ψ )(x)≤A+(x)≤B+(x). (3.5) T T T T T T Hence we have A−(x)≤B−(x)<(λ ∧ψ )(x)<A+(x)≤B+(x) (3.6) T T T T T T or A−(x)≤B−(x)<(λ ∧ψ )(x)=A+(x)≤B+(x). (3.7) T T T T T T The case (3.6) induces a contradiction. The case (3.7) induces (cid:161) (cid:162) (λ ∧ψ )(x)∈/ (A ∩B )−(x),(A ∩B )+(x) . T T T T T T Therefore the P-intersection A ∩ B = (A∩B,Λ∧Ψ) is a T-external neu- P trosophic cubic set in X. Similarly, we have the following theorems. Theorem 3.2. Let A = (A,Λ) and B = (B,Ψ) be I-external neutrosophic cubic sets in X such that (cid:169) (cid:170) max min{A+(x),B−(x)},min{A−(x),B+(x)} <(λ ∧ψ )(x) (cid:169) I I I I (cid:170) I I (3.8) ≤min max{A+(x),B−(x)},max{A−(x),B+(x)} I I I I for all x ∈ X. Then the P-intersection A ∩ B = (A∩B,Λ∧Ψ) is an P I-external neutrosophic cubic set in X. Theorem 3.3. Let A = (A,Λ) and B = (B,Ψ) be F-external neutrosophic cubic sets in X such that (cid:169) (cid:170) max min{A+(x),B−(x)},min{A−(x),B+(x)} <(λ ∧ψ )(x) (cid:169) F F F F (cid:170) F F (3.9) ≤min max{A+(x),B−(x)},max{A−(x),B+(x)} F F F F for all x ∈ X. Then the P-intersection A ∩ B = (A∩B,Λ∧Ψ) is an P F-external neutrosophic cubic set in X. P-UNION AND P-INTERSECTION OF NEUTROSOPHIC CUBIC SETS 5 Corollary 3.4. Let A = (A,Λ) and B = (B,Ψ) be external neutrosophic cubic sets in X. Then the P-intersection of A = (A,Λ) and B = (B,Ψ) is an external neutrosophic cubic set in X when the conditions (3.1), (3.8) and (3.9) are valid. The following example shows that the P-union of F-external (resp. I- external and T-external) neutrosophic cubic sets may not be an F-external (resp. I-external and T-external) neutrosophic cubic set. Example3.5. (1)LetA =(A,Λ)andB =(B,Ψ)beneutrosophiccubicsets in X =[0,1] with the tabular representations in Tables 1 and 2, respectively. Table 1: Tabular representation of A =(A,Λ) X A(x) Λ(x) 0≤x<0.5 ([0.25,0.26],[0.2,0.3],[0.15,0.25]) (0.25,0.15,0.5x+0.5) 0.5≤x≤1 ([0.5,0.7],[0.5,0.6],[0.6,0.7]) (0.55,0.75,0.30) Table 2: Tabular representation of B =(B,Ψ) X B(x) Ψ(x) 0≤x<0.5 ([0.25,0.26],[0.2,0.3],[0.8,0.9]) (0.25,0.15,0.40) 0.5≤x≤1 ([0.5,0.7],[0.5,0.6],[0.1,0.2]) (0.55,0.75,x) Then A = (A,Λ) and B = (B,Ψ) are F-external neutrosophic cubic sets in X = [0,1], and the P-union A ∪ B = (A∪B,Λ∨Ψ) of A = (A,Λ) and P B =(B,Ψ) is given by Table 3. Table 3: Tabular representation of A ∪ B =(A∪B,Λ∨Ψ) P X (A∪B)(x) (Λ∨Ψ)(x) 0≤x<0.5 ([0.25,0.26],[0.2,0.3],[0.8,0.9]) (0.25,0.15,0.5x+0.5) 0.5≤x≤1 ([0.5,0.7],[0.5,0.6],[0.6,0.7]) (0.55,0.75,x) 6 Young Bae Jun, Florentin Smarandacheand Chang Su Kim Then (λ ∨ψ )(0.67)=0.67∈(0.6,0.7) F F (cid:161) (cid:162) = (A ∪B )−(0.67),(A ∪B )+(0.67) , F F F F and so the P-union A ∪ B = (A∪B,Λ∨Ψ) is not an F-external neutro- P sophic cubic set in X =[0,1]. (2) Let A = (A,Λ) and B = (B,Ψ) be neutrosophic