1 Introduction

Many theories have been developed for uncertainties, including the theory of probability, the theory of fuzzy sets, the theory of intuitionistic fuzzy sets, the theory of rough sets, and so on. Although many new techniques have been developed as a result of these theories there are still some difficulties. Major difficulties arise due to inadequacy of parameters. In 1999, Smarandache [1,2] proposed the concept of the neutrosophic set (NS) by adding an independent indeterminacy-membership function, which is a generalization of a classic set, a fuzzy set [3], an intuitionistic fuzzy set [4], and so on. In an NS, the indeterminacy is quantified explicitly and truth-membership function (T), indeterminacy-membership function (I), and false-membership function (F) are completely independent and from a scientific or engineering point of view, the NS operators need to be specified. Therefore, Wang, et al. [5] defined the single valued neutrosophic set (SVNS) and then provided the set-theoretic operations and various properties of single valued neutrosophic sets. Wang, et al. [6] proposed the set-theoretic operations on an instance of the neutrosophic set, called the interval valued neutrosophic set (IVNS), which is more flexible and practical than the NS. Works on single valued neutrosophic sets (SVNS) and interval valued neutrosophic sets (IVNS) and their hybrid structure in theory and applications have progressed rapidly (e.g., [7-23]). Also, neutrosophic sets have been extended to neutrosophic models both in theory [24-28] and in applications [29-32].

Received March 27th, 2017, 1st Revision August 10th, 2017, 2nd Revision September 25th, 2017, Accepted for publication October 5th, 2017.

Copyright © 2018 Published by ITB Journal Publisher, ISSN: 2337-5760, DOI: 10.5614/j.math.fund.sci.2018.50.2.1

The organization of this paper is as follows. In Section 2, some basic properties of neutrosophic sets are redefined. In Section 3, the \((\alpha, \beta, \gamma)\) interval cut sets of interval valued neutrosophic sets are defined and some properties of these \((\alpha, \beta, \gamma)\) interval cut sets are given. In Section 4, the operations of addition, subtraction, multiplication and division over neutrosophic multi-sets are defined based on the interval valued neutrosophic set. Finally, the conclusion is given.

2 Preliminaries

In this section, we recall some concepts of neutrosophic set theory and interval valued neutrosophic set theory. See the refences for more detailed explanations related to this subsection and its background.

Definition 2.1 [4]: An intuitionistic fuzzy set (IFS) A in X is:

\[A = \{ \langle x, \mu_A(x), \nu_A(x) \rangle : x \in X \},\] where the functions \(\mu_A(x)\), \(\nu_A(x): X \to [0,1]\) with the condition \(0 \le \mu_A(x) + \nu_A(x) \le 1\). The numbers \(\mu_A(x)\) and \(\nu_A(x)\) represent, respectively, the degree of membership and degree of non-membership of the element \(x \in X\).

Definition 2.2 [33]: Let \(A = \{\langle x, \mu_A(x), \nu_A(x) \rangle : x \in X \}\) be any intuitionistic fuzzy set on the universal set X. For any ordered pair \(\langle \alpha, \beta \rangle\), where \(\alpha \in [0, 1]\), \(\beta \in [0, 1]\), and \(0 \le \alpha + \beta \le 1\), let:

\[A_{\langle \alpha, \beta \rangle} = \{x : \mu_A(x) \ge \alpha, \nu_A(x) \le \beta : x \in X \},\] which is called the \(\langle \alpha, \beta \rangle\)-cut set (or level set) of intuitionistic fuzzy set A.

Definition 2.3 [34]: Let X be an universe of discourse with a generic element in X, denoted by x. Then, a neutrosophic (NS) set A is an object with the following form:

\[A = \{\langle x: T_{\Delta}(x), I_{\Delta}(x), F_{\Delta}(x) \rangle, x \in X\}\]

A neutrosophic set A in X is characterized by truth-membership function \(T_A\), indeterminacy membership function \(I_A\), and falsity-membership function \(F_A\). \(T_A(x)\), \(I_A(x)\) and \(F_A(x)\) are real standard or non-standard subsets of \(]^-0\), \(1^+[\). That is:

\[T_A(x): X \to ]^-0, 1^+[\]
\(I_A(x): X \to ]^-0, 1^+[\)
\(F_A(x): X \to ]^-0, 1^+[\)

There is no restriction on the sum of \(T_A(x)\), \(I_A(x)\) and \(F_A(x)\), so:

\[^-0 \le \sup T_A(x) + \sup I_A(x) + \sup F_A(x) \le 3^+\].

Instead of \(]^-0\), \(1^+[\), we need to take the interval [0, 1] for technical applications, because \(]^-0\), \(1^+[\) will be difficult to apply in real applications, for example in scientific and engineering problems.

For two NS:

\[A_{NS} = \{\langle x, T_A(x), I_A(x), F_A(x) \rangle | x \in X\}\] and

\[B_{NS} = \{ \langle x, T_B(x), I_B(x), F_B(x) \rangle | x \in X \}\] the two relations are defined as follows:

(1) \(A_{NS} \subseteq B_{NS}\) if and only if

\[T_A(x) \le T_B(x), \ I_A(x) \ge I_B(x), \ F_A(x) \ge F_B(x)\]

(2) \(A_{NS} = B_{NS}\) if and only if

\[T_A(x) = T_B(x), I_A(x) = I_B(x), F_A(x) = F_B(x).\]

Definition 2.4: [5,34] Let X be a space of points (objects) with a generic element in X denoted by x. An interval valued neutrosophic set (IVNS) A in X is characterized by truth-membership function \(T_A\), indeterminacy-membership function \(I_A\), and falsity-membership function \(F_A\). For each point \(x \in X\), \(T_A(x)\), \(I_A(x)\), \(F_A(x) \subseteq [0,1]\).

