ivnpiv-Neutrosophic soft sets and their decision making based on similarity measure

Abstract

In recent years, soft sets and neutrosophic sets have become a subject of great interest for researchers and have been widely studied based on decision-making problems. In this paper, we propose a new concept of the soft sets that is called interval-valued neutrosophic parameterized interval-valued neutrosophic soft sets (ivnpivn-soft sets). It is a generalization of the other soft sets such as fuzzy soft sets, intuitionistic fuzzy soft sets, neutrosophic soft sets, fuzzy parameterized soft sets, intuitionistic fuzzy parameterized soft sets, neutrosophic parameterized neutrosophic soft sets. Also, we proposed ivnpivn-soft matrices which are representative of the ivnpivn-soft sets. We then developed a decision-making method on the ivnpivn-soft sets and ivnpivn-soft matrices. Then, we proposed a numerical example to verify validity and feasibility of the developed method. Finally, the proposed method is compared with several different methods to verify its feasibility.

1 Introduction

Smarandache [31, 32] introduced the notation of neutrosophic set to reflect the truth, indeterminate and false information simultaneously in real problems. Additionally, since neutrosophic sets were difficult to be applied in practical problems, single-valued neutrosophic sets which are an extension of intuitionistic fuzzy sets [4] and fuzzy sets [40] are introduced by Wang et al. [38]. The theory is characterized by a truth-membership, indeterminacy-membership and falsity-membership that describe by crisp numbers in real numbers. Then, interval neutrosophic sets defined by Wang et al. [37] which can be described by three real unit interval in [0, 1]. However, because of the ambiguity and complexity of in the real world, it is difficult for decision-makers to precisely express their preference by using these sets. Therefore, Molodtsov developed the concept of soft sets by using parameter set for the inadequacy of the parameterization tool of the theories. In the literature, many conclusions and propositions are obtained to use in decision-making problems; some of them are given in [1–3, 20, 29, 34–36, 39, 41].

After Molodtsov, many researchers combines the soft sets with the fuzzy sets, intuitionistic fuzzy sets, interval-valued fuzzy sets and neutrosophic sets. For example, fuzzy soft sets [22], fuzzy parameterized soft sets [6, 8, 9], intuitionistic fuzzy soft sets [7, 23], intuitionistic fuzzy parameterized soft sets [12], interval-valued intuitionistic fuzzy parameterized soft sets [13], intuitionistic fuzzy parameterized fuzzy soft sets [16], neutrosophic soft sets [11, 21], interval-valued neutrosophic soft sets [14], interval-valued neutrosophic parameterized soft sets [10] and neutrosophic parameterized neutrosophic soft sets [15] have been studied by researchers. In these set theories, many conclusions and propositions are obtained to use in decision-making problems; some of them are given in [7, 11, 18, 26, 30, 33].

In this paper, we define a kind of soft sets called interval-valued neutrosophic parameterized interval-valued neutrosophic soft sets (ivnpivn-soft sets) which is generalization of npn-soft set [15]. The rest of paper is organized as follows. In Sect. 2, we review basic notions about neutrosophic sets, interval-valued neutrosophic sets, soft sets and neutrosophic parameterized neutrosophic soft sets. In Sect. 3, ivnpivn-soft sets and their operations are given. In Sect. 4, ivnpivn-soft matrices which are representative of the ivnpivn-soft sets have been introduced. In Sect. 5, we proposed a similarity measure and distance measures on ivnpivn-soft sets. In Sect. 6, we developed a decision-making method for ivnpivn-soft sets and give two illustrative example. In Sect. 7, the proposed method is compared with several extant methods to verify its feasibility. Finally, the conclusions are drawn.

2 Preliminary

In this section, we give the basic definitions and results of neutrosophic sets [31, 38], interval-valued neutrosophic sets [37], soft sets [28] and neutrosophic parameterized neutrosophic soft sets [15].

Definition 1

[38] Let U be a universe. A neutrosophic set $\widetilde{A}$ in U is characterized by a truth-membership function $T_{\widetilde{A}}$, an indeterminacy-membership function $I_{\widetilde{A}}$ and a falsity-membership function $F_{\widetilde{A}}$. $T_{\widetilde{A}}(u)$; $I_{\widetilde{A}}(u)$ and $F_{\widetilde{A}}(u)$ are real standard or nonstandard element of $^-[0,1]^+$. It can be written as

$$\begin{aligned} \widetilde{A}=\left\{ \langle u,(T_{\widetilde{A}}(u),I_{\widetilde{A}}(u),F_{\widetilde{A}}(u))\rangle :u\in U\right\} \end{aligned}$$

There is no restriction on the sum of $T_{\widetilde{A}}(u)$; $I_{\widetilde{A}}(u)$ and $F_{\widetilde{A}}(u)$, so $^-0\le T_{\widetilde{A}}(u) + I_{\widetilde{A}}(u) + F_{\widetilde{A}}(u)\le 3^+$.

Example 1

Suppose that the universe of discourse $U=\{u_1, u_2, u_3\}$. It may be further assumed that the values of $u_1$, $u_2$ and $u_3$ are in $^-[0,1]^+$. Then, $\widetilde{A}$ is a neutrosophic set of U, such that,

$$\begin{aligned} \widetilde{A}=\left\{ \langle u_1,(0.5,0.6,0.9)\rangle , \langle u_2,(0.4,0.2,0.7)\rangle , \langle u_3,(0.8,0.3,0.6)\rangle \right\} \end{aligned}$$

Definition 2

[37] Let U be a universe. An interval value neutrosophic set A in U is characterized by truth-membership function $T_A$, a indeterminacy-membership function $I_A$ and a falsity-membership function $F_A$. For each point $u \in U$; $T_A$, $I_A$ and $F_A \subseteq [0,1]$.

Thus, an interval value neutrosophic set A over U can be represented by

$$\begin{aligned} A= \{\langle T_A(u), I_A(u), F_A(u)\rangle /u:u\in U\} \end{aligned}$$

Here, $(T_A(u), I_A(u), F_A(u))$ is called interval value neutrosophic number for all $u \in U$ and all interval value neutrosophic numbers over U will be denoted by IVN(U).

Example 2

Suppose that the universe of discourse $U=\{u_1, u_2\}$ where $u_1$ and characterizes the quality, $u_2$ indicates the prices of the objects. It may be further assumed that the values of $u_1$ and $u_2$ are subset of [ 0, 1 ] and they are obtained from a expert person. The expert construct an interval value neutrosophic set the characteristics of the objects as follows;

$$\begin{aligned} A=\{\langle [0.1,1.0], [0.1,0.4], [0.4,0.7]\rangle /u_1,\langle [0.6,0.9], [0.8,1.0], [0.4,0.6] \rangle /u_2\} \end{aligned}$$

Definition 3

[37] Let A and B be two interval-valued neutrosophic sets. Then, for all $u \in U$,

• 1. A is empty, denoted by $A=\widehat{{\emptyset }}$, and is defined by $\widehat{{\emptyset }}= \{ \langle [ 0, 0],[ 1, 1],[ 1,1] \rangle /u:u\in U\}$

• 2. A is universal, denoted by $A=\widehat{{X}}$, and is defined by $\widehat{{X}}= \{\langle [ 1, 1],[ 0, 0],[ 0,0] \rangle /u:u\in U\}$

• 3. The complement of A, denoted by $A^c$, is defined by

  $$\begin{aligned} A^{\widehat{c}}= \{\langle [\hbox {inf}F_A(u),\hbox {sup}F_A(u)],[ 1-\hbox {sup} I_A(u),1-\hbox {inf} I_A(u)],[\hbox {inf}T_A(u),\hbox {sup}T_A(u)] \rangle /u:u\in U\} \end{aligned}$$

• 4. $A\widehat{\subseteq }B$ $\Leftrightarrow$ $[ \hbox {inf} T_A(u) \le \hbox {inf} T_B(u),$ $\hbox {sup} T_A(u)\le \hbox {sup} T_B(u),$ $\hbox {inf} I_A(u)\ge \hbox {inf} I_B(u),$ $\hbox {sup} I_A(u)\ge \hbox {sup} I_B(u),$ $\hbox {inf} F_A(u)\ge \hbox {inf} F_B(u),$ $\hbox {sup}F_A(u)\ge \hbox {sup}F_B(u)]$.

• 5. Intersection of A and B, denoted by $A\,\widehat{\cap }\, B$, is defined by

  $$\begin{aligned} A\,\widehat{\cap } \,B&= \{\langle [ \hbox {min}(\hbox {inf} T_A(u), \hbox {inf} T_B(u)),\hbox {min}(\hbox {sup} T_A(u), \hbox {sup} T_B(u))],\\&[ \hbox {max}(\hbox {inf} I_A(u), \hbox {inf} I_B(u)), \hbox {max}(\hbox {sup} IA(x), \hbox {sup} I_B(u)) ],\\&[ \hbox {max}(\hbox {inf} F_A(u), \hbox {inf} F_B(u)), \hbox {max}(\hbox {sup}F_A(u),\hbox {sup} F_B(u))] \rangle /u:u\in U\} \end{aligned}$$

• 6. Union of A and B, denoted by $A\,\widehat{\cup } \,B$, is defined by

  $$\begin{aligned} A\,\widehat{\cup } \,B &= \{ \langle [ \hbox {max}(\hbox {inf} T_A(u), \hbox {inf} T_B(u)),\hbox {max}(\hbox {sup} T_A(u), \hbox {sup} T_B(u))],\\&[\hbox {min}(\hbox {inf} I_A(u), \hbox {inf} I_B(u)), \hbox {min}(\hbox {sup} I_A(u), \hbox {sup} I_B(u))],\\&[\hbox {min}(\hbox {inf} F_A(u), \hbox {inf} F_B(u)), \hbox {min}(\hbox {sup}F_A(u), \hbox {sup} F_B(u))] \rangle /u:u\in U\} \end{aligned}$$

Definition 4

[17] t-norms are associative, monotonic and commutative two valued functions t that map from $[0,1]\times [0,1]$ into [0, 1]. These properties are formulated with the following conditions: $\forall a,b,c,d \in [0,1],$

• 1. $t(0,0)=0$ and $t(a,1)=t(1,a)=a,$

• 2. If $a\le c$ and $b\le d$, then $t(a,b)\le t(c,d)$

• 3. $t(a,b)= t(b,a)$

• 4. $t(a,t(b,c))=t(t(a,b),c)$

Definition 5

[17] t-conorms (s-norm) are associative, monotonic and commutative two placed functions s which map from $[0,1]\times [0,1]$ into [0, 1]. These properties are formulated with the following conditions: $\forall a,b,c,d \in [0,1],$

• 1. $s(1,1)=1 \,\hbox {and} \, s(a,0)=s(0,a)=a,$

• 2. if $a\le c$ and $b\le d$, then $s(a,b)\le s(c,d)$

• 3. $s(a,b)= s(b,a)$

• 4. $s(a,s(b,c))=s(s(a,b),c)$

Definition 6

[28] Let U be a universe, X be a set of parameters that are describe the elements of U and P(U) be the power set of U. Then, a soft set f over U is a function defined by

$$\begin{aligned} f:{X}\rightarrow P(U) \end{aligned}$$

In other words, the soft set is a parameterized family of subsets of the set U. A soft set over U can be represented by the set of ordered pairs

$$\begin{aligned} f=\{ (x,f(x)):x\in {X}\} \end{aligned}$$

Example 3

Suppose that $U=\{u_1,u_2,u_3,u_4,u_5,u_6\}$ is the universe contains six house under consideration in a real agent and $X=\{x_1=\hbox {cheap}, x_2=\hbox {beatiful}, x_3=\hbox {green surroundings}, x_4=\hbox {costly}, x_5= \hbox {large}\}$.

If a customer to select a house from the real agent, then he/she can construct a soft set f that describes the characteristic of houses according to own requests. Assume that $f(x_1)= \{u_1,u_2\}$, $f(x_2)= \{u_1 \}$, $f(x_3)= \emptyset$, $f(x_4)= U$, $\{u_1,u_2,u_3,u_4,u_5\}$ then the soft set f is written by

$$\begin{aligned} f=\{(x_1, \{u_1,u_2\}),(x_2, \{u_1,u_4,u_5,u_6\}), (x_4,U), (x_5,\{u_1,u_2,u_3,u_4,u_5\})\} \end{aligned}$$

Definition 7

[28] Let f and g be two soft sets over U. Then,

• 1. f is called an empty soft set, denoted by $\Phi _{X}$, if $f(x)=\emptyset ,$ for all $x\in X$.

• 2. f is called a universal soft set, denoted by $f_{\widetilde{X}}$, if $f(x)=U$, for all $x\in X$.

• 3. Im$\left( f\right) =\left\{ f(x):x\in X\right\}$ is called image of f.

• 4. f is a soft subset of g, denoted by $f\,\overline{\subseteq }\,g$, if $f(x)\subseteq g(x)$ for all $x\in X$.

• 5. f and g are soft equal, denoted by $f=g$, if and only if $f(x)=g(x)$ for all $x\in X$.