cubic sets in X = [0,1] with the tabular representations in Tables 4 and 5, respectively. Table 4: Tabular representation of A =(A,Λ) X A(x) Λ(x) 0≤x≤0.3 ([0.3,0.6],[0.3,0.5],[0.6,1]) (x+0.6,0.15,1x+ 1) 2 2 0.3<x≤1 ([0.4,0.9],[0.5,0.6],[0.6,0.7]) (−2x+0.4,0.75,0.30) 5 Table 5: Tabular representation of B =(B,Ψ) X B(x) Ψ(x) 0≤x≤0.3 ([0.4,0.8],[0.2,0.3],[0.8,0.9]) (1x+0.8,0.15,0.40) 2 0.3<x≤1 ([0.3,0.5],[0.5,0.6],[0.1,0.2]) (1x+0.5,0.75,x) 3 Then A = (A,Λ) and B = (B,Ψ) are T-external neutrosophic cubic sets in X =[0,1]. Note that (cid:189) [0.4, 0.8] if 0≤x≤0.3, (A ∪B )−(x)= T T [0.4,0.9] if 0.3<x≤1, (cid:189) 1x+0.8 if 0≤x≤0.3, (λ ∨ψ )(x)= 2 T T 1x+0.5 if 0.3<x≤1, 3 andsotheP-unionA∪ B =(A∪B,Λ∨Ψ)isnotaT-externalneutrosophic P cubic set in X =[0,1] since (cid:161) (cid:162) (λ ∨ψ )(0.6)=0.7∈(0.4,0.9)= (A ∪B )−(0.6),(A ∪B )+(0.6) . T T T T T T (3) Let A = (A,Λ) and B = (B,Ψ) be neutrosophic cubic sets in X = [0,1] with the tabular representations in Tables 6 and 7, respectively. P-UNION AND P-INTERSECTION OF NEUTROSOPHIC CUBIC SETS 7 Table 6: Tabular representation of A =(A,Λ) X A(x) Λ(x) 0≤x≤0.5 ([0.3,0.6],[0.2,0.7],[0.6,1.0]) (0.4,1x+0.7,1x+ 1) 5 2 2 0.5<x≤1 ([0.4,0.9],[0.3,1.0],[0.6,0.7]) (0.3,− 1 x+0.3,0.30) 10 Table 7: Tabular representation of B =(B,Ψ) X B(x) Ψ(x) 0≤x≤0.5 ([0.4,0.8],[0.3,0.8],[0.8,0.9]) (0.3,−1x+0.3,0.40) 5 0.5<x≤1 ([0.3,0.5],[0.5,0.9],[0.1,0.2]) (0.5,− 1 x+1.0,x) 10 It is routine to verify that A =(A,Λ) and B =(B,Ψ) are I-external neutrosophic cubic sets in X =[0,1], but their P-union is not an I-external neutrosophic cubic sets in X =[0,1] since (cid:161) (cid:162) (λ ∨ψ )(0.7)=0.93∈(0.5,1.0)= (A ∪B )−(0.7),(A ∪B )+(0.7) . I I I I I I We consider a condition for the P-union of T-external (resp. I-external and F-external) neutrosophic cubic sets to be a T-external (resp. I-external and F-external) neutrosophic cubic set. Theorem 3.6. Let A = (A,Λ) and B = (B,Ψ) be F-external neutrosophic cubic sets in X such that (cid:169) (cid:170) max min{A+(x),B−(x)},min{A−(x),B+(x)} ≤(λ ∨ψ )(x) (cid:169) F F F F (cid:170) F F (3.10) <min max{A+(x),B−(x)},max{A−(x),B+(x)} F F F F for all x∈X. Then the P-union A ∪ B =(A∪B,Λ∨Ψ) is an F-external P neutrosophic cubic set in X. Proof. For any x∈X, let (cid:169) (cid:170) a :=min max{A+(x),B−(x)},max{A−(x),B+(x)} x F F F F and (cid:169) (cid:170) b :=max min{A+(x),B−(x)},min{A−(x),B+(x)} . x F F F F 8 Young Bae Jun, Florentin Smarandacheand Chang Su Kim Then a =A−(x), a =B−(x), a =A+(x), or a =B+(x). It is possible to x F x F x F x F consider the cases a = A−(x) and a = A+(x) only because the remaining x F x F cases are similar to these cases. If a =A−(x), then x F B−(x)≤B+(x)≤A−(x)≤A+(x), F F F F and so b =B+(x). Thus x F (A ∪B )−(x)=A−(x)=a >(λ ∨ψ )(x), F F F x F F and