This can be written as:

\[A_{\text{IVNS}} = \left\{ \langle x, [\inf T_A(x), \sup T_A(x)], [\inf I_A(x), \sup I_A(x)], \rangle : x \in X \right\}\]

Definition 2.5 [5]: For two IVNS,

\[A_{\text{IVNS}} = \left\{ \left\langle x, \left[\inf T_A(x), \sup T_A(x)\right], \left[\inf I_A(x), \sup I_A(x)\right], \right\rangle : x \in X \right\}\] and

\[B_{\text{IVNS}} = \left\{ \langle x, [\inf T_B(x), \sup T_B(x)], [\inf I_B(x), \sup I_B(x)], \rangle : x \in X \right\},\]

we now present the set-theoretic operators on the interval neutrosophic set.

1. An interval valued neutrosophic set A is contained in another interval valued neutrosophic set B, \(A_{IVNS} \subseteq B_{IVNS}\), if and only if

\[\inf T_A(x) \le \inf T_B(x), \sup T_A(x) \le \sup T_B(x),\]
\(\inf I_A(x) \ge \inf I_B(x), \sup I_A(x) \ge \sup I_B(x),\)
\(\inf F_A(x) \ge \inf F_B(x), \sup F_A(x) \ge \sup F_B(x),\)

for all \(x \in X\).

2. Two interval valued neutrosophic sets A and B are equal, written as \(A_{\text{IVNS}} = B_{\text{IVNS}}\), if and only if \(A \subseteq B\) and \(B \subseteq A\), i.e.

\[\inf T_A(x) = \inf T_B(x), \sup T_A(x) = \sup T_B(x),\]
\[\inf I_A(x) = \inf I_B(x), \sup I_A(x) = \sup I_B(x),\]
\[\inf F_A(x) = \inf F_B(x), \sup F_A(x) = \sup F_B(x),\] for any \(x \in X\).

3. An interval neutrosophic set A is empty if and only if

\[inf T_A(x) = sup T_A(x) = 0,\]
\(inf I_A(x) = sup I_A(x) = 1\) and
\(inf F_A(x) = sup F_A(x) = 0,\)

for all \(x \in X\).

4. The complement of an interval neutrosophic set A is denoted by \(A^c\) and is defined by

\[A_{\text{IVNS}}c = \begin{cases} x, [\inf F_A(x), \sup F_A(x)], \\ [1 - \sup I_A(x), 1 - \inf I_A(x)], : x \in X \\ [\inf T_A(x), \sup T_A(x)] \end{cases},\] for all x in X.

5. The intersection of two neutrosophic sets A and B is a neutrosophic set \(A \cap B\), whose truth-membership, indeterminacy-membership and falsity-membership functions are related to those of A and B by

\[A_{IVNS} \cap B_{IVNS} = \begin{cases} x, [\inf T_A(x) \wedge \inf T_B(x), \sup T_A(x) \wedge \sup T_B(x), \\ \langle [\inf I_A(x) \vee \inf I_B(x), \sup I_A(x) \vee \sup I_B(x)], \rangle : x \in X \\ [\inf F_A(x) \vee \inf F_B(x), \sup F_A(x) \vee \sup F_B(x) \end{cases},\] for all x in X.

6. The union of two interval neutrosophic sets A and B is an interval neutrosophic set \(A_{\rm IVNS} \cup B_{\rm IVNS}\), whose truth-membership, indeterminacy-membership and false-membership are related to those of A and B by

\[A_{\text{IVNS}} \cup B_{\text{IVNS}} = \begin{cases} x, [\inf T_A(x) \vee \inf T_B(x), \sup T_A(x) \vee \sup T_B(x)] \\ \langle [\inf I_A(x) \wedge \inf I_B(x), \sup I_A(x) \wedge \sup I_B(x)], \rangle : x \in X \\ [\inf F_A(x) \wedge \inf F_B(x), \sup F_A(x) \wedge \sup F_B(x)] \end{cases}\] for all ∈.

7. The difference of two interval neutrosophic sets and is an interval neutrosophic set ୍ୗ\ ୍ୗ, whose truth-membership, indeterminacymembership and falsity-membership functions are related to those of and by

୍ୗ\୍ୗ ൌ

\[\begin{cases} x, [\min\{\inf T_A(x), \inf F_B(x)\}, \min\{\sup T_A(x), \sup F_B(x)\}], \\ \langle [\max(\inf I_A(x), 1 - \sup I_B(x)), \max(\sup I_A(x), 1 - \inf I_B(x))], \rangle \ x \in X \\ [\max(\inf F_A(x), \inf T_B(x)), \max(\sup F_A(x), \sup T_B(x))] \end{cases}\] 8. The addition of two interval neutrosophic sets A and B is an interval neutrosophic set ୍ୗ ୍ୗ, whose truth-membership, indeterminacymembership and falsity-membership functions are related to those of A and B by

୍ୗ ୍ୗ ൌ

\[\left\{ \begin{array}{l} x, \left[ \min(\inf T_A(x) + \inf T_B(x), 1 \right), \min(\sup T_A(x) + \sup T_B(x), 1) \right], \\ \left\langle \ \left[ \min(\inf I_A(x) + \inf I_B(x), 1 \right), \min(\sup I_A(x) + \sup I_B(x), 1) \right], \ \right\rangle x \in X \\ \left[ \min(\inf F_A(x) + \inf F_B(x), 1 \right), \min(\sup F_A(x) + \sup F_B(x), 1) \right] \end{array} \right\}\] for all ∈.