• 6. $(f\,\overline{\cup }\,g)(x)=f(x)\cup g(x) \text { for all } x\in X$ is called union of f and g.

• 7. $(f\,\overline{\cap }\,g)(x)=f(x)\cap g(x) \text { for all } x\in X$ is called intersection of f and g.

• 8. $f^{\overline{c}}(x)=U\setminus f(x) \text { for all } x\in X$ is called complement of f.

Definition 8

[15] Let U be a universe, N(U) be the set of all neutrosophic sets on U, X be a set of parameters that describe the elements of U and K be a neutrosophic set over X. Then, a neutrosophic parameterized neutrosophic soft set (npn-soft set) N over U is a set defined by a set valued function $f_N$ representing a mapping

$$\begin{aligned} f_N: K \rightarrow N(U) \end{aligned}$$

where $f_N$ is called approximate function of the npn-soft set N. For $x\in X$, the set $f_N(x)$ is called x-approximation of the npn-soft set N which may be arbitrary; some of them may be empty and some may have a nonempty intersection. It can be written a set of ordered pairs,

$$\begin{aligned} N= \big \{(<x,T_{{N}}(x),I_{{N}}(x),F_{{N}}(x)>, \{<u,T_{f_N(x)}(u),I_{f_N(x)}(u),F_{f_N(x)}(u)>:u\in U\}): x\in X \big \} \end{aligned}$$

where

$$\begin{aligned} F_{{N}}(x),I_{{N}}(x),T_{{N}}(x),T_{f_N(x)}(u),I_{f_N(x)}(u),F_{f_N(x)}(u)\in [0,1] \end{aligned}$$

Example 4

Let $U=\{u_1, u_2, u_3,u_4\}$, $X=\{x_1, x_2 \}$. N be a npn-soft sets as

$$\begin{aligned} N &= \bigg \{(\langle x_1,(0.1,0.2,0.3)\rangle ,\{\langle u_1,(0.4,0.6,0.6)\rangle , \langle u_2,(0.6,0.0,0.1)\rangle , \langle u_3,(0.3,0.4,0.4)\rangle ,\\&\langle u_4,(0.5,0.6,0.2)\rangle \}),(\langle x_2,(0.9,0.1,0.5)\rangle , \langle u_1,(0.3,0.8,0.6)\rangle , \langle u_2,(0.7,0.6,0.5)\rangle ,\\&\langle u_3,(0.6,0.6,0.9)\rangle , \langle u_4,(0.4,0.1,0.5)\rangle ,\})\bigg \} \end{aligned}$$

Definition 9

[15] Let N, $N_1$ and $N_2$ be three npn-soft sets. Then,

• 1. Complement of an npn-soft set N, denoted by $N^{\tilde{c}}$, is defined by

  $$\begin{aligned} N^{\tilde{c}}= \big \{(\langle x,F_{{N}}(x),1-I_{{N}}(x),T_{{N}}(x) \rangle , \{\langle u,F_{f_N(x)}(u),1-I_{f_N(x)}(u),T_{f_N(x)}(u) \rangle :u\in U\}): x\in X \big \} \end{aligned}$$

• 2. Union of $N_1$ and $N_2$, denoted by $N_3=N_1\tilde{\cup } N_2$, is defined by

  $$\begin{aligned} {N_3}= \big \{(\langle x,T_{{N_3}}(x),I_{{N_3}}(x),F_{{N_3}}(x)\rangle , \{\langle u,T_{f_{N_3(x)}}(u),I_{f_{N_3(x)}}(u),F_{f_{N_3(x)}}(u)\rangle :u\in U\}): x\in X\} \end{aligned}$$

  where

  $$\begin{aligned} &T_{{N_3}}(x)= s(T_{{N_1}}(x),T_{{N_2}}(x)),\quad T_{f_{N_3(x)}}(u)=s(T_{f_{N_1(x)}}(u),T_{f_{N_2(x)}}(u)),\\ &I_{{N_3}}(x)= t(I_{{N_1}}(x),I_{{N_2}}(x)), \quad I_{f_{N_3(x)}}(u)=t(I_{f_{N_1(x)}}(u),I_{f_{N_2(x)}}(u)),\\& F_{{N_3}}(x)= t(F_{{N_1}}(x),F_{{N_2}}(x)), \quad F_{f_{N_3(x)}}(u)=t(F_{f_{N_1(x)}}(u),F_{f_{N_2(x)}}(u)) \end{aligned}$$

• 3. Intersection of $N_1$ and $N_2$, denoted by $N_4=N_1\tilde{\cap } N_2$, is defined by

  $$\begin{aligned} {N_4}= \big \{(\langle x,T_{{N_4}}(x),I_{{N_4}}(x),F_{{N_4}}(x)\rangle , \{\langle u,T_{f_{N_4(x)}}(u),I_{f_{N_4(x)}}(u),F_{f_{N_4(x)}}(u)\rangle :u\in U\}): x\in X\} \end{aligned}$$

  where

  $$\begin{aligned} T_{{N_4}}(x)&= t(T_{{N_1}}(x),T_{{N_2}}(x)),\quad T_{f_{N_4(x)}}(u)=t(T_{f_{N_1(x)}}(u),T_{f_{N_2(x)}}(u)),\\ I_{{N_4}}(x)&= s(I_{{N_1}}(x),I_{{N_2}}(x)), \quad I_{f_{N_4(x)}}(u)=s(I_{f_{N_1(x)}}(u),I_{f_{N_2(x)}}(u)),\\ F_{{N_4}}(x)&= s(F_{{N_1}}(x),F_{{N_2}}(x)), \quad F_{f_{N_4(x)}}(u)=s(F_{f_{N_1(x)}}(u),F_{f_{N_2(x)}}(u)) \end{aligned}$$

3 ivnpivn-soft sets

In this section, we present concept of “interval-valued neutrosophic parameterized interval-valued neutrosophic soft sets” is abbreviated as “ivnpivn-soft sets”. Then, we introduce some definitions and operations on ivnpivn-soft set and some properties of the sets which are connected to operations have been established.

In the following, some definitions and operations are defined on npn-soft set in [15] and on interval-valued neutrosophic soft set in [14] and we extended these definitions and operations to ivnpivn-soft sets.

Definition 10

Let U be a universe, IVN(U) be a set of all interval-valued neutrosophic sets over U, X be a set of parameters that are describe the elements of U and K be a interval-valued neutrosophic set over X. Then, an ivnpivn-soft set ${\mathcal {F}}$ over U is a set defined by a set valued function f representing a mapping

$$\begin{aligned} f: K \rightarrow IVN(U) \end{aligned}$$

where f is called approximate function of the ivnpivn-soft set ${\mathcal {F}}$. For $x\in X$, the set f(x) is called x-approximation of the ivnpivn-soft set ${\mathcal {F}}$ which may be arbitrary; some of them may be empty and some may have a nonempty intersection. It can be written a set of ordered pairs,

$$\begin{aligned} {\mathcal {F}}= \big \{(\langle x,T_{{\mathcal {F}}}(x),I_{{\mathcal {F}}}(x),F_{{\mathcal {F}}}(x)\rangle , \{\langle T_{f_{\mathcal {F}}(x)}(u),I_{f_{\mathcal {F}}(x)}(u),F_{f_{\mathcal {F}}(x)}(u)\rangle /u:u\in U \}): x\in X \big \} \end{aligned}$$

where

$$\begin{aligned} T_{\mathcal {F}}(x),F_{\mathcal {F}}(x),I_{\mathcal {F}}(x),T_{f_{\mathcal {F}}(x)}(u),I_{f_{\mathcal {F}}(x)}(u),F_{f_{\mathcal {F}}(x)}(u)\subseteq [0,1] \end{aligned}$$

and

$$\begin{aligned} &T_{\mathcal {F}}(x)= [\hbox {inf}T_{\mathcal {F}}(x), \hbox {sup}T_{\mathcal {F}}(x)],\\& I_{\mathcal {F}}(x)= [\hbox {inf}I_{\mathcal {F}}(x), \hbox {sup}I_{\mathcal {F}}(x)],\\& F_{\mathcal {F}}(x)= [\hbox {inf}F_{\mathcal {F}}(x), \hbox {sup}F_{\mathcal {F}}(x)],\\& T_{f_{\mathcal {F}}(x)}(u)= [\hbox {inf}T_{f_{\mathcal {F}}(x)}(u), \hbox {sup}T_{f_{\mathcal {F}}(x)}(u)],\\& I_{f_{\mathcal {F}}(x)}(u)= [\hbox {inf}I_{f_{\mathcal {F}}(x)}(u), \hbox {sup}I_{f_{\mathcal {F}}(x)}(u)],\\& F_{f_{\mathcal {F}}(x)}(u)=[\hbox {inf}F_{f_{\mathcal {F}}(x)}(u), \hbox {sup}F_{f_{\mathcal {F}}(x)}(u)] \end{aligned}$$

From now on, the set of all ivnpivn-soft sets over U will be denoted by $\widetilde{\mathcal {F}}$.

Example 5

Assume that $U=\{u_1, u_2, u_3\}$ and $X=\{x_1, x_2, x_3\}$, then an ivnpivn-soft set can be written as

$$\begin{aligned} {\mathcal {F}}&= \big \{(\langle x_1,[0.3,0.4],[0.5,0.6],[0.4,0.5]\rangle ,\{\langle [0.5,0.6],[0.6,0.7],[0.3,0.4]\rangle /u_1,\\&\langle [0.4,0.5],[0.7,0.8],[0.2,0.3]\rangle /u_2, \langle [0.6,0.7],[0.2,0.3],[0.3,0.5]\rangle /u_3 \}),\\&(\langle x_2,[0.1,0.2],[0.3,0.4],[0.6,0.7]\rangle ,\{\langle [0.7,0.8],[0.3,0.4],[0.2,0.4]\rangle /u_1,\\&\langle [0.8,0.4],[0.2,0.6],[0.3,0.4]\rangle /u_2, \langle [0.4,0.5],[0.1,0.3],[0.2,0.4] \rangle /u_3 \}),\\&(\langle x_3,[0.2,0.4],[0.4,0.5],[0.4,0.6]\rangle ,\{\langle [0.2,0.3],[0.1,0.4],[0.3,0.6]\rangle /u_1,\\&\langle [0.2,0.5],[0.1,0.6],[0.5,0.8]\rangle /u_2, \langle [0.3,0.7],[0.1,0.3],[0.6,0.7]\rangle /u_3 \})\big \} \\ \end{aligned}$$

Definition 11

Let ${\mathcal {F}}\in \widetilde{\mathcal {F}}$. Then, ${\mathcal {F}}$ is called

• 1. An empty ivnpivn-soft set, denoted by $\mathcal {O}$, is defined as:

  $$\begin{aligned} \mathcal {O}= \big \{(\langle x,[0,0],[1,1],[1,1]\rangle, \{\langle [0,0],[1,1],[1,1]\rangle/u:u\in U\}): x\in X \big \} \end{aligned}$$

• 2. a universal ivnpivn-soft set, denoted by ${\mathcal {U}}$,

  $$\begin{aligned} {\mathcal {U}}= \big \{(\langle x,[1,1],[0,0],[0,0]\rangle, \{\langle [1,1],[0,0],[0,0]\rangle /u:u\in U\}): x\in X \big \} \end{aligned}$$

Definition 12

Let ${\mathcal {F}},{\mathcal {F}}_1,{\mathcal {F}}_2\in \widetilde{\mathcal {F}}$. Then,

• 1. ${\mathcal {F}}_1$ is a sub$-ivnpivn$-soft set of ${\mathcal {F}}_2$, denoted by ${\mathcal {F}}_1 \widetilde{\subseteq }{\mathcal {F}}_2$, if and only if

  $$\begin{aligned} \begin{array}{ll} \hbox {inf}T_{{\mathcal {F}_1}}(x)\le \hbox {inf}T_{{\mathcal {F}_2}}(x), & \hbox {sup}T_{{\mathcal {F}_1}}(x)\le \hbox {sup}T_{{\mathcal {F}_2}}(x), \\ \hbox {inf}I_{{\mathcal {F}_1}}(x)\ge \hbox {inf}I_{{\mathcal {F}_2}}(x),& \hbox {sup}I_{{\mathcal {F}_1}}(x)\ge \hbox {sup}I_{{\mathcal {F}_2}}(x),\\ \hbox {inf}F_{{\mathcal {F}_1}}(x)\ge \hbox {inf}F_{{\mathcal {F}_2}}(x),& \hbox {sup}F_{{\mathcal {F}_1}}(x)\ge \hbox {sup}F_{{\mathcal {F}_2}}(x), \\ \hbox {inf}T_{f_{\mathcal {F}_1(x)}}(u)\le \hbox {inf}T_{f_{\mathcal {F}_2(x)}}(u),& \hbox {sup}T_{f_{\mathcal {F}_1(x)}}(u)\le \hbox {sup}T_{f_{\mathcal {F}_2(x)}}(u), \\ \hbox {inf}I_{f_{\mathcal {F}_1(x)}}(u)\ge \hbox {inf}I_{f_{\mathcal {F}_2(x)}}(u),& \hbox {sup}I_{f_{\mathcal {F}_1(x)}}(u)\ge \hbox {sup}I_{f_{\mathcal {F}_2(x)}}(u), \\ \hbox {inf}F_{f_{\mathcal {F}_1(x)}}(u)\ge \hbox {inf}F_{f_{\mathcal {F}_2(x)}}(u),& \hbox {sup}F_{f_{\mathcal {F}_1(x)}}(u)\ge \hbox {sup}F_{f_{\mathcal {F}_2(x)}}(u). \end{array} \end{aligned}$$