hence (λ ∨ψ )(x)∈/ ((A ∪B )−(x),(A ∪B )+(x)). If a =A+(x), F F F F F F x F then B−(x) ≤ A+(x) ≤ B+(x) and thus b = max{A−(x),B−(x)}. Suppose F F F x F F that b =A−(x). Then x F B−(x)≤A−(x)≤(λ ∨ψ )(x)<A+(x)≤B+(x), (3.11) F F F F F F and so B−(x)≤A−(x)<(λ ∨ψ )(x)<A+(x)≤B+(x) (3.12) F F F F F F or B−(x)≤A−(x)=(λ ∨ψ )(x)<A+(x)≤B+(x). (3.13) F F F F F F The case (3.12) induces a contradiction. The case (3.13) implies that (cid:161) (cid:162) (λ ∨ψ )(x)∈/ (A ∪B )−(x),(A ∪B )+(x) F F F F F F since (λ ∨ψ )(x)=A−(x)=(A ∪B )−(x). Now, if b =B−(x), then F F F F F x F A−(x)≤B−(x)≤(λ ∨ψ )(x)≤A+(x)≤B+(x). (3.14) F F F F F F Hence we have A−(x)≤B−(x)<(λ ∨ψ )(x)≤A+(x)≤B+(x) (3.15) F F F F F F or A−(x)≤B−(x)=(λ ∨ψ )(x)≤A+(x)≤B+(x). (3.16) F F F F F F The case (3.15) induces a contradiction. The case (3.16) induces (cid:161) (cid:162) (λ ∨ψ )(x)∈/ (A ∪B )−(x),(A ∪B )+(x) . F F F F F F ThereforetheP-unionA∪ B =(A∪B,Λ∨Ψ)isanF-externalneutrosophic P cubic set in X. P-UNION AND P-INTERSECTION OF NEUTROSOPHIC CUBIC SETS 9 Similarly, we have the following theorems. Theorem 3.7. Let A = (A,Λ) and B = (B,Ψ) be T-external neutrosophic cubic sets in X such that (cid:169) (cid:170) max min{A+(x),B−(x)},min{A−(x),B+(x)} ≤(λ ∨ψ )(x) (cid:169) T T T T (cid:170) T T (3.17) <min max{A+(x),B−(x)},max{A−(x),B+(x)} T T T T for all x ∈ X. Then the P-union A ∪ B = (A∪B,Λ∨Ψ) is a T-external P neutrosophic cubic set in X. Theorem 3.8. Let A = (A,Λ) and B = (B,Ψ) be I-external neutrosophic cubic sets in X such that (cid:169) (cid:170) max min{A+(x),B−(x)},min{A−(x),B+(x)} ≤(λ ∨ψ )(x) (cid:169) I I I I (cid:170) I I (3.18) <min max{A+(x),B−(x)},max{A−(x),B+(x)} I I I I for all x ∈ X. Then the P-union A ∪ B = (A∪B,Λ∨Ψ) is an I-external P neutrosophic cubic set in X. We give a condition for the P-intersection of two neutrosophic cubic sets to be both a T-internal (resp. I-internal and F-internal) neutrosophic cubic set and a T-external (resp. I-external and F-external) neutrosophic cubic set. Theorem 3.9. If neutrosophic cubic sets A = (A,Λ) and B = (B,Ψ) X satisfy the following condition (cid:169) (cid:170) min max{A+(x),B−(x)},max{A−(x),B+(x)} =(λ ∧ψ )(x) (cid:169) F F F F (cid:170) F F (3.19) =max min{A+(x),B−(x)},min{A−(x),B+(x)} F F F F for all x∈X, then the P-intersection of A =(A,Λ) and B =(B,Ψ) is both an F-internal neutrosophic cubic set and an F-external neutrosophic cubic set in X. Proof. For any x∈X, take (cid:169) (cid:170) a :=min max{A+(x),B−(x)},max{A−(x),B+(x)} x F F F F and (cid:169) (cid:170) b :=max min{A+(x),B−(x)},min{A−(x),B+(x)} . x F F F F Then a is one of A−(x), B−(x), A+(x) and B+(x). Weconsider a =A−(x) x F F F F x F or a = A+(x) only. For remaining cases, it is similar to these cases. If x F a =A−(x), then x F B−(x)≤B+(x)≤A−(x)≤A+(x) F F F F