9. The scalar multiplication of interval neutrosophic set A is ୍ୗ. , whose truth-membership, indeterminacy-membership and falsity-membership functions are related to those of A by

\[A_{\text{IVNS}}. a = \begin{cases} x, [\min(\inf T_A(x). \ a \ , 1) \ , \min(\sup T_A(x). \ a \ , 1)], \\ \langle \ [\min(\inf I_A(x). \ a \ , 1) \ , \min(\sup I_A(x). \ a \ , 1)], \ \rangle : x \in X \end{cases}\] \[[\min(\inf F_A(x). \ a \ , 1) \ , \min(\sup F_A(x). \ a \ , 1)]\] for all ∈, a ∈ R+ .

10. The scalar division of interval neutrosophic set A is ୍ୗ/a, whose truthmembership, indeterminacy-membership and falsity-membership functions are related to those of A by

\[A_{\text{IVNS}}/a = \begin{cases} x, [\min(\inf T_A(x)/a, 1), \min(\sup T_A(x)/a, 1)], \\ \langle [\min(\inf I_A(x)/a, 1), \min(\sup I_A(x)/a, 1)], \rangle : x \in X \\ [\min(\inf F_A(x)/a, 1), \min(\sup F_A(x).a, 1)] \end{cases}\] for all ∈, ∈ା.

3 \((\alpha, \beta, \gamma)\) Interval Cut Set of Interval Valued Neutrosophic Sets

Definition 3.1 Let

\[A_{\text{IVNS}} = \begin{cases} x, [\inf T_A(x), \sup T_A(x)], \\ \langle [\inf I_A(x), \sup I_A(x)], \rangle : x \in X \\ [\inf F_A(x), \sup F_A(x)] \end{cases}\] be any interval valued neutrosophic set on X. For any ordered \(\alpha = [\alpha_1, \alpha_2]\), \(\beta = [\beta_1, \beta_2]\) and \(\gamma = [\gamma_1, \gamma_2] \in [0,1]\), such that \(0 \le \alpha + \beta + \gamma \le 3\). We define the \((\alpha, \beta, \gamma)\) cut of interval valued neutrosophic set A, denoted by \(A_{([\alpha_1, \alpha_2], [\beta_1, \beta_2], [\gamma_1, \gamma_2])}\):

\[A_{\left(\left[\alpha_{1},\alpha_{2}\right],\left[\beta_{1},\beta_{2}\right],\left[\gamma_{1},\gamma_{2}\right]\right)} = \begin{cases} x: \inf T_{A}(x) \geq \alpha_{1}, \sup T_{A}(x) \geq \alpha_{2}, \\ \inf I_{A}(x) \leq \beta_{1}, \sup I_{A}(x) \leq \beta_{2}, x \in X \\ \inf F_{A}(x) \leq \gamma_{1}, \sup F_{A}(x) \leq \gamma_{2}; \end{cases}\]

That is,

\[A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])} = \begin{cases} x : \inf T_A(x) \land \alpha_1 = \alpha_1, \sup T_A(x) \land \alpha_2 = \alpha_2, \\ \langle \inf I_A(x) \lor \beta_1 = \beta_1, \sup I_A(x) \lor \beta_2 = \beta_2, \rangle x \in X \\ \inf F_A(x) \lor \gamma_1 = \gamma_1, \sup F_A(x) \lor \gamma_2 = \gamma_2 \end{cases}.\]

We define

\(A_{([\alpha_1,\alpha_2])} = \{x : \inf T_A(x) \ge \alpha_1, \sup T_A(x) \ge \alpha_2; x \in X\}\) the \(\alpha\)-cut of the truth membership function generated by A.

\(A_{([\beta_1,\beta_2])} = \{x: inf \ I_A(x) \le \beta_1, \ sup \ I_A(x) \le \beta_2; x \in X\}\) the \(\beta\)-cut of the indeterminacy membership function generated by A and

\(A_{([\gamma_1,\gamma_2])} = \{x: inf F_A(x) \le \gamma_1, sup F_A(x) \le \gamma_2; x \in X\}\) the \(\gamma\)-cut of the falsity membership function generated by A.

Example 3.2 Let \(= \{x_1, x_2, x_3\}\). Let us consider an interval valued neutrosophic set:

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] \[x_2: ([0.5, 0.8]; [0.4, 0.6]; [0.4, 0.5]);\] \[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\]

Let \(\alpha_1 = 0.3\); \(\beta_1 = 0.4\); \(\gamma_1 = 0.5 \in [0,1]\) and \(\alpha_2 = 0.5\); \(\beta_2 = 0.6\); \(\gamma_2 = 0.7 \in [0,1]\).