• 2. Complement of ${\mathcal {F}}$, denoted by ${\mathcal {F}}^{\widetilde{c}}$, is defined by

  $$\begin{aligned} {\mathcal {F}}^{\widetilde{c}} = \big \{(\langle x,F_{\mathcal {F}}(x),1-I_{\mathcal {F}}(x),T_{\mathcal {F}}(x) \rangle , \{\langle F_{f_{\mathcal {F}}(x)}(u),1-I_{f_{\mathcal {F}}(x)}(u),T_{f_{\mathcal {F}}(x)}(u) \rangle /u:u\in U\}): x\in X \big \} \end{aligned}$$

  where

  $$\begin{aligned} &T_{\mathcal {F}}(x)= [\hbox {inf}T_{\mathcal {F}}(x), \hbox {sup}T_{\mathcal {F}}(x)],\\& 1-I_{\mathcal {F}}(x)= [1-\hbox {sup}I_{\mathcal {F}}(x), 1-infI_{\mathcal {F}}(x)],\\& F_{\mathcal {F}}(x)= [\hbox {inf}F_{\mathcal {F}}(x), \hbox {sup}F_{\mathcal {F}}(x)],\\& T_{f_{\mathcal {F}}(x)}(u)= [\hbox {inf}T_{f_{\mathcal {F}}(x)}(u), \hbox {sup}T_{f_{\mathcal {F}}(x)}(u)],\\& 1-I_{f_{\mathcal {F}}(x)}(u)= [1-\hbox {sup}I_{f_{\mathcal {F}}(x)}(u), 1-\hbox {inf}I_{f_{\mathcal {F}}(x)}(u)],\\& F_{f_{\mathcal {F}}(x)}(u)= [\hbox {inf}F_{f_{\mathcal {F}}(x)}(u),\hbox {sup}F_{f_{\mathcal {F}}(x)}(u)]. \end{aligned}$$

• 3. Union of ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$, denoted by ${\mathcal {F}}_3={\mathcal {F}}_1\tilde{\cup } {\mathcal {F}}_2$, is defined by

  $$\begin{aligned} {\mathcal {F}}_3= \big \{(\langle x,T_{{\mathcal {F}_3}}(x),I_{{\mathcal {F}_3}}(x),F_{{\mathcal {F}_3}}(x) \rangle , \{\langle T_{f_{\mathcal {F}_3(x)}}(u),I_{f_{\mathcal {F}_3(x)}}(u),F_{f_{\mathcal {F}_3(x)}}(u) \rangle /u:x\in U\}): x\in X\} \end{aligned}$$

  where

  $$\begin{aligned} &T_{{\mathcal {F}_3}}(x)= [s(\hbox {inf}T_{{\mathcal {F}_1}}(x),\hbox {inf}T_{{\mathcal {F}_2}}(x)),s(\hbox {sup}T_{{\mathcal {F}_1}}(x),\hbox {sup}T_{{\mathcal {F}_2}}(x))],\\& I_{{\mathcal {F}_3}}(x)= [t(\hbox {inf}I_{{\mathcal {F}_1}}(x),\hbox {inf}I_{{\mathcal {F}_2}}(x)),t(\hbox {sup}I_{{\mathcal {F}_1}}(x),\hbox {sup}I_{{\mathcal {F}_2}}(x))],\\& F_{{\mathcal {F}_3}}(x)= [t(\hbox {inf}F_{{\mathcal {F}_1}}(x),\hbox {inf}F_{{\mathcal {F}_2}}(x)),t(\hbox {sup}F_{{\mathcal {F}_1}}(x),\hbox {sup}F_{{\mathcal {F}_2}}(x))],\\& T_{f_{\mathcal {F}_3(x)}}(u)= [s(\hbox {inf}T_{f_{\mathcal {F}_1(x)}}(u),\hbox {inf}T_{f_{\mathcal {F}_2(x)}}(u)),s(\hbox {sup}T_{f_{\mathcal {F}_1(x)}}(u),\hbox {sup}T_{f_{\mathcal {F}_2(x)}}(u))],\\& I_{f_{\mathcal {F}_3(x)}}(u)= [t(\hbox {inf}I_{f_{\mathcal {F}_1(x)}}(u),\hbox {inf}I_{f_{\mathcal {F}_2(x)}}(u)),t(\hbox {sup}I_{f_{\mathcal {F}_1(x)}}(u),\hbox {sup}I_{f_{\mathcal {F}_2(x)}}(u))],\\& F_{f_{\mathcal {F}_3(x)}}(u)= [t(\hbox {inf}F_{f_{\mathcal {F}_1(x)}}(u),\hbox {inf}F_{f_{\mathcal {F}_2(x)}}(u)),t(\hbox {sup}F_{f_{\mathcal {F}_1(x)}}(u),\hbox {sup}F_{f_{\mathcal {F}_2(x)}}(u))]. \end{aligned}$$

• 4. Intersection of ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$, denoted by ${\mathcal {F}}_4={\mathcal {F}}_1\tilde{\cap } {\mathcal {F}}_2$, is defined by

  $$\begin{aligned} {\mathcal {F}}_4= \big \{(\langle x,T_{{\mathcal {F}_4}}(x),I_{{\mathcal {F}_4}}(x),F_{{\mathcal {F}_4}}(x)\rangle , \{\langle T_{f_{\mathcal {F}_4(x)}}(u),I_{f_{\mathcal {F}_4(x)}}(u),F_{f_{\mathcal {F}_4(x)}}(u) \rangle /u:x\in U\}): x\in X\} \end{aligned}$$

  where

  $$\begin{aligned} &T_{{\mathcal {F}_4}}(x)= [t(\hbox {inf}T_{{\mathcal {F}_1}}(x),\hbox {inf}T_{{\mathcal {F}_2}}(x)),t(\hbox {sup}T_{{\mathcal {F}_1}}(x),\hbox {sup}T_{{\mathcal {F}_2}}(x))],\\& I_{{\mathcal {F}_4}}(x)= [s(\hbox {inf}I_{{\mathcal {F}_1}}(x),\hbox {inf}I_{{\mathcal {F}_2}}(x)),s(\hbox {sup}I_{{\mathcal {F}_1}}(x),\hbox {sup}I_{{\mathcal {F}_2}}(x))],\\& F_{{\mathcal {F}_4}}(x)= [s(\hbox {inf}F_{{\mathcal {F}_1}}(x),\hbox {inf}F_{{\mathcal {F}_2}}(x)),s(\hbox {sup}F_{{\mathcal {F}_1}}(x),\hbox {sup}F_{{\mathcal {F}_2}}(x))],\\& T_{f_{\mathcal {F}_4(x)}}(u)= [t(\hbox {inf}T_{f_{\mathcal {F}_1(x)}}(u),\hbox {inf}T_{f_{\mathcal {F}_2(x)}}(u)),t(\hbox {sup}T_{f_{\mathcal {F}_1(x)}}(u),\hbox {sup}T_{f_{\mathcal {F}_2(x)}}(u))],\\& I_{f_{\mathcal {F}_4(x)}}(u)= [s(\hbox {inf}I_{f_{\mathcal {F}_1(x)}}(u),\hbox {inf}I_{f_{\mathcal {F}_2(x)}}(u)),s(\hbox {sup}I_{f_{\mathcal {F}_1(x)}}(u),\hbox {sup}I_{f_{\mathcal {F}_2(x)}}(u))],\\& F_{f_{\mathcal {F}_4(x)}}(u)= [s(\hbox {inf}F_{f_{\mathcal {F}_1(x)}}(u),\hbox {inf}F_{f_{\mathcal {F}_2(x)}}(u)),s(\hbox {sup}F_{f_{\mathcal {F}_1(x)}}(u),\hbox {sup}F_{f_{\mathcal {F}_2(x)}}(u))]. \end{aligned}$$

Example 6

Let $U=\{u_1, u_2, u_3\}$, $X=\{x_1, x_2, x_3\}$, ${\mathcal {F}}_1$ be given as in Example 5 and ${\mathcal {F}}_2$ be given as follows

$$\begin{aligned} {\mathcal {F}}_2&= \bigg \{(\langle x_1,[0.6,0.7],[0.2,0.9],[0.7,0.8]\rangle ,\{\langle [0.1,0.5],[0.7,0.9],[0.4,0.7]\rangle /u_1,\\&\langle [0.1,0.3],[0.2,0.4],[0.8,0.9]\rangle /u_2, \langle [0.3,0.5],[0.7,0.9],[0.6,0.7] \rangle /u_3 \}),\\&(\langle x_2,[0.1,0.4],[0.2,0.4],[0.5,0.8]\rangle ,\{\langle [0.2,0.3],[0.6,0.8],[0.1,0.2]\rangle /u_1,\\&\langle [0.5,0.9],[0.4,0.6],[0.7,0.9]\rangle /u_2, \langle [0.1,0.2],[0.3,0.4],[0.5,0.6]\rangle /u_3 \}),\\&(\langle x_3,[0.3,0.9],[0.5,0.7],[0.3,0.6]\rangle ,\{\langle [0.3,0.5],[0.2,0.6],[0.4,0.8]\rangle /u_1,\\&\langle [0.3,0.7],[0.2,0.8],[0.6,0.9]\rangle /u_2, \langle [0.4,0.9],[0.2,0.5],[0.6,0.9]\rangle /u_3 \})\bigg \} \\ \end{aligned}$$

Then,

$$\begin{aligned} ({\mathcal {F}}_1)^{\widetilde{c}}&= \bigg \{(\langle x_1,[0.4,0.5],[0.5,0.4],[0.3,0.4] \rangle ,\{\langle [0.3,0.4],[0.4,0.3],[0.5,0.6]\rangle /u_1,\\&\langle [0.2,0.3],[0.3,0.2],[0.4,0.5]\rangle /u_2, \langle [0.3,0.5],[0.8,0.7],[0.6,0.7]\rangle /u_3 \}),\\&(\langle x_2,[0.6,0.7],[0.7,0.6],[0.1,0.2]\rangle ,\{\langle [0.2,0.4],[0.7,0.6],[0.7,0.8] \rangle /u_1,\\&\langle [0.3,0.4],[0.8,0.4],[0.8,0.4]\rangle /u_2, \langle [0.2,0.4],[0.9,0.7],[0.4,0.5]\rangle /u_3 \}),\\&(\langle x_3,[0.4,0.6],[0.6,0.5],[0.2,0.4]\rangle ,\{\langle [0.3,0.6],[0.9,0.6],[0.2,0.3]\rangle /u_1,\\&\langle [0.5,0.8],[0.9,0.4],[0.2,0.5] \rangle /u_2, \langle [0.6,0.7],[0.9,0.7],[0.3,0.7] \rangle /u_3 \})\bigg \} \end{aligned}$$

Let us consider the t-norm $min\{a,b\}$ and s-norm $max\{a,b\}$. Then,

$$\begin{aligned} {\mathcal {F}}_1\,\widetilde{\cup }\,{\mathcal {F}}_2&= \bigg \{(\langle x_1,[0.6,0.7],[0.2,0.6],[0.4,0.5]\rangle ,\{\langle [0.5,0.6],[0.6,0.7],[0.3,0.4]\rangle /u_1,\\&\langle [0.4,0.5],[0.2,0.4],[0.2,0.3]\rangle /u_2, \langle [0.6,0.7],[0.2,0.3],[0.3,0.5] \rangle /u_3 \}),\\&(\langle x_2,[0.1,0.4],[0.2,0.4],[0.5,0.7]\rangle ,\{\langle [0.7,0.8],[0.3,0.4],[0.1,0.2]\rangle /u_1,\\&\langle [0.8,0.9],[0.2,0.6],[0.3,0.4] \rangle /u_2, \langle [0.4,0.5],[0.1,0.3],[0.2,0.4]\rangle /u_3 \}),\\&(\langle x_3,[0.3,0.9],[0.4,0.5],[0.3,0.6]\rangle ,\{\langle [0.3,0.5],[0.1,0.4],[0.3,0.6]\rangle /u_1,\\&\langle [0.3,0.7],[0.1,0.6],[0.5,0.8]\rangle /u_2,\langle [0.4,0.9],[0.1,0.3],[0.6,0.7]\rangle /u_3 \})\bigg \} \\ \end{aligned}$$

and

$$\begin{aligned} {\mathcal {F}}_1\,\widetilde{\cap }\,{\mathcal {F}}_2&= \bigg \{(\langle x_1,[0.3,0.4],[0.5,0.9],[0.7,0.8]\rangle ,\{\langle [0.1,0.5],[0.7,0.9],[0.4,0.7]\rangle /u_1,\\&\langle [0.1,0.3],[0.7,0.8],[0.8,0.9]\rangle /u_2, \langle [0.3,0.5],[0.7,0.9],[0.6,0.7]\rangle /u_3 \}),\\&(\langle x_2,[0.1,0.2],[0.3,0.4],[0.5,0.8]\rangle ,\{\langle [0.2,0.3],[0.6,0.8],[0.2,0.4]\rangle /u_1,\\&\langle [0.5,0.4],[0.4,0.6],[0.7,0.9]\rangle /u_2,\langle [0.1,0.2],[0.3,0.4],[0.5,0.6]\rangle /u_3 \}),\\&(\langle x_3,[0.2,0.4],[0.5,0.7],[0.4,0.6]\rangle ,\{\langle [0.2,0.3],[0.2,0.6],[0.4,0.8]\rangle /u_1,\\&\langle [0.2,0.5],[0.2,0.8],[0.6,0.9]\rangle /u_2, \langle [0.3,0.7],[0.2,0.5],[0.6,0.9]\rangle /u_3 \})\bigg \} \\ \end{aligned}$$

Proposition 1

Let ${\mathcal {F}}\in \widetilde{\mathcal {F}}$. Then,

• 1. $({\mathcal {F}}^{\widetilde{c}} \big )^{\widetilde{c}} ={\mathcal {F}}$

• 2. $\mathcal {O}^{\widetilde{c}} ={\mathcal {U}}$

• 3. ${\mathcal {F}}\,\widetilde{\subseteq } \,{\mathcal {U}}$

• 4. $\mathcal {O} \,\widetilde{\subseteq }\,{\mathcal {F}}$

• 5. ${\mathcal {F}}\,\widetilde{\subseteq }\,{\mathcal {F}}$

Proposition 2

Let ${\mathcal {F}}_1, {\mathcal {F}}_2, {\mathcal {F}}_3\in \widetilde{\mathcal {F}}$. Then,

• 1. ${\mathcal {F}}_1 \,\widetilde{\subseteq }\,{\mathcal {F}}_2 \wedge {\mathcal {F}}_2 \,\widetilde{\subseteq }\,{\mathcal {F}}_3$ $\Rightarrow$ ${\mathcal {F}}_1\,\widetilde{\subseteq }\,{\mathcal {F}}_3$

• 2. ${\mathcal {F}}_1={\mathcal {F}}_2\wedge {\mathcal {F}}_2={\mathcal {F}}_3$ $\Leftrightarrow$ ${\mathcal {F}}_1={\mathcal {F}}_3$

• 3. ${\mathcal {F}}_1 \,\widetilde{\subseteq }\,{\mathcal {F}}_2 \wedge {\mathcal {F}}_2 \,\widetilde{\subseteq }\,{\mathcal {F}}_1$ $\Rightarrow$ ${\mathcal {F}}_1={\mathcal {F}}_2$

Proposition 3

Let ${\mathcal {F}}_1, {\mathcal {F}}_2, {\mathcal {F}}_3 \in \widetilde{\mathcal {F}}$. Then,

• 1. ${\mathcal {F}}_1 \,\widetilde{\cup }\,{\mathcal {F}}_1 ={\mathcal {F}}_1$

• 2. ${\mathcal {F}}_1\, \widetilde{\cup }\,\mathcal {O}={\mathcal {F}}_1$

• 3. ${\mathcal {F}}_1\,\widetilde{\cup }\,{\mathcal {U}}={\mathcal {U}}$

• 4. ${\mathcal {F}}_1\,\widetilde{\cup }\,{\mathcal {F}}_2={\mathcal {F}}_2\,\widetilde{\cup }\,{\mathcal {F}}_1$

• 5. $({\mathcal {F}}_1\, \widetilde{\cup }\,{\mathcal {F}}_2)\widetilde{\cup }{\mathcal {F}}_3= {\mathcal {F}}_1\,\widetilde{\cup }\,({\mathcal {F}}_2\widetilde{\cup }{\mathcal {F}}_3)$

Proposition 4

Let ${\mathcal {F}}_1, {\mathcal {F}}_2, {\mathcal {F}}_3 \in \widetilde{{\mathcal {F}}}$. Then,

• 1. ${\mathcal {F}}_1\,\widetilde{\cap }\,{\mathcal {F}}_1={\mathcal {F}}_1$

• 2. ${\mathcal {F}}_1\,\widetilde{\cap }\,\mathcal {O}=\mathcal {O}$

• 3. ${\mathcal {F}}_1\,\widetilde{\cap }\,{\mathcal {U}}={\mathcal {F}}_1$

• 4. ${\mathcal {F}}_1\,\widetilde{\cap }\,{\mathcal {F}}_2={\mathcal {F}}_2\,\widetilde{\cap }\,{\mathcal {F}}_1$

• 5. $({\mathcal {F}}_1\,\widetilde{\cap }\,{\mathcal {F}}_2)\widetilde{\cap }{\mathcal {F}}_3= {\mathcal {F}}_1\,\widetilde{\cap }\,({\mathcal {F}}_2\widetilde{\cap }{\mathcal {F}}_3)$

Proposition 5

Let ${\mathcal {F}}_1, {\mathcal {F}}_2\in \widetilde{\mathcal {F}}$. Then, De Morgan’s laws are valid

• 1. $({\mathcal {F}}_1\,\widetilde{\cup }\,{\mathcal {F}}_2)^{\widetilde{c}} ={\mathcal {F}}_1^{\widetilde{c}} \,\widetilde{\cap }\,{\mathcal {F}}_2^{\widetilde{c}}$

• 2. $({\mathcal {F}}_1\,\widetilde{\cap }\,{\mathcal {F}}_2)^{\widetilde{c}} ={\mathcal {F}}_1^{\widetilde{c}} \,\widetilde{\cup }\,{\mathcal {F}}_2^{\widetilde{c}}$

Proposition 6

Let ${\mathcal {F}}_1, {\mathcal {F}}_2, {\mathcal {F}}_3 \in \widetilde{\mathcal {F}}$. Then,

• 1. ${\mathcal {F}}_1\,\widetilde{\cap }\,({\mathcal {F}}_2\,\widetilde{\cup }\,{\mathcal {F}}_3)= ({\mathcal {F}}_1\,\widetilde{\cap }\,{\mathcal {F}}_2)\,\widetilde{\cup }\,({\mathcal {F}}_1\,\widetilde{\cap }\,{\mathcal {F}}_3)$

• 2. ${\mathcal {F}}_1\,\widetilde{\cup }\,({\mathcal {F}}_2\,\widetilde{\cap }\,{\mathcal {F}}_3)= ({\mathcal {F}}_1\,\widetilde{\cup }\,{\mathcal {F}}_2)\,\widetilde{\cap }\,({\mathcal {F}}_1\,\widetilde{\cup }\,{\mathcal {F}}_3)$

4 ivnpivn-soft matrices

In this section, we presented ivnpivn-soft matrices which are representative of the ivnpivn-soft sets. The matrix is useful for storing an ivnpivn-soft sets in computer memory which are very useful and applicable. In the following, some definitions and operations on npn-soft sets are defined in [15]; we extend these definitions and operations to ivnpivn-soft sets.

Definition 13

Let $U=\{u_1,u_2,\ldots ,u_m\}$, $X=\{x_1,x_2,\ldots ,x_n\}$ and

$$\begin{aligned} {\mathcal {F}}= \big \{(\langle x_i,T_{\mathcal {F}}(x_i),I_{\mathcal {F}}(x_i),F_{\mathcal {F}}(x_i) \rangle , \{\langle T_{f_{\mathcal {F}}(x_i)}(u_j),I_{f_{\mathcal {F}}(x_i)}(u_j),F_{f_{\mathcal {F}}(x_i)}(u_j) \rangle /u_j:\ \ u_j\in U \}): x_i\in X \big \} \end{aligned}$$

be an ivnpivn-soft set over U. Where

$$\begin{aligned} &T_{\mathcal {F}}(x_i)= [\hbox {inf}T_{\mathcal {F}}(x_i), \hbox {sup}T_{\mathcal {F}}(x_i)]\subseteq [0,1],\\ &I_{\mathcal {F}}(x_i)= [\hbox {inf}I_{\mathcal {F}}(x_i), \hbox {sup}I_{\mathcal {F}}(x_i)]\subseteq [0,1],\\ &F_{\mathcal {F}}(x_i)= [\hbox {inf}F_{\mathcal {F}}(x_i), \hbox {sup}F_{\mathcal {F}}(x_i)]\subseteq [0,1],\\& T_{f_{\mathcal {F}}(x_i)}(u_j)= [\hbox {inf}T_{f_{\mathcal {F}}(x_i)}(u_j), \hbox {sup}T_{f_{\mathcal {F}}(x_i)}(u_j)]\subseteq [0,1],\\& I_{f_{\mathcal {F}}(x_i)}(u_j)= [\hbox {inf}I_{f_{\mathcal {F}}(x_i)}(u_j), \hbox {sup}I_{f_{\mathcal {F}}(x_i)}(u_j)]\subseteq [0,1],\\ &F_{f_{\mathcal {F}}(x_i)}(u_j)= [\hbox {inf}F_{f_{\mathcal {F}}(x_i)}(u_j), \hbox {sup}F_{f_{\mathcal {F}}(x_i)}(u_j)]\subseteq [0,1]. \end{aligned}$$

If $\alpha _i=\langle T_{\mathcal {F}}(x_i),I_{\mathcal {F}}(x_i),F_{\mathcal {F}}(x_i) \rangle$ and $V_{ij}=\langle T_{f_{\mathcal {F}}(x_i)}(u_j),I_{f_{\mathcal {F}}(x_i)}(u_j),F_{f_{\mathcal {F}}(x_i)}(u_j)\rangle$, then the ${\mathcal {F}}$ can be represented by a matrix as in the following form

$$\begin{aligned} {\mathcal {F}}[n\times m]=\left[ \begin{array}{c|ccccc} \alpha _{1} & V_{11} & V_{12} & \cdots & V_{1m} \\ \alpha _{2} & V_{21} & V_{22} & \cdots & V_{2m} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \alpha _{n} & V_{n1} & V_{n2} & \cdots & V_{nm} \end{array}\right] \end{aligned}$$

which is called an $n\times m$ ivnpivn-soft matrix (or ivnpivns-matrix) of the ivnpivn-soft set ${\mathcal {F}}$ over U.

According to this definition, an ivnpivn-soft set ${\mathcal {F}}$ is uniquely characterized by matrix ${\mathcal {F}}[n\times m]$. Therefore, we shall identify any ivnpivn-soft set with its ivnpivns-matrix and use these two concepts as interchangeable.

From now on, the set of all $n\times m$ ivnpivns-matrix over U will be denoted by $\widetilde{\mathcal {F}}_{n\times m}$.

Example 7

Let $U=\{u_1, u_2, u_3\}$, $X=\{x_1, x_2, x_3\}$, ${\mathcal {F}}_1$ be given as in Example 5. Then, the ivnpivns-matrix of ${\mathcal {F}}_1$ is written by

$$\begin{aligned} {\mathcal {F}}_1[3\times 3]=\left[ \begin{array}{c|cccc} \langle [.3,.4],[.5,.6],[.4,.5]\rangle & \langle [.5,.6],[.6,.7],[.3,.4]\rangle & \langle [.4,.5],[.7,.8],[.2,.3]\rangle & \langle [.6,.7],[.2,.3],[.3,.5]\rangle \\ \langle [.1,.2],[.3,.4],[.6,.7]\rangle & \langle [.7,.8],[.3,.4],[.2,.4]\rangle & \langle [.8,.4],[.2,.6],[.3,.4]\rangle & \langle [.4,.5],[.1,.3],[.2,.4]\rangle \\ \langle [.2,.4],[.4,.5],[.4,.6]\rangle & \langle [.2,.3],[.1,.4],[.3,.6]\rangle & \langle [.2,.5],[.1,.6],[.5,.8]\rangle & \langle [.3,.7],[.1,.3],[.6,.7]\rangle \end{array}\right] \end{aligned}$$

Definition 14

Let ${\mathcal {F}}[n\times m]\in \widetilde{\mathcal {F}}_{n\times m}$. Then, ${\mathcal {F}}[n\times m]$ is called

• 1. A zero ivnpivn-soft matrix, denoted by $\mathcal {O}[n\times m]$, if $\alpha _i=\langle [0,0],[1,1],[1,1]\rangle$ and $V_{ij}=\langle [0,0],[1,1],[1,1]\rangle$ for $i=1,2,\ldots ,m$ and $j=1,2,\ldots ,n$.

• 2. A universal ivnpivn-soft matrix, denoted by ${\mathcal {U}}[n\times m]$, if $\alpha _i=\langle [1,1],[0,0],[0,0]\rangle$ and $V_{ij}=\langle [1,1],[0,0],[0,0]\rangle$ for $i=1,2,\ldots ,m$ and $j=1,2,\ldots ,n$.