Then,

ሺሾఈభ,ఈమሿ,ሾఉభ,ఉమሿ,ሾఊభ,ఊమሿሻ ൌ ሼଵ, ଶሽ. Here, ଷ is not an element of this cut set. Moreover,

\[A_{([\alpha_1,\alpha_2])} = \{x_1, x_2\}\] \[A_{([\beta_1,\beta_2])} = \{x_1, x_2, x_3\}\] \[A_{([\gamma_1,\gamma_2])} = \{x_1, x_2\}\]

Definition 3.3 Let ୍ୗ be any interval valued neutrosophic set on . We define the ሺ, , ሻ strong cut of interval valued neutrosophic set , denoted by ሺሾఈభ,ఈమሿ,ሾఉభ,ఉమሿ,ሾఊభ,ఊమሿሻା, where ൌ ሾଵ, ଶሿ, ൌ ሾଵ, ଶሿ and ൌ ሾଵ, ଶሿ ∈ ሾ0,1ሿ such that 03,

\[A_{([\alpha_{1},\alpha_{2}],[\beta_{1},\beta_{2}],[\gamma_{1},\gamma_{2}])+} = \begin{cases} x \colon \inf T_{A}(x) > \alpha_{1} = \alpha_{1}, \sup T_{A}(x) > \alpha_{2} = \alpha_{2}, \\ \langle \inf I_{A}(x) < \beta_{1} = \beta_{1}, \sup I_{A}(x) < \beta_{2} = \beta_{2}, \rangle; x \in X \\ \inf F_{A}(x) < \gamma_{1} = \gamma_{1}, \sup F_{A}(x) < \gamma_{2} = \gamma_{2} \end{cases}\]

we define

• 1. ሺሾఈభ,ఈమሿሻା ൌ ሼ〈: inf ሺሻ ଵ, sup ሺሻ ଶ〉; ∈ ሽ the strong cut of the truth membership function generated by ,
• 2. ሺሾఉభ,ఉమሿሻା ൌ ሼ〈: inf ሺሻ ൏ ଵ, sup ሺሻ ൏ ଶ〉; ∈ ሽ the strong -cut of the indeterminacy membership function generated by and
• 3. ሺሾఊభ,ఊమሿሻା ൌ ሼ⟨: inf ሺሻ ൏ ଵ, sup ሺሻ ൏ ଶ⟩; ∈ ሽ the strong -cut of the falsity membership function generated by .

Example 3.4 Let ൌ ሼଵ, ଶ, ଷሽ. Consider an interval valued neutrosophic set:

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] \[x_2 : ([0.5, 0.8]; [0.4, 0.6]; [0.4, 0.5]);\] \[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] Let \(\alpha_1 = 0.3; \beta_1 = 0.4; \gamma_1 = 0.5 \in [0,1],\) \[\alpha_2 = 0.5; \beta_2 = 0.6; \gamma_2 = 0.7 \in [0,1].\] Then, \[A_{([\alpha_1, \alpha_2], [\beta_1, \beta_2], [\gamma_1, \gamma_2]) +} = \{x_1\}\] \[A_{([\alpha_1, \alpha_2]) +} = \{x_1, x_2\}\]

\[A_{([\beta_1, \beta_2])+} = \{x_1\}\] \[A_{([\gamma_1, \gamma_2])+} = \{x_1, x_2\}\]

Moreover, we can define the cut sets where for any ordered ൌ ሾଵ, ଶሿ, ൌ ሾଵ, ଶሿ and ൌ ሾଵ, ଶሿ ∈ ሾ0,1ሿ such that 03, as:

\[A_{([\alpha_1,\alpha_2]+,[\beta_1,\beta_2],[\gamma_1,\gamma_2])} =\]

ሼ〈: ሺሻ ଵ, ሺሻ ଶ, ሺሻ ଵ, ሺሻ ଶ, ሺሻ ଵ, ሺሻ ଶ〉; ∈ ሽ is ሺ, , ሻ,

\[A_{([\alpha_1,\alpha_2]+,[\beta_1,\beta_2],[\gamma_1,\gamma_2])} = \begin{cases} x: \inf T_A(x) > \alpha_1, \sup T_A(x) > \alpha_2, \\ \langle \inf I_A(x) \leq \beta_1, \sup I_A(x) \leq \beta_2, \rangle x \in X \\ \inf F_A(x) \leq \gamma_1, \sup F_A(x) \leq \gamma_2 \end{cases}\] is ሺ, , ሻ and

\[A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2]+)} = \begin{cases} x : \inf T_A(x) \ge \alpha_1, \sup T_A(x) \ge \alpha_2, \\ \langle \inf I_A(x) \le \beta_1, \sup I_A(x) \le \beta_2, \rangle x \in X \\ \inf F_A(x) < \gamma_1, \sup F_A(x) < \gamma_2 \end{cases}\] is a ሺ, , ሻ-cut set of , respectively. Similarly, we can define other cut sets of .

Proposition 3.5 The cut sets of interval valued neutrosophic sets satisfy the following properties: for an interval valued neutrosophic set A on , any ordered ൌ ሾଵ, ଶሿ, ൌ ሾଵ, ଶሿ and ൌ ሾଵ, ଶሿ ∈ ሾ0,1ሿ, ൌ ሾଵ, ଶሿ, ൌ ሾଵ, ଶሿ, ൌ ሾଵ, ଶሿ such that 03, and such that 0 3,

• (1) ሺሾఈభ,ఈమሿ,ሾఉభ,ఉమሿ,ሾఊభ,ఊమሿሻ ൌ ሺሾఈభ,ఈమሿሻ ∩ ሺሾఉభ,ఉమሿሻ ∩ ሺሾఊభ,ఊమሿሻ,
• (2) If ሾଵ, ଶሿ ൌ ሾଵ, ଶሿ, ሺሾఈభ,ఈమሿሻ ൌ ሺሾఊభ,ఊమሿାሻ,