Example 8

Let $U=\{u_1, u_2, u_3\}$, $X=\{x_1, x_2, x_3\}$. Then, a zero ivnpivn-soft matrix is written by

$$\begin{aligned} \mathcal {O}[3\times 3]=\left[ \begin{array}{c|cccc} \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle \\ \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle \\ \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle & \langle [0,0],[1,1],[1,1]\rangle \\ \end{array}\right] \end{aligned}$$

and a universal ivnpivn-soft matrix is written by

$$\begin{aligned} {\mathcal {U}}[3\times 3]=\left[ \begin{array}{c|cccc} \langle [1,1],[0,0],[0,0]\rangle & \langle [1,1],[0,0],[0,0]\rangle & \langle [1,1],[0,0],[0,0]\rangle & \langle [1,1],[0,0],[0,0]\rangle \\ \langle [1,1],[0,0],[0,0]\rangle & \langle [1,1],[0,0],[0,0]\rangle & \langle [1,1],[0,0],[0,0]\rangle & \langle [1,1],[0,0],[0,0]\rangle \\ \langle [1,1],[0,0],[0,0]\rangle & \langle [1,1],[0,0],[0,0]\rangle & \langle [1,1],[0,0],[0,0]\rangle & <[1,1],[0,0],[0,0]\rangle \\ \end{array}\right] \end{aligned}$$

Definition 15

Let ${\mathcal {F}}_1[n\times m],{\mathcal {F}}_2[n\times m]\in \widetilde{\mathcal {F}}_{n\times m}$. Then

• 1. ${\mathcal {F}}_1[n\times m]$ is a sub$-ivnpivn$-matrix of ${\mathcal {F}}_2[n\times m]$, denoted by ${\mathcal {F}}_1[n\times m]\,\widetilde{\subseteq }\,{\mathcal {F}}_2[n\times m]$, if

  $$\begin{aligned} \begin{array}{ll} \hbox {inf}T_{{\mathcal {F}_1}}(x_i)\le \hbox {inf}T_{{\mathcal {F}_2}}(x_i)& \hbox {sup}T_{{\mathcal {F}_1}}(x_i)\le \hbox {sup}T_{{\mathcal {F}_2}}(x_i) \\ \hbox {inf}I_{{\mathcal {F}_1}}(x_i)\ge \hbox {inf}I_{{\mathcal {F}_2}}(x_i)& \hbox {sup}I_{{\mathcal {F}_1}}(x_i)\ge \hbox {sup}I_{{\mathcal {F}_2}}(x_i),\\ \hbox {inf}F_{{\mathcal {F}_1}}(x_i)\ge \hbox {inf}F_{{\mathcal {F}_2}}(x_i)& \hbox {sup}F_{{\mathcal {F}_1}}(x_i)\ge \hbox {sup}F_{{\mathcal {F}_2}}(x_i) \\ \hbox {inf}T_{f_{\mathcal {F}_1(x_i)}}(u_j)\le \hbox {inf}T_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}T_{f_{\mathcal {F}_1(x_i)}}(u_j)\le \hbox {sup}T_{f_{\mathcal {F}_2(x_i)}}(u_j), \\ \hbox {inf}I_{f_{\mathcal {F}_1(x_i)}}(u_j)\ge \hbox {inf}I_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}I_{f_{\mathcal {F}_1(x_i)}}(u_j)\ge \hbox {sup}I_{f_{\mathcal {F}_2(x_i)}}(u_j), \\ \hbox {inf}F_{f_{\mathcal {F}_1(x_i)}}(u_j)\ge \hbox {inf}F_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}F_{f_{\mathcal {F}_1(x_i)}}(u_j)\ge \hbox {sup}F_{f_{\mathcal {F}_2(x_i)}}(u_j). \end{array} \end{aligned}$$

  for $i=1,2,\ldots ,m$ and $j=1,2,\ldots ,n$.

• 2. ${\mathcal {F}}_1[n\times m]$ is a proper sub$-ivnpivn$-matrix of ${\mathcal {F}}_2[n\times m]$, denoted by ${\mathcal {F}}_1[n\times m]\widetilde{\subseteq }{\mathcal {F}}_2[n\times m]$, if

  $$\begin{aligned} \begin{array}{ll} \hbox {inf}T_{{\mathcal {F}_1}}(x_i)< \hbox {inf}T_{{\mathcal {F}_2}}(x_i)& \hbox {sup}T_{{\mathcal {F}_1}}(x_i)< \hbox {sup}T_{{\mathcal {F}_2}}(x_i) \\ \hbox {inf}I_{{\mathcal {F}_1}}(x_i)> \hbox {inf}I_{{\mathcal {F}_2}}(x_i)& \hbox {sup}I_{{\mathcal {F}_1}}(x_i)> \hbox {sup}I_{{\mathcal {F}_2}}(x_i),\\ \hbox {inf}F_{{\mathcal {F}_1}}(x_i)> \hbox {inf}F_{{\mathcal {F}_2}}(x_i)& \hbox {sup}F_{{\mathcal {F}_1}}(x_i)> \hbox {sup}F_{{\mathcal {F}_2}}(x_i) \\ \hbox {inf}T_{f_{\mathcal {F}_1(x_i)}}(u_j)< \hbox {inf}T_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}T_{f_{\mathcal {F}_1(x_i)}}(u_j)< \hbox {sup}T_{f_{\mathcal {F}_2(x_i)}}(u_j), \\ \hbox {inf}I_{f_{\mathcal {F}_1(x_i)}}(u_j)> \hbox {inf}I_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}I_{f_{\mathcal {F}_1(x_i)}}(u_j)> \hbox {sup}I_{f_{\mathcal {F}_2(x_i)}}(u_j), \\ \hbox {inf}F_{f_{\mathcal {F}_1(x_i)}}(u_j)> \hbox {inf}F_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}F_{f_{\mathcal {F}_1(x_i)}}(u_j)> \hbox {sup}F_{f_{\mathcal {F}_2(x_i)}}(u_j). \end{array} \end{aligned}$$

  for $i=1,2,\ldots ,m$ and $j=1,2,\ldots ,n$.

• 3. ${\mathcal {F}}_1[n\times m]$ and ${\mathcal {F}}_2[n\times m]$ are equal $-ivnpivn$-matrix, denoted by ${\mathcal {F}}_1[n\times m]={\mathcal {F}}_2[n\times m]$, if

  $$\begin{aligned} \begin{array}{ll} \hbox {inf}T_{{\mathcal {F}_1}}(x_i)= \hbox {inf}T_{{\mathcal {F}_2}}(x_i)& \hbox {sup}T_{{\mathcal {F}_1}}(x_i)= \hbox {sup}T_{{\mathcal {F}_2}}(x_i) \\ \hbox {inf}I_{{\mathcal {F}_1}}(x_i)= \hbox {inf}I_{{\mathcal {F}_2}}(x_i)& \hbox {sup}I_{{\mathcal {F}_1}}(x_i)= \hbox {sup}I_{{\mathcal {F}_2}}(x_i),\\ \hbox {inf}F_{{\mathcal {F}_1}}(x_i)= \hbox {inf}F_{{\mathcal {F}_2}}(x_i)& \hbox {sup}F_{{\mathcal {F}_1}}(x_i)= \hbox {sup}F_{{\mathcal {F}_2}}(x_i) \\ \hbox {inf}T_{f_{\mathcal {F}_1(x_i)}}(u_j)= \hbox {inf}T_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}T_{f_{\mathcal {F}_1(x_i)}}(u_j)= \hbox {sup}T_{f_{\mathcal {F}_2(x_i)}}(u_j), \\ \hbox {inf}I_{f_{\mathcal {F}_1(x_i)}}(u_j)= \hbox {inf}I_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}I_{f_{\mathcal {F}_1(x_i)}}(u_j)= \hbox {sup}I_{f_{\mathcal {F}_2(x_i)}}(u_j), \\ \hbox {inf}F_{f_{\mathcal {F}_1(x_i)}}(u_j)= \hbox {inf}F_{f_{\mathcal {F}_2(x_i)}}(u_j)& \hbox {sup}F_{f_{\mathcal {F}_1(x_i)}}(u_j)= \hbox {sup}F_{f_{\mathcal {F}_2(x_i)}}(u_j). \end{array} \end{aligned}$$

  for $i=1,2,\ldots ,m$ and $j=1,2,\ldots ,n$.

Definition 16

Let ${\mathcal {F}}[n\times m],{\mathcal {F}}_1[n\times m],{\mathcal {F}}_2[n\times m]\in \widetilde{{\mathcal {F}}}_{n\times m}$. Then

• 1 Union of ${\mathcal {F}}_1[n\times m]$ and ${\mathcal {F}}_2[n\times m]$, denoted by ${\mathcal {F}}_3[n\times m]={\mathcal {F}}_1[n\times m]\,\widetilde{\cup }\,{\mathcal {F}}_2[n\times m]$, if

  $$\begin{aligned} &T_{{\mathcal {F}_3}}(x_i)= [s(\hbox {inf}T_{{\mathcal {F}_1}}(x_i),\hbox {inf}T_{{\mathcal {F}_2}}(x_i)),s(\hbox {sup}T_{{\mathcal {F}_1}}(x_i),\hbox {sup}T_{{\mathcal {F}_2}}(x_i))],\\& I_{{\mathcal {F}_3}}(x_i)= [t(\hbox {inf}I_{{\mathcal {F}_1}}(x_i),\hbox {inf}I_{{\mathcal {F}_2}}(x_i)),t(\hbox {sup}I_{{\mathcal {F}_1}}(x_i),\hbox {sup}I_{{\mathcal {F}_2}}(x_i))],\\& F_{{\mathcal {F}_3}}(x_i)= [t(\hbox {inf}F_{{\mathcal {F}_1}}(x_i),\hbox {inf}F_{{\mathcal {F}_2}}(x_i)),t(\hbox {sup}F_{{\mathcal {F}_1}}(x_i),\hbox {sup}F_{{\mathcal {F}_2}}(x_i))],\\& T_{f_{\mathcal {F}_3(x_i)}}(u_j)= [s(\hbox {inf}T_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {inf}T_{f_{\mathcal {F}_2(x_i)}}(u_j)),s(\hbox {sup}T_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {sup}T_{f_{\mathcal {F}_2(x_i)}}(u_j))],\\& I_{f_{\mathcal {F}_3(x_i)}}(u_j)= [t(\hbox {inf}I_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {inf}I_{f_{\mathcal {F}_2(x_i)}}(u_j)),t(\hbox {sup}I_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {sup}I_{f_{\mathcal {F}_2(x_i)}}(u_j))],\\& F_{f_{\mathcal {F}_3(x_i)}}(u_j)= [t(\hbox {inf}F_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {inf}F_{f_{\mathcal {F}_2(x_i)}}(u_j)),t(\hbox {sup}F_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {sup}F_{f_{\mathcal {F}_2(x_i)}}(u_j))]. \end{aligned}$$

  for $i=1,2,\ldots ,m$ and $j=1,2,\ldots ,n$.

• 2. Intersection of ${\mathcal {F}}_1[n\times m]$ and ${\mathcal {F}}_2[n\times m]$, denoted by ${\mathcal {F}}_4[n\times m]={\mathcal {F}}_1[n\times m]\,\widetilde{\cap }\,{\mathcal {F}}_2[n\times m]$, if

  $$\begin{aligned} &T_{{\mathcal {F}_4}}(x_i)= [t(\hbox {inf}T_{{\mathcal {F}_1}}(x_i),\hbox {inf}T_{{\mathcal {F}_2}}(x_i)),t(\hbox {sup}T_{{\mathcal {F}_1}}(x_i),\hbox {sup}T_{{\mathcal {F}_2}}(x_i))],\\& I_{{\mathcal {F}_4}}(x_i)= [s(\hbox {inf}I_{{\mathcal {F}_1}}(x_i),\hbox {inf}I_{{\mathcal {F}_2}}(x_i)),s(\hbox {sup}I_{{\mathcal {F}_1}}(x_i),\hbox {sup}I_{{\mathcal {F}_2}}(x_i))],\\& F_{{\mathcal {F}_4}}(x_i)= [s(\hbox {inf}F_{{\mathcal {F}_1}}(x_i),\hbox {inf}F_{{\mathcal {F}_2}}(x_i)),s(\hbox {sup}F_{{\mathcal {F}_1}}(x_i),\hbox {sup}F_{{\mathcal {F}_2}}(x_i))],\\& T_{f_{\mathcal {F}_4(x_i)}}(u_j)= [t(\hbox {inf}T_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {inf}T_{f_{\mathcal {F}_2(x_i)}}(u_j)),t(\hbox {sup}T_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {sup}T_{f_{\mathcal {F}_2(x_i)}}(u_j))],\\& I_{f_{\mathcal {F}_4(x_i)}}(u_j)= [s(\hbox {inf}I_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {inf}I_{f_{\mathcal {F}_2(x_i)}}(u_j)),s(\hbox {sup}I_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {sup}I_{f_{\mathcal {F}_2(x_i)}}(u_j))],\\& F_{f_{\mathcal {F}_4(x_i)}}(u_j)= [s(\hbox {inf}F_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {inf}F_{f_{\mathcal {F}_2(x_i)}}(u_j)),s(\hbox {sup}F_{f_{\mathcal {F}_1(x_i)}}(u_j),\hbox {sup}F_{f_{\mathcal {F}_2(x_i)}}(u_j))]. \end{aligned}$$

  for $i=1,2,\ldots ,m$ and $j=1,2,\ldots ,n$.