(3) \[A^{c}_{([\beta_{1},\beta_{2}])} = 1 - A^{c}_{([\beta_{1},\beta_{2}]+)} \text{ and } A^{c}_{([\gamma_{1},\gamma_{2}]+)} = A^{c}_{([\alpha_{1},\alpha_{2}])},\] \[A_{([\alpha_{1},\alpha_{2}],[\beta_{1},\beta_{2}],[\gamma_{1},\gamma_{2}])+} \subseteq A_{([\alpha_{1},\alpha_{2}],[\beta_{1},\beta_{2}],[\gamma_{1},\gamma_{2}])},\]

(4) For \[\alpha_1 \geq \lambda_1\] and \(\alpha_2 \geq \lambda_2\), \(\beta_1 \leq \mu_1\) and \(\beta_2 \leq \mu_2\), \(\gamma_1 \leq \delta_1\) and \(\gamma_2 \leq \delta_2\), \(A_{([\alpha_1,\alpha_2])} \subseteq A_{([\lambda_1,\lambda_2])}\), \(A_{([\beta_1,\beta_2])} \subseteq A_{([\mu_1,\mu_2])}\), \(A_{([\gamma_1,\gamma_2])} \subseteq A_{([\delta_1,\delta_2])}\) and \(A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])} \subseteq A_{([\lambda_1,\lambda_2],[\mu_1,\mu_2],[\delta_1,\delta_2])}\).

Proof. Let ୍ୗ be any interval valued neutrosophic set on for any ordered ൌ ሾଵ, ଶሿ, ൌ ሾଵ, ଶሿ and ൌ ሾଵ, ଶሿ ∈ ሾ0,1ሿ, ൌ ሾଵ, ଶሿ ,

\(\mu = [\mu_1, \mu_2], \, \delta = [\delta_1, \delta_2]\) such that \(0 \le \alpha + \beta + \gamma \le 3\), and such that \(0 \le \lambda + \mu + \delta \le 3\),

• (1) Follows directly from Definition 3.1.
• (2) We know

\[A^{c} = \begin{cases} x: [\inf F_{A}(x), \sup F_{A}(x)], \\ \langle [1 - \sup I_{A}(x), 1 - \inf I_{A}(x)], \rangle x \in X \\ [\inf T_{A}(x), \sup T_{A}(x)] \end{cases}\]

SO

\[A^{c}_{([\alpha_1,\alpha_2])} = \{\langle x: \inf F_A(x) \geq \gamma_1, \sup F_A(x) \geq \gamma_2 \rangle; x \in X\}.\]

By definition

\[A_{([\gamma_1, \gamma_2])+} = \{ \langle x : \inf F_A(x) < \gamma_1, \sup T_A(x) < \gamma_2 \rangle; x \in X \},\]
\[A^c_{([\gamma_1, \gamma_2]+)} = \{ \langle x : \inf F_A(x) \ge \gamma_1, \sup F_A(x) \ge \gamma_2 \rangle; x \in X \}.\]

Therefore, \(A^c_{([\alpha_1,\alpha_2])}=A^c_{([\gamma_1,\gamma_2]+)}\). Similarly, \(A^c_{([\beta_1,\beta_2])}=1-A^c_{([\beta_1,\beta_2]+)}\) and \(A^c_{([\gamma_1,\gamma_2]+)}=A^c_{([\alpha_1,\alpha_2])}\) can be also proved.

(3) \(A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])+}\)

\[= \begin{cases} x : \inf T_A(x) > \alpha_1, \sup T_A(x) > \alpha_2, \\ \langle \inf I_A(x) < \beta_1, \sup I_A(x) < \beta_2, \rangle x \in X \\ \inf F_A(x) < \gamma_1, \sup F_A(x) < \gamma_2 \end{cases}\]

\[\subseteq \left\{ \begin{aligned} x &: \inf T_A(x) \ge \alpha_1, \sup T_A(x) \ge \alpha_2, \\ \langle &: \inf I_A(x) \le \beta_1, \sup I_A(x) \le \beta_2, \ \rangle \, x \in X \\ &: \inf F_A(x) \le \gamma_1, \sup F_A(x) \le \gamma_2 \end{aligned} \right\}\]

\[= A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])}\]

Hence, \(A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])+} \subseteq A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])}\).

(4) By Definition 3.1, for any element x of \(A_{([\alpha_1,\alpha_2])}\), \(\alpha_1 \geq \lambda_1\) and \(\alpha_2 \geq \lambda_2\) inf \(T_A(x) \geq \alpha_1 \geq \lambda_1\), sup \(T_A(x) \geq \alpha_2 \geq \lambda_2\) imply that \(A_{([\alpha_1,\alpha_2])} \subseteq A_{([\lambda_1,\lambda_2])}\).

For any element x of \(A_{([\beta_1,\beta_2])}\), \(\beta_1 \leq \mu_1\) and \(\beta_2 \leq \mu_2\), \(\inf I_A(x) \leq \beta_1 \leq \mu_1\), \(\sup I_A(x) \leq \beta_2 \leq \mu_2\) show that \(A_{([\beta_1,\beta_2])} \subseteq A_{([\mu_1,\mu_2])}\).

Similarly, for \(\gamma_1 \leq \delta_1\) and \(\gamma_2 \leq \delta_2\), we obtain \(A_{([\gamma_1, \gamma_2])} \subseteq A_{([\delta_1, \delta_2])}\). Therefore, we can say that \(A_{([\alpha_1, \alpha_2], [\beta_1, \beta_2], [\gamma_1, \gamma_2])} \subseteq A_{([\lambda_1, \lambda_2], [\mu_1, \mu_2], [\delta_1, \delta_2])}\).

Example 3.6: Let us consider set A defined in Example 3.2. Let \(X = \{x_1, x_2, x_3\}\) be an interval valued neutrosophic set

\[A = \begin{cases} x_1 : ([0.5,0.6], [0.3,0.4], [0.2,0.3]), \\ \langle x_2 : ([0.5,0.8], [0.4,0.6], [0.4,0.5]), \rangle \\ x_3 : ([0.1,0.2], [0.4,0.5], [0.7,0.8]) \end{cases}\]

Let \[\alpha_1 = 0.3\]; \(\beta_1 = 0.4\); \(\gamma_1 = 0.5 \in [0,1]\)
\(\alpha_2 = 0.5\); \(\beta_2 = 0.6\); \(\gamma_2 = 0.7 \in [0,1]\).

Then, \(A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])} = \{x_1,x_2\}\) and \(A_{([\alpha_1,\alpha_2])} = \{x_1,x_2\}\), \(A_{([\beta_1,\beta_2])} = \{x_1,x_2,x_3\}\), \(A_{([\gamma_1,\gamma_2])} = \{x_1,x_2\}\).

We can realize that \(A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])} = \{x_1,x_2\}\) is actually the intersection of \(A_{([\alpha_1,\alpha_2])}\), \(A_{([\beta_1,\beta_2])}\) and \(A_{([\gamma_1,\gamma_2])}\), i.e. \(A_{([\alpha_1,\alpha_2],[\beta_1,\beta_2],[\gamma_1,\gamma_2])} = A_{([\alpha_1,\alpha_2])} \cap A_{([\beta_1,\beta_2])} \cap A_{([\gamma_1,\gamma_2])}\).

4 Algebraic Operations over Interval Valued Neutrosophic Sets

Let A and B be two interval valued neutrosophic sets on a universal set X. We define algebraic operations such as addition, subtraction, multiplication, division and scalar multiplication over the interval neutrosophic sets A and B as follows:

• 1. \(A + B = \{z, \max[\inf T_A(x) + \inf T_B(x) \inf T_A(x) \cdot \inf T_B(x) \sup T_A(x) + \sup T_B(x) \sup T_A(x) \cdot \sup T_B(x)] \min[\inf I_A(x) \cdot \inf I_B(x), (\sup I_A(x) \cdot \sup I_B(x)], \min[\inf F_A(x) \cdot \inf F_B(x), (\sup F_A(x) \cdot \sup F(x)] : z = x + y, (x, y) \in X \times Y\}.\)
• 2. \(A B = \{z, \max[\inf T_A(x) + \inf T_B(x) \inf T_A(x).\inf T_B(x), \sup T_A(x) + \sup T_B(x) \sup T_A(x).\sup T_B(x)], \min[\inf I_A(x).\inf I_B(x), (\sup I_A(x).\sup I_B(x)], \min[\inf F_A(x).\inf F_B(x), (\sup F_A(x).\sup F_B(x)] : z = x y, (x, y) \in X \times Y\}.\)
• 3. \(A.B = \{\langle z, \max[\inf T_A(x).\inf T_B(x), \sup T_A(x).\sup T_B(x)], \min[\inf I_A(x) + \inf I_B(x) \inf I_A(x).\inf I_B(x), \sup I_A(x) + \sup I_B(x) \sup I_A(x).\sup I_B(x)], \min[\inf F_A(x) + \inf F_B(x) \inf F_A(x).\inf F_B(x), \min[\inf F_A(x) + \inf F_B(x)]\}\)

\sup F_A(x) + \sup F_B(x) - \sup F_A(x) \cdot \sup F_B(x) \}
: z = x, y, (x, y) \in X \times Y \}.

• 4. \(A/B = \{z, \max[\inf T_A(x).\inf T_B(x), \sup T_A(x).\sup T_B(x)], \min[\inf I_A(x) + \inf I_B(x) \inf I_A(x).\inf I_B(x), \sup I_A(x)\sup I_B(x) \sup I_A(x).\sup I_B(x)], \min[\inf F_A(x) + \inf F_B(x) \inf F_A(x).\inf F_B(x), \sup F_A(x) + \sup F_B(x) \sup F_A(x).\sup F_B(x)] : z = x/y, (x, y) \in X \times Y\},\)
• 5. \(\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\) \(min[inf I_B(y)^{\lambda}, sup I_B(y)^{\lambda}],\) \(\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\)

where \(\lambda\) is any nonzero real number.

Example 4.1 Let

A = \{3; ([0.2,0.3], [0.1,0.4], [0.2,0.5]), 2; ([0.3,0.5], [0.2,0.6], [0.3,0.4])\} and

\(B = \{1; ([0.1,0.2], [0.2,0.3], [0.3,0.5]), 5; ([0.2,0.4], [0.1,0.3], [0.2,0.5])\}\) be two interval valued neutrosophic sets.