• 3. Complement of $\mathcal {F}[n\times m]$, denoted by $\mathcal {F}^{\widetilde{c}} [n\times m]$, if

  $$\begin{aligned}& T_{\mathcal {F}^{\widetilde{c}} }(x_i)= [\hbox {inf}F_{\mathcal {F}}(x_i), \hbox {sup}F_{\mathcal {F}}(x_i)],\\ &I_{\mathcal {F}^{\widetilde{c}} }(x_i)= [1-\hbox {sup}I_{\mathcal {F}}(x_i), 1-\hbox {inf}I_{\mathcal {F}}(x_i)],\\& F_{\mathcal {F}^{\widetilde{c}} }(x_i)= [\hbox {inf}T_{\mathcal {F}}(x_i), \hbox {sup}T_{\mathcal {F}}(x_i)],\\& T_{f_{\mathcal {F}^{\widetilde{c}} }(x_i)}(u_j)= [\hbox {inf}F_{f_{\mathcal {F}}(x_i)}(u_j), \hbox {sup}F_{f_{\mathcal {F}}(x_i)}(u_j)],\\& I_{f_{\mathcal {F}^{\widetilde{c}} }(x_i)}(u_j)= [1-\hbox {sup}I_{f_{\mathcal {F}}(x_i)}(u_j), 1-\hbox {inf}I_{f_{\mathcal {F}}(x_i)}(u_j)],\\& F_{f_{\mathcal {F}^{\widetilde{c}}}(x_i)}(u_j)= [\hbox {inf}T_{f_{\mathcal {F}}(x_i)}(u_j), \hbox {sup}T_{f_{\mathcal {F}^{\widetilde{c}} }(x_i)}(u_j)]. \end{aligned}$$

  for $i=1,2,\ldots ,m$ and $j=1,2,\ldots ,n$.

Example 9

Consider Example 6. For t-norm and s-norm we use $min\{a,b\}$ and $max\{a,b\}$, respectively. Then, ${\mathcal {F}}_3[3\times 3]={\mathcal {F}}_1[3\times 3]\,\widetilde{\cup }\,{\mathcal {F}}_2[3\times 3]$, ${\mathcal {F}}_4[3\times 3]={\mathcal {F}}_1[3\times 3]\,\widetilde{\cap }\,{\mathcal {F}}_2[3\times 3]$ and ${\mathcal {F}}_1^{\widetilde{c}} [3\times 3]$ is given as

$$\begin{aligned} {\mathcal {F}}_3[3\times 3]=\left[ \begin{array}{c|cccc} \langle [.6,.7],[.2,.6],[.4,.5]\rangle & \langle [.5,.6],[.6,.7],[.3,.4]\rangle & \langle [.4,.5],[.2,.4],[.2,.3]\rangle & \langle [.6,.7],[.2,.3],[.3,.5]\rangle \\ \langle [.1,.4],[.2,.4],[.5,.7]\rangle & \langle [.7,.8],[.3,.4],[.1,.2]\rangle & \langle [.8,.9],[.2,.6],[.3,.4]\rangle & \langle [.4,.5],[.1,.3],[.2,.4]\rangle \\ \langle [.3,.9],[.4,.5],[.3,.6]\rangle & \langle [.3,.5],[.1,.4],[.3,.6]\rangle & \langle [.3,.7],[.1,.6],[.5,.8]\rangle & \langle [.4,.9],[.1,.3],[.6,.7]\rangle \end{array}\right] \\ {\mathcal {F}}_4[3\times 3]=\left[ \begin{array}{c|cccc} \langle [.3,.4],[.5,.9],[.7,.8]\rangle & \langle [.1,.5],[.7,.9],[.4,.7]\rangle & \langle [.1,.3],[.7,.8],[.8,.9]\rangle & \langle [.3,.5],[.7,.9],[.6,.7]\rangle \\ \langle [.1,.2],[.3,.4],[.5,.8]\rangle & \langle [.2,.3],[.6,.8],[.2,.4]\rangle & \langle [.5,.4],[.4,.6],[.7,.9]\rangle & \langle [.1,.2],[.3,.4],[.5,.6]\rangle \\ \langle [.2,.4],[.5,.7],[.4,.6]\rangle & \langle [.2,.3],[.2,.6],[.4,.8]\rangle & \langle [.2,.5],[.2,.8],[.6,.9]\rangle & \langle [.3,.7],[.2,.5],[.6,.9]\rangle \end{array}\right] \end{aligned}$$

and

$$\begin{aligned} {\mathcal {F}}_1^{\widetilde{c}} [3\times 3]=\left[ \begin{array}{c|cccc} \langle [.4,.5],[.4,.5],[.3,.4]\rangle & \langle [.3,.4],[.4,.3],[.5,.6]\rangle & \langle [.2,.3],[.3,.2],[.4,.5]\rangle & \langle [.3,.5],[.8,.7],[.6,.7]\rangle \\ \langle [.6,.7],[.6,.7],[.1,.2]\rangle & \langle [.2,.4],[.7,.6],[.7,.8]\rangle & \langle [.3,.4],[.8,.4],[.8,.4]\rangle & \langle [.2,.4],[.9,.7],[.4,.5]\rangle \\ \langle [.4,.6],[.5,.6],[.2,.4]\rangle & \langle [.3,.6],[.9,.6],[.2,.3]\rangle & \langle [.5,.8],[.9,.4],[.2,.5]\rangle & \langle [.6,.7],[.9,.7],[.3,.7]\rangle \end{array}\right] . \end{aligned}$$

Proposition 7

Let ${\mathcal {F}}\in \widetilde{\mathcal {F}}_{n\times m}$. Then

• 1. $\big ({\mathcal {F}}^{\widetilde{c}}[m\times n]\big )^{\widetilde{c}} ={\mathcal {F}}[m\times n]$

• 2. $\mathcal {O}^{\widetilde{c}} [m\times n]={\mathcal {U}}[m\times n]$

Proposition 8

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n],{\mathcal {F}}_3[m\times n]\in \widetilde{\mathcal {F}}_{n\times m}$. Then

• 1. ${\mathcal {F}}_1[m\times n]\,\subseteq \,{\mathcal {U}}[m\times n]$

• 2. $\mathcal {O}[m\times n]\,\widetilde{\subseteq }\,{\mathcal {F}}_1[m\times n]$

• 3. ${\mathcal {F}}_1[m\times n]\,\widetilde{\subseteq }\,{\mathcal {F}}_1[m\times n]$

• 4. ${\mathcal {F}}_1[m\times n]\,\widetilde{\subseteq }\,{\mathcal {F}}_2[m\times n]\wedge {\mathcal {F}}_2[m\times n]\,\widetilde{\subseteq }\,{\mathcal {F}}_3[m\times n]$ $\Rightarrow$ ${\mathcal {F}}_1[m\times n]\,\widetilde{\subseteq }\,{\mathcal {F}}_3[m\times n]$

Proposition 9

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n],{\mathcal {F}}_3[m\times n]\in \widetilde{\mathcal {F}}_{n\times m}$. Then

• 1. ${\mathcal {F}}_1[m\times n]={\mathcal {F}}_2[m\times n]$ and ${\mathcal {F}}_2[m\times n]={\mathcal {F}}_3[m\times n]$ $\Leftrightarrow$ ${\mathcal {F}}_1[m\times n]={\mathcal {F}}_3[m\times n]$

• 2. ${\mathcal {F}}_1[m\times n]\,\widetilde{\subseteq }\,{\mathcal {F}}_2[m\times n]$ and ${\mathcal {F}}_2[m\times n]\,\widetilde{\subseteq }\,{\mathcal {F}}_1[m\times n]$ $\Leftrightarrow$ ${\mathcal {F}}_1[m\times n]={\mathcal {F}}_2[m\times n]$

Proposition 10

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n],{\mathcal {F}}_3[m\times n]\in \widetilde{{\mathcal {F}}}_{n\times m}$. Then

• 1. ${\mathcal {F}}_1[m\times n]\widetilde{\cup }{\mathcal {F}}_1[m\times n]={\mathcal {F}}_1[m\times n]$

• 2. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,\mathcal {O}[m\times n]={\mathcal {F}}_1[m\times n]$

• 3. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,{\mathcal {U}}[m\times n]={\mathcal {U}}[m\times n]$

• 4. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,{\mathcal {F}}_2[m\times n]={\mathcal {F}}_2[m\times n]\widetilde{\cup }{\mathcal {F}}_1[m\times n]$

• 5. $({\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,{\mathcal {F}}_2[m\times n])\widetilde{\cup }{\mathcal {F}}_3[m\times n]= {\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,({\mathcal {F}}_2[m\times n]\,\widetilde{\cup }\,{\mathcal {F}}_3[m\times n])$

Proposition 11

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n],{\mathcal {F}}_3[m\times n] \,\in \widetilde\,{\mathcal {F}}_{n\times m}$. Then

• 1. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,{\mathcal {F}}_1[m\times n]={\mathcal {F}}_1[m\times n]$

• 2. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,\mathcal {O}[m\times n]=\mathcal {O}[m\times n]$

• 3. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,{\mathcal {U}}[m\times n]={\mathcal {F}}_1[m\times n]$

• 4. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,{\mathcal {F}}_2[m\times n]={\mathcal {F}}_2[m\times n]\,\widetilde{\cap }\,{\mathcal {F}}_1[m\times n]$

• 5. $({\mathcal {F}}_1[m\times n]\widetilde{\cap }{\mathcal {F}}_2[m\times n])\,\widetilde{\cap }\,{\mathcal {F}}_3[m\times n]= {\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,({\mathcal {F}}_2[m\times n]\,\widetilde{\cap }\,{\mathcal {F}}_3[m\times n])$

Proposition 12

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]\in \widetilde{{\mathcal {F}}}_{n\times m}$. Then, De Morgan’s laws are valid

• 1. $({\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,{\mathcal {F}}_2[m\times n])^{\widetilde{c}} ={\mathcal {F}}_1[m\times n]^{\widetilde{c}} \,\widetilde{\cap }\,{\mathcal {F}}_2[m\times n]^{\widetilde{c}}$

• 2. $({\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,{\mathcal {F}}_2[m\times n])^{\widetilde{c}} ={\mathcal {F}}_1[m\times n]^{\widetilde{c}} \,\widetilde{\cup }\,{\mathcal {F}}_2[m\times n]^{\widetilde{c}}$

Proposition 13

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n],{\mathcal {F}}_3[m\times n]\in \widetilde{\mathcal {F}}_{n\times m}$. Then

• 1. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,({\mathcal {F}}_2[m\times n]\,\widetilde{\cup }\,{\mathcal {F}}_3[m\times n])= ({\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,{\mathcal {F}}_2[m\times n])\,\widetilde{\cup }\,({\mathcal {F}}_1[m\times n]\,\widetilde{\cap }\,{\mathcal {F}}_3[m\times n])$

• 2. ${\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,({\mathcal {F}}_2[m\times n]\,\widetilde{\cap }\,{\mathcal {F}}_3[m\times n])= ({\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,{\mathcal {F}}_2[m\times n])\,\widetilde{\cap }\,({\mathcal {F}}_1[m\times n]\,\widetilde{\cup }\,{\mathcal {F}}_3[m\times n])$

5 Similarity measure on ivnpivn-soft sets

In the following, some definitions and operations on soft set is defined in [24], we extend these definitions and operations to ivnpivn-soft sets.