• 1. \(A + B = \{4; ([0.28,0.44], [0.02,0.12], [0.06,0.25]), 8; ([0.36,0.58], [0.01,0.12], [0.04,0.25]), 3; ([0.37,0.6], [0.04,0.18], [0.09,0.2]), 7; ([0.44,0.7], [0.02,0.18], [0.06,0.2])\}.\)
• 2. \(A B = \{2; ([0.28,0.44], [0.02,0.12], [0.06,0.25]), -2; ([0.36,0.58], [0.01,0.12], [0.04,0.25]), 1; ([0.37,0.6], [0.04,0.18], [0.09,0.2]), -3; ([0.44,0.7], [0.02,0.18], [0.06,0.2])\}.\)
• 3. \(A.B = \{3; ([0.02,0.06], [0.28,0.58], [0.44,0.7]), \\ 15; ([0.06,0.2], [0.19,0.58], [0.36,0.75]), \\ 2; ([0.03,0.1], [0.36,0.72], [0.51,0.58]), \\ 10; ([0.06,0.2], [0.28,0.72], [0.51,0.7])\}.\)
• 4. \(A/B = \{3; ([0.02,0.06], [0.28,0.58], [0.44,0.7]), 3/5; ([0.06,0.2], [0.19,0.58], [0.36,0.75]), 2; ([0.03,0.1], [0.36,0.72], [0.51,0.58]), 2/5; ([0.06,0.2], [0.28,0.72], [0.51,0.7])\}.\)
• 5. \(B = \{3; ([0.001,0.027], [0.008,0.065], [0.097,0.025]), \\ 15; ([0.008,0.016], [0.001,0.027], [0.008,0.025])\}.\)

4.1 New Operations of Interval Valued Neutrosophic Sets

4.1.1 Disjunctive Sum

Let A and B be two interval valued neutrosophic sets on a universal set X. We define the disjunctive sum of two interval valued neutrosophic sets such that

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] for all x in X. Then, we present the following on the disjunctive sum for the truth-membership function,

T_{A \cap B}c = \{ \min[\inf T_A(x), \inf F_B(x)], \min[\sup T_A(x), \sup F_B(x)] \},
T_{A^C \cap B} = \{ \min[\inf F_A(x), \inf T_B(x)], \min[\sup F_A(x), \sup T_B(x)] \},
T_{A \oplus B} = \{ \max(\min[\inf T_A(x), \inf F_B(x)], \min[\inf F_A(x), \inf T_B(x)] \},
\max(\min[\sup T_A(x), \sup F_B(x)], \min[\sup F_A(x), \sup T_B(x)] \},

Next, for the indeterminacy membership function,

\begin{split} I_{A \cap B^c} &= \{ \max[\inf I_A(x), 1 - \sup I_B(x)], \max[\sup I_A(x), 1 - \inf I_B(x)] \}, \\ I_{A^c \cap B} &= \{ \max[1 - \sup I_A(x), \inf I_B(x)], \max[1 - \inf I_A(x), \sup I_B(x)] \}, \\ I_{A \oplus B} &= \{ \min(\max[\inf I_A(x), 1 - \sup I_B(x)], \max[1 - \sup I_A(x), \inf I_B(x)] \}, \\ \min(\max[\sup I_A(x), 1 - \inf I_B(x)], \max[1 - \inf I_A(x), \sup I_B(x)] \}, \end{split}

Finally, for the falsity-membership function,

\begin{split} F_{A \cap B^c} &= \{ \max[\inf F_A(x), \inf T_B(x)], \max[\sup F_A(x), \sup T_B(x)] \}, \\ F_{A^c \cap B} &= \{ \max[\inf T_A(x), \inf F_B(x)], \max[\sup T_A(x), \sup F_B(x)] \}, \\ F_{A \oplus B} &= \{ \min(\max[\inf F_A(x), \inf T_B(x)], \max[\inf T_A(x), \inf F_B(x)] \}, \\ \min(\max[\sup F_A(x), \sup T_B(x)], \max[\sup T_A(x), \sup F_B(x)] ) \}. \end{split}

Therefore, we conclude that

A \oplus B = \{x; [\max(\min[\inf T_A(x), \inf F_B(x)], \min[\inf F_A(x), \inf T_B(x)]), \max(\min[\sup T_A(x), \sup F_B(x)], \min[\sup F_A(x), \sup T_B(x)])\}, 

[\min(\max[\inf I_A(x), 1 - \sup I_B(x)], \max[1 - \sup I_A(x), \inf I_B(x)]), 

\min(\max[\sup I_A(x), 1 - \inf I_B(x)], \max[1 - \inf I_A(x), \sup I_B(x)])\},

ሾሺሾ ሺሻ , ሺሻሿ, ሾ ሺሻ , ሺሻሿሻ, ሺሾ ሺሻ , ሺሻሿ, ሾ ሺሻ , ሺሻሿሻሿሽ.

Example 4.2.2.: Let ൌ ሼ ଵ; ሺሾ0.2,0.3ሿ, ሾ0.1,0.4ሿ, ሾ0.2,0.5ሿሻሽ and ൌ ሼଶ; ሺሾ0.1,0.2ሿ, ሾ0.2,0.3ሿ, ሾ0.3,0.5ሿሻሽ be two interval valued neutrosophic sets. Then, the disjunctive sum is equal to

\[A \oplus B\]
= \(\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\)
\([min(max[0.1,0.7] max[0.6,0.2]), min(max[0.4,0.8], max[0.9,0.3])],\)
\(\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\)

Hence, we obtain from the above equality

\[A \oplus B = \{x; ([0.2,0.3], [0.6,0.8], [0.2,0.5])\}.\]

4.1.2 Difference of Interval Valued Neutrosophic Sets

Let and be two interval valued neutrosophic sets on a universal set . We define the difference of two interval valued neutrosophic sets such that ⊝ ൌ ∩ , where

\[A = \begin{cases} x, [\inf T_{A}(x), \sup T_{A}(x)], [\inf I_{A}(x), \sup I_{A}(x)], \\ [\inf F_{A}(x), \sup F_{A}(x)] : x \in X \end{cases},\] \[B = \begin{cases} x, [\inf T_{B}(x), \sup T_{B}(x)], [\inf I_{B}(x), \sup I_{B}(x)], \\ [\inf F_{B}(x), \sup F_{B}(x)] : x \in X \end{cases},\] \[B^{c} = \begin{cases} \langle x: [\inf F_{B}(x), \sup F_{B}(x)], [1 - \sup I_{B}(x), 1 - \inf I_{B}(x)], \\ [\inf T_{B}(x), \sup T_{B}(x)] \end{cases}\] for all in . Then, we present the following on the difference for the truthmembership function, the indeterminacy membership function and the falsitymembership function, respectively.