Definition 17

Let $U=\{u_1, u_2, \ldots ,u_m\}$ be a universe, $E=\{x_1, x_2, \ldots ,x_n\}$ be a set of parameters ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]\in \widetilde{\mathcal {F}}_{n\times m}$. If $\overrightarrow{\Vert {T_{ij}}\Vert }^2\not = 0$ or $\overrightarrow{\Vert {I_{ij}}\Vert }^2\not = 0$ or $\overrightarrow{\Vert {F_{ij}}\Vert }^2\not = 0$ for at least one $i \in \{1,2,\ldots ,n\}$ and $j \in \{1,2,\ldots ,m\}$, then similarity between ${\mathcal {F}}_1[m\times n]$ and ${\mathcal {F}}_2[m\times n]$ is defined by

$$\begin{aligned} S({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n])= \frac{\sum _{i=1}^n\sum _{j=1}^m \overrightarrow{\Vert V^2_i\Vert }\cdot \overrightarrow{\Vert T_{ij}\Vert }\cdot \overrightarrow{\Vert I_{ij}\Vert }\cdot \overrightarrow{\Vert F_{ij}\Vert }}{\sum _{i=1}^n\sum _{j=1}^m max\{\overrightarrow{\Vert {T_{ij}}}\Vert ^2, \overrightarrow{\Vert {I_{ij}}}\Vert ^2,\overrightarrow{\Vert {F_{ij}}}\Vert ^2\}} \end{aligned}$$

where

$$\begin{aligned} &\overrightarrow{V^2_i}= \left( \hbox {inf}T^1_{i}\cdot \hbox {inf}T^2_{i}+\hbox {sup}T^1_{i}\cdot \hbox {sup}T^2_{i},\hbox {inf}I^1_{i}\cdot \hbox {inf}I^2_{i}+\hbox {sup}I^1_{i}\cdot \hbox {sup}I^2_{i},\hbox {inf}F^1_{i}\cdot \hbox {inf}F^2_{i}+\hbox {sup}F^1_{i}\cdot \hbox {sup}F^2_{i}\right) \\ &\overrightarrow{T_{ij}}= \left( \hbox {inf}T^1_{i1}+\hbox {sup}T^1_{i1}-\hbox {inf}T^2_{i1}-\hbox {sup}T^2_{i1}, \hbox {inf}T^1_{i2}+\hbox {sup}T^1_{i2}-\hbox {inf}T^2_{i2}-\hbox {sup}T^2_{i2},\ldots ,\right. \\ &\left. \hbox {inf}T^1_{im}+\hbox {sup}T^1_{im}-\hbox {inf}T^2_{im}-\hbox {sup}T^2_{im}\right) \\ & \overrightarrow{I_{ij}}= \left( \hbox {inf}I^1_{i1}+\hbox {sup}I^1_{i1}-\hbox {inf}I^2_{i1}-\hbox {sup}I^2_{i1}, \hbox {inf}I^1_{i2}+\hbox {sup}I^1_{i2}-\hbox {inf}I^2_{i2}-\hbox {sup}I^2_{i2},\ldots ,\right. \\&\left. \hbox {inf}I^1_{im}+\hbox {sup}I^1_{im}-\hbox {inf}I^2_{im}-\hbox {sup}I^2_{im}\right) \\& \overrightarrow{F_{ij}}= \left( \hbox {inf}F^1_{i1}+\hbox {sup}F^1_{i1}-\hbox {inf}F^2_{i1}-\hbox {sup}F^2_{i1}, \hbox {inf}F^1_{i2}+\hbox {sup}F^1_{i2}-\hbox {inf}F^2_{i2}-\hbox {sup}F^2_{i2},\ldots ,\right. \\&\left. \hbox {inf}F^1_{im}+\hbox {sup}F^1_{im}-\hbox {inf}F^2_{im}-\hbox {sup}F^2_{im}\right) \end{aligned}$$

Note: If $\overrightarrow{\Vert {T_{ij}}\Vert }^2,\overrightarrow{\Vert {I_{ij}}\Vert }^2,\overrightarrow{{F_{ij}}\Vert }^2= 0$ for all $i \in \{1,2,\ldots ,n\}$ and $j \in \{1,2,\ldots ,m\}$ or ${\mathcal {F}}_1[m\times n]={\mathcal {F}}_2[m\times n]$, then $S({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n])=1$.

Definition 18

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]\in \widetilde{{\mathcal {F}}}_{n\times m}$. Then, ${\mathcal {F}}_1[m\times n]$ and ${\mathcal {F}}_2[m\times n]$ are said to be $\alpha$-similar, denoted ${\mathcal {F}}_1[m\times n] \approx ^\alpha {\mathcal {F}}_2[m\times n]$, if and only if $S({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]) = \alpha$ for $\alpha \in (0,1).$

We call the two ivnpivn-soft sets significantly similar if $S ({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]) > \frac{1}{2}$.

Example 10

Consider Example 6. If two ivnpivn-matrices ${\mathcal {F}}_1[3\times 3]$ and ${\mathcal {F}}_2[3\times 3]$ are written by

$$\begin{aligned} {\mathcal {F}}_1[3\times 3]=\left[ \begin{array}{c|cccc} \langle [.3,.4],[.5,.6],[.5,.6]\rangle & \langle [.5,.6],[.6,.7],[.3,.4]\rangle & \langle [.4,.5],[.7,.8],[.2,.3]\rangle & \langle [.6,.7],[.2,.3],[.3,.5]\rangle \\ \langle [.1,.2],[.3,.4],[.6,.7]\rangle & \langle [.7,.8],[.3,.4],[.2,.4]\rangle & \langle [.4,.8],[.2,.6],[.3,.4]\rangle & \langle [.4,.5],[.1,.3],[.2,.4]\rangle \\ \langle [.2,.4],[.4,.5],[.4,.6]\rangle & \langle [.2,.3],[.1,.4],[.3,.6]\rangle & \langle [.2,.5],[.1,.6],[.5,.8]\rangle & \langle [.3,.7],[.1,.3],[.6,.7]\rangle \end{array}\right] , \end{aligned}$$

and

$$\begin{aligned} {\mathcal {F}}_2[3\times 3]=\left[ \begin{array}{c|cccc} \langle [.6,.7],[.2,.9],[.7,.8]\rangle & \langle [.1,.5],[.7,.9],[.4,.7]\rangle & \langle [.1,.3],[.2,.4],[.8,.9]\rangle & \langle [.3,.5],[.7,.9],[.6,.7]\rangle \\ \langle [.1,.4],[.2,.4],[.5,.8]\rangle & \langle [.2,.3],[.6,.8],[.1,.2]\rangle & \langle [.5,.9],[.4,.6],[.7,.9]\rangle & \langle [.1,.2],[.3,.4],[.5,.6]\rangle \\ \langle [.3,.9],[.5,.7],[.3,.6]\rangle & \langle [.3,.5],[.2,.6],[.4,.8]\rangle & \langle [.3,.7],[.2,.8],[.6,.9]\rangle & \langle [.4,.9],[.2,.5],[.6,.9]\rangle \end{array}\right] \end{aligned}$$

Then, we can obtain

• $i=1, \ \ ( 0.5,0.5,0.5)$   $\Rightarrow \ \ \Vert T_{1j}\Vert =\sqrt{0.75}$ ,

• $i=2, \ \ ( 1.0,-0.2,0.6)$   $\Rightarrow \ \ \Vert T_{2j}\Vert =\sqrt{1.4}$,

• $i=3, \ \ ( -0.3,-0.3,-0.3)$   $\Rightarrow \ \ \Vert T_{3j}\Vert =\sqrt{0.27}$,

• $i=1, \ \ ( -0.3,0.9,-1.1)$   $\Rightarrow \ \ \Vert I_{1j}\Vert =\sqrt{2.11}$ ,

• $i=2, \ \ ( -0.7,-0.2,-0.3)$   $\Rightarrow \ \ \Vert I_{2j}\Vert =\sqrt{0.62}$,

• $i=3, \ \ ( -0.3,-0.3,-0.3)$   $\Rightarrow \ \ \Vert I_{3j}\Vert =\sqrt{0.27}$,

• $i=1, \ \ ( -0.4,-1.2,-0.5)$   $\Rightarrow \ \ \Vert F_{1j}\Vert =\sqrt{1.85}$,

• $i=2, \ \ ( 0.3,-0.9,-0.5)$   $\Rightarrow \ \ \Vert F_{2j}\Vert =\sqrt{1.15}$,

• $i=3, \ \ ( -0.3,-0.2,-0.2)$   $\Rightarrow \ \ \Vert F_{3j}\Vert =\sqrt{0.17}$.

and

• $i=1, \ \ V^2_{1}=( 0.46,0.64,0.82)$   $\Rightarrow \ \ \Vert V^2_{1}\Vert =\sqrt{1.2936}$,

• $i=2, \ \ V^2_{2}=( 0.09,0.22,0.86)$   $\Rightarrow \ \ \Vert V^2_{2}\Vert =\sqrt{0.7961}$,

• $i=3, \ \ V^2_{3}=( 0.38,0.55,0.48)$   $\Rightarrow \ \ \Vert V^2_{3}\Vert =\sqrt{0.6773}$,

Now the similarity between ${\mathcal {F}}_1[3\times 3]$ and ${\mathcal {F}}_2[3\times 3]$ is calculated as

$$\begin{aligned} S({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3]) &= \frac{\sqrt{1.2936}\sqrt{0.75}\sqrt{2.11}\sqrt{1.85}\ +\sqrt{0.7961}\sqrt{1.4}\sqrt{0.62}\sqrt{1.15}\ +\sqrt{0.6773}\sqrt{0.27}\sqrt{0.27}\sqrt{0.17}}{\hbox {max}\{0.75,2.11,1.85\}+\hbox {max}\{1.4,0.63,1.15\}+\hbox {max}\{0.27,0.27,0.17\}}\\&= \frac{1.95 +0.89 +0.09}{3.78}=0.78 \end{aligned}$$

Proposition 14

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n],{\mathcal {F}}_3[m\times n]\in \widetilde{\mathcal {F}}_{n\times m}$. Then, the followings hold;

• (i.) $S({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n])=S({\mathcal {F}}_2[m\times n],{\mathcal {F}}_1[m\times n])$,

• (ii.) $0\le S({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]) \le 1$,

• (iii.) $S({\mathcal {F}}_1[m\times n],{\mathcal {F}}_1[m\times n])=1$.

Proof

Proof easily can be made by using Definition 10.

Now we give distance measures between ${\mathcal {F}}_1[m\times n]$ and ${\mathcal {F}}_2[m\times n]$ with propositions by using the study of Jiang et al. [19]. $\square$

Definition 19

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]\in \widetilde{{\mathcal {F}}}_{n\times m}$. Then, the distances between ${\mathcal {F}}_1[m\times n]$ and ${\mathcal {F}}_2[m\times n]$ are defined as,

• 1. Hamming measure,

  $$\begin{aligned} d({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n])=\frac{1}{2}\sum _{i=1}^m\sum _{j=1}^n|V_i.V_{ij}| \end{aligned}$$

  where

  $$\begin{aligned} \widetilde{{V_{i}}} &= \hbox {inf}T^1_{i}+supT^1_{i}-\hbox {inf}T^2_{i}-\hbox {sup}T^2_{i}+\hbox {inf}I^1_{i}+\hbox {sup}I^1_{i}-\hbox {inf}I^2_{i}-\hbox {sup}I^2_{i}+\\&\hbox {inf}F^1_{i}+\hbox {sup}F^1_{i}-\hbox {inf}F^2_{i}-\hbox {sup}F^2_{i} \end{aligned}$$

  and

  $$\begin{aligned} \widetilde{{V_{ij}}}&= \hbox {inf}T^1_{ij}+\hbox {sup}T^1_{ij}-\hbox {inf}T^2_{ij}-\hbox {sup}T^2_{ij}+\hbox {inf}I^1_{ij}+\hbox {sup}I^1_{ij}-\hbox {inf}I^2_{ij}-\hbox {sup}I^2_{ij}+\\&\hbox {inf}F^1_{ij}+\hbox {sup}F^1_{ij}-\hbox {inf}F^2_{ij}-\hbox {sup}F^2_{ij} \end{aligned}$$

• 2. Normalized Hamming measure,

  $$\begin{aligned} l ({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n] )=\frac{1}{ mn}d({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]) \end{aligned}$$

• 3. Euclidean distance,

  $$\begin{aligned} e ({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n] )=\left( \sum _{i=1}^m\sum _{j=1}^n|\overline{V}_i.\overline{V}_{ij}|\right) ^{\frac{1}{2}} \end{aligned}$$

  where

  $$\begin{aligned} {{\overline{V}_{i}}}&= \left( \hbox {inf}T^1_{i}+\hbox {sup}T^1_{i}-\hbox {inf}T^2_{i}-\hbox {sup}T^2_{i}\right) ^2+\left( \hbox {inf}I^1_{i}+\hbox {sup}I^1_{i}-\hbox {inf}I^2_{i}-\hbox {sup}I^2_{i}\right) ^2+\\&\left( \hbox {inf}F^1_{i}+\hbox {sup}F^1_{i}-\hbox {inf}F^2_{i}-\hbox {sup}F^2_{i}\right) ^2 \end{aligned}$$

  and

  $$\begin{aligned} {{\overline{V}_{ij}}}&= \left( \hbox {inf}T^1_{ij}+\hbox {sup}T^1_{ij}-\hbox {inf}T^2_{ij}-\hbox {sup}T^2_{ij}\right) ^2+\left( \hbox {inf}I^1_{ij}+\hbox {sup}I^1_{ij}-infI^2_{ij}-\hbox {sup}I^2_{ij}\right) ^2+\\&\left( \hbox {inf}F^1_{ij}+\hbox {sup}F^1_{ij}-\hbox {inf}F^2_{ij}-\hbox {sup}F^2_{ij}\right) ^2 \end{aligned}$$

• 4. Normalized Euclidean distance,

  $$\begin{aligned} q ({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n] )= \frac{1}{mn} e ({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n] ) \end{aligned}$$

Example 11

Consider Example 10. Now we give distance measures between ${\mathcal {F}}_1[3\times 3]$ and ${\mathcal {F}}_2[3\times 3]$ as,