\[T_{A\cap B^c} = \{ \min[\inf T_A(x), \inf F_B(x)], \min[\sup T_A(x), \sup F_B(x)] \},\] \[I_{A\cap B^c} = \{ \max[\inf I_A(x), 1 - \sup I_B(x)], \max[\sup I_A(x), 1 - \inf I_B(x)] \}\] and

\[F_{A \cap B^c} = \{ \max[\inf F_A(x), \inf T_B(x)], \max[\sup F_A(x), \sup T_B(x)] \}.\]

Hence, we have that

\[A \ominus B = \{x; (min[inf T_A(x), inf F_B(x)], min[sup T_A(x), sup F_B(x)]\},\]
\(\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\)
\(\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\)

Example 4.2.4: Using the values of example 4.2.2, we calculate the difference values and we obtain the following result:

\[A \ominus B = \begin{cases} x; (\min[0.2,0.3], \min[0.3,0.5]), \\ \max[0.4,0.8]), (\max[0.2,0.1], \max[0.5,0.2]) \end{cases}\]

Therefore.

\[A \ominus B = \{x; ([0.2,0.3], [0.7,0.8], [0.2,0.5])\}.\]

4.1.3 Cartesian Product of Interval Valued Neutrosophic Sets

Let A be an interval valued neutrosophic set on a universal set X. We define the power of the interval valued neutrosophic set A such that

\[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] \[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] \[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] \[\text{[rumus tidak dapat ditampilkan dengan baik — lihat PDF asli]}\] where \(\inf T_A^2(x) = [\inf T_A(x)]^2\), \(\sup T_A^2(x) = [\sup T_A(x)]^2\) and so are the other functions. Similarly, the \(m^{th}\) power of interval valued neutrosophic set \(A^m\) may be computed as

\[A^{m} = \{x, [\inf T_{A}^{m}(x), \sup T_{A}^{m}(x)], [\inf I_{A}^{m}(x), \sup I_{A}^{m}(x)], [\inf F_{A}^{m}(x), \sup F_{A}^{m}(x)] : x \in X\}.\]

Denoting \(T_{A_1}(x), T_{A_2}(x), ..., T_{A_n}(x), I_{A_1}(x), I_{A_2}(x), ..., I_{A_n}(x)\) and \(F_{A_1}(x), F_{A_2}(x), ..., F_{A_n}(x)\) as the truth-membership function, indeterminacy membership function and falsity membership function, respectively \(A_1, A_2, ..., A_n\) for all \(x_1, x_2, ..., x_n \in X\). Then, the probability for n-tuple \((x_1, x_2, ..., x_n)\) to be involved in a Cartesian product neutrosophic set is

\[\begin{split} A_1 \times A_2 \times ... \times A_n &= \{x; [min[inf \ T_{A_1}(x) \, , inf \ T_{A_2}(x) \, , ... \, , inf \ T_{A_n}(x)], \\ min[sup \ T_{A_1}(x) \, , sup \ T_{A_2}(x) \, , ... \, , sup \ T_{A_n}(x)] \, ], \\ [max[inf \ I_{A_1}(x) \, , inf \ I_{A_2}(x) \, , ... \, , inf \ I_{A_n}(x)], \\ max[sup \ I_{A_1}(x) \, , sup \ I_{A_2}(x) \, , ... \, , sup \ I_{A_n}(x)] \, ], \\ [max[inf \ F_{A_1}(x) \, , inf \ F_{A_2}(x) \, , ... \, , inf \ F_{A_n}(x)], \\ max[sup \ F_{A_1}(x) \, , sup \ F_{A_2}(x) \, , ... \, , sup \ F_{A_n}(x)] \, ]\}. \end{split}\]

Example 4.2.6: Let \(A = \{x_1; ([0.2,0.3], [0.1,0.4], [0.2,0.5])\}\) be an interval valued neutrosophic set on X. Then, the triple Cartesian product for A is

\[A^{3} = A \times A \times A = \{x; [min[0.2, 0.04, 0.008], min[0.3, 0.09, 0.027]], \\ [max[0.1, 0.01, 0.001], max[0.4, 0.16, 0.064]], \\ [max[0.2, 0.04, 0.008], max[0.5, 0.25, 0.125]]\}\]

Thus, we can easily see that

ଷ ൌ ሼ; ሾ0.008,0.027ሿ, ሾ0.1, 0.4ሿ, ሾ0.2, 0.5ሿሽ.

What we conclude from the result is that the values of the indeterminacy membership function and falsity membership function in the Cartesian product are the same with A, whereas the value of the truth membership is decreased since the values are between 0 and 1 and the powers are less than themselves.

5 Conclusion

In this work, some new concepts have been put forward, such as the disjunctive sum, difference ሺ, , ሻ-cut interval valued neutrosophic sets and the ሺ, , ሻcut strong interval valued neutrosophic sets of interval valued neutrosophic sets. Some related properties have been established with examples and counter examples. It is hoped that our work will enhance the understanding of interval valued neutrosophic sets.