• 1. Hamming measure,

  $$\begin{aligned} &\widetilde{{V_{1}}} = -0.6+0-0.4=-1\\ &\widetilde{{V_{2}}} = -0.2+0.1+0=-0.1\\& \widetilde{{V_{3}}} = -0.6-0.3+0.1=-0.8\\ \end{aligned}$$

  and

  $$\begin{aligned} &\widetilde{{V_{11}}} = 0.5-0.3-0.4=-0.2\\& \widetilde{{V_{12}}} = 0.5+0.9-0.1=0.2\\& \widetilde{{V_{13}}} = 0.5-1.1-0.5=-1.1\\ \\& \widetilde{{V_{21}}} = 1-0.7+0.3=0.6\\ &\widetilde{{V_{22}}} = -0.2-0.2-0.9=-1.3\\ &\widetilde{{V_{23}}} = 0.6-0.3-0.5=-0.2\\ \\ &\widetilde{{V_{31}}} = -0.3-0.3-0.3=-0.9\\& \widetilde{{V_{32}}} = -0.3-0.3-0.2=-0.8\\& \widetilde{{V_{31}}} = -0.3-0.3-0.2=-0.8\\ \end{aligned}$$

  Then, we can obtain Hamming measure between ${\mathcal {F}}_1[3\times 3]$ and ${\mathcal {F}}_2[3\times 3]$ as,

  $$\begin{aligned} d({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3])=\frac{1}{2}\sum _{i=1}^3\sum _{j=1}^3|V_i.V_{ij}|=\frac{1}{2}(0.2+0.2+1.1+0.6+0.13+0.02+0.72+0.64+0.64) =1.855 \end{aligned}$$

• 2. Normalized Hamming measure,

  $$\begin{aligned} l({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )=\frac{1}{ 3.3}d({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3])=\frac{1}{ 9}1.855\cong 0.2061 \end{aligned}$$

• 3. Euclidean distance,

  $$\begin{aligned}& {{\overline{V}_{1}}}= 0.36+0+0.16=0.52\\ &{{\overline{V}_{2}}}= 0.04+0.01+0=0.05\\& {{\overline{V}_{3}}}= 0.36+0.09+0.01=0.46\\ \end{aligned}$$

  and

  $$\begin{aligned} &{{\overline{V}_{11}}} = 0.25+0.09+0.16=0.50\\ &{{\overline{V}_{12}}} = 0.25+0.81+1.44=2.50\\& {{\overline{V}_{13}}} = 0.25+1.21+0.25=1.71\\ \\ &{{\overline{V}_{21}}} = 1+0.49+0.09=1.58\\& {{\overline{V}_{22}}} = 0.04+0.04+0.81=0.89\\ &{{\overline{V}_{23}}} = 0.36+0.09+0.25=0.70\\ \\& {{\overline{V}_{31}}} = 0.09+0.09+0.09=0.27\\ &{{\overline{V}_{32}}} = 0.09+0.09+0.04=0.22\\& {{\overline{V}_{33}}} = 0.09+0.09+0.04=0.22\\ \end{aligned}$$

  Then, we can obtain Euclidean distance between ${\mathcal {F}}_1[3\times 3]$ and ${\mathcal {F}}_2[3\times 3]$ as,

  $$\begin{aligned} e ({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )=\left(\sum _{i=1}^3\sum _{j=1}^3|\overline{V}_i.\overline{V}_{ij}|\right)^{\frac{1}{2}}=(2.4492+0.1585+0.3266)^{\frac{1}{2}}=(2.9343)^{\frac{1}{2}}\,\cong \,1.713 \end{aligned}$$

• 4. Normalized Euclidean distance,

  $$\begin{aligned} q ({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )= \frac{1}{3.3} e ({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )=\frac{1}{9}1.713\cong 0.19 \end{aligned}$$

Theorem 1

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]\in \widetilde{\mathcal {F}}_{n\times m}$. Then, the followings hold;

• (i.) $d({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]) \le mn$

• (ii.) $l({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]) \le 1$

• (iii.) $e ({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]) \le \sqrt{mn}$

• (iv.) $q ({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n])\le 1$

Proof

Proof easily can be made by using Definition 6. $\square$

Definition 20

Let ${\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n]\in \widetilde{{\mathcal {F}}}_{n\times m}$. Then, by using the distances, similarity measure of ${\mathcal {F}}_1[m\times n]$ and ${\mathcal {F}}_2[m\times n]$ is defined as,

$$\begin{aligned} S_K({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n])=\frac{1}{1+K ({\mathcal {F}}_1[m\times n],{\mathcal {F}}_2[m\times n])} \end{aligned}$$

where $K\in \{d,l,e,q\}$.

Example 12

Consider Example 11. Now we give similarity measure of ${\mathcal {F}}_1[3\times 3]$ and ${\mathcal {F}}_2[3\times 3]$ by using the distances of Example 11 as,

$$\begin{aligned}& S_d({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )= \frac{1}{1+d ({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )}=\frac{1}{1+1.855}\,\cong \,0.35 \\& S_l({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )= \frac{1}{1+l ({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )}=\frac{1}{1+0.2061}\,\cong \,0.83 \\& S_e({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )= \frac{1}{1+e ({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )}=\frac{1}{1+1.713}\,\cong \,0.37 \\& S_q({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )= \frac{1}{1+q ({\mathcal {F}}_1[3\times 3],{\mathcal {F}}_2[3\times 3] )}=\frac{1}{1+0.19}\,\cong \,0.84 \end{aligned}$$

6 Decision-making method

In this section, we construct a decision-making method that is based on the similarity measure of two ivnpivn-soft sets. The algorithm of decision-making method can be given as:

Step 1. :

Construct an ivnpivn-soft set ${\mathcal {F}}_1$ over U for problem with the help of a expert,

Step 2. :

Construct an ivnpivn-soft set ${\mathcal {F}}_2$ based on a responsible person for the problem,

Step 3. :

Write ivnpivn-matrices ${\mathcal {F}}_1[m\times n]$ and ${\mathcal {F}}_2[m\times n]$ for ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$ according to Definition 13, respectively,

Step 4. :

Calculate the similarity between ${\mathcal {F}}_1[m\times n]$ and ${\mathcal {F}}_2[m\times n]$ according to Definition 17,

Step 5. :

Determine result by using the similarity.

Now, we can give an application for the decision-making method. The similarity measure can be applied to detect whether an ill person is suffering from a certain disease or not.

6.1 Application

Let us consider the decision-making problem adopted from [24]. In this applications, we will try to estimate the possibility that an ill person having certain visible symptoms is suffering from cancer. For this, we first construct an ivnpivn-soft set for the illness and an ivnpivn-soft set for the ill person. We then find the similarity measure of these two ivnpivn-soft sets. If they are significantly similar, then we conclude that the person is possibly suffering from cancer.

Example 13

Assume that our universal set contain only two elements cancer and not cancer, i.e. $U=\{u_1, u_2\}$. Here the set of parameters X is the set of certain visible symptoms, let us say, $X=\{x_1, x_2, x_3,x_4,x_5,$ $x_6, x_7,x_8,x_9 \}$ where $x_1=$ jaundice, $x_2=$ bone pain, $x_3=$ headache, $x_4=$ loss of appetite, $x_5=$ weight loss, $x_6=$ heal wounds , $x_7=$ handle and shoulder pain, $x_8=$ lump anywhere on the body for no reason and $x_9=$ chest pain.

Step 1. :

We construct an ivnpivn-soft set ${\mathcal {F}}_1$ over U for cancer with the help of a medical person as:

$$\begin{aligned} {\mathcal {F}}_1&= \bigg \{(\langle x_1,[0.5,0.7],[0.1,0.2],[0.7,0.8]\rangle ,\{\langle [0.5,0.6],[0.1,0.3],[0.8,0.9]\rangle /u_1,\langle [0.4,0.6],[0.1,0.3],[0.4,0.5]\rangle /u_2\}),\\&(\langle x_2,[0.7,0.8],[0.0,0.1],[0.1,0.2]\rangle ,\{\langle [0.6,0.7],[0.1,0.2],[0.1,0.3]\rangle /u_1,\langle [0.6,0.7],[0.3,0.4],[0.8,0.9]\rangle /u_2\}),\\&(\langle x_3,[0.3,0.6],[0.2,0.3],[0.3,0.4]\rangle ,\{\langle [0.5,0.6],[0.2,0.3],[0.3,0.4]\rangle /u_1,\langle [0.4,0.5],[0.2,0.4],[0.7,0.9]\rangle /u_2\}),\\&(\langle x_4,[0.6,0.7],[0.1,0.2],[0.2,0.3]\rangle ,\{\langle [0.6,0.7],[0.1,0.2],[0.2,0.3]\rangle /u_1,\langle [0.3,0.6],[0.3,0.5],[0.8,0.9]\rangle /u_2\}),\\&(\langle x_5,[0.4,0.5],[0.2,0.3],[0.3,0.4]\rangle ,\{\langle [0.4,0.6],[0.1,0.3],[0.2,0.4]\rangle /u_1,\langle [0.7,0.9],[0.2,0.3],[0.4,0.5]\rangle /u_2\}),\\&(\langle x_6,[0.3,0.5],[0.7,0.8],[0.2,0.6]\rangle ,\{\langle [0.5,0.7],[0.3,0.5],[0.4,0.8]\rangle /u_1,\langle [0.2,0.6],[0.5,0.6],[0.3,0.7]\rangle /u_2\}),\\&(\langle x_7,[0.3,0.8],[0.6,0.7],[0.5,0.9]\rangle ,\{\langle [0.5,0.9],[0.5,0.8],[0.7,0.9]\rangle /u_1,\langle [0.3,0.7],[0.8,0.9],[0.4,0.5]\rangle /u_2\}),\\&(\langle x_8,[0.2,0.6],[0.3,0.4],[0.5,0.7]\rangle ,\{\langle [0.4,0.7],[0.7,0.9],[0.3,0.6]\rangle /u_1,\langle [0.6,0.7],[0.2,0.4],[0.1,0.5]\rangle /u_2\}),\\&(\langle x_9,[0.1,0.2],[0.5,0.6],[0.1,0.6]\rangle ,\{\langle [0.3,0.4],[0.2,0.3],[0.4,0.6]\rangle /u_1,\langle [0.6,0.9],[0.4,0.7],[0.6,0.8]\rangle /u_2\}) \bigg \} \\ \end{aligned}$$

Step 2. :

We construct an ivnpivn-soft sets ${\mathcal {F}}_2$ based on data of ill person as:

$$\begin{aligned} {\mathcal {F}}_2&= \bigg \{(\langle x_1,[0.3,0.4],[0.5,0.6],[0.4,0.5]\rangle ,\{\langle [0.1,0.9],[0.1,0.5],[0.2,0.6]\rangle /u_1,\langle [0.0,0.9],[0.1,0.2],[0.0,0.1]\rangle /u_2\}),\\&(\langle x_2,[0.1,0.2],[0.3,0.4],[0.6,0.7]\rangle ,\{\langle [0.0,0.1],[0.5,0.7],[0.8,0.9]\rangle /u_1,\langle [0.1,0.3],[0.0,0.2],[0.8,0.9]\rangle /u_2\}),\\&(\langle x_3,[0.2,0.4],[0.4,0.5],[0.4,0.6]\rangle ,\{\langle [0.1,0.3],[0.4,0.6],[0.6,0.7]\rangle /u_1,\langle [0.8,0.9],[0.9,1.0],[0.6,0.7]\rangle /u_2\}),\\&(\langle x_4,[0.5,0.6],[0.6,0.7],[0.3,0.4]\rangle ,\{\langle [0.0,0.9],[0.2,0.3],[0.0,0.1]\rangle /u_1,\langle [0.5,0.8],[0.1,0.4],[0.7,0.9]\rangle /u_2\}),\\&(\langle x_5,[0.3,0.5],[0.7,0.8],[0.2,0.3]\rangle ,\{\langle [0.9,0.1],[0.5,0.8],[0.1,0.2]\rangle /u_1,\langle [0.8,0.9],[0.1,0.3],[0.2,0.4]\rangle /u_2\}),\\&(\langle x_6,[0.6,0.7],[0.2,0.3],[0.3,0.5]\rangle ,\{\langle [0.1,0.3],[0.8,0.9],[0.9,1.0]\rangle /u_1,\langle [0.8,1.0],[0.8,0.9],[0.1,0.2]\rangle /u_2\}),\\&(\langle x_7,[0.7,0.8],[0.3,0.4],[0.2,0.4]\rangle ,\{\langle [0.8,1.0],[0.7,0.8],[0.0,0.1]\rangle /u_1,\langle [0.6,0.7],[0.0,0.1],[0.6,0.9]\rangle /u_2\}),\\&(\langle x_8,[0.8,0.9],[0.2,0.6],[0.3,0.4]\rangle ,\{\langle [0.6,0.9],[0.8,1.0],[0.3,0.4]\rangle /u_1,\langle [0.0,0.1],[0.0,0.2],[0.9,1.0]\rangle /u_2\}),\\&(\langle x_9,[0.3,0.4],[0.7,0.9],[0.1,0.2]\rangle ,\{\langle [0.8,0.9],[0.7,0.8],[0.5,0.6]\rangle /u_1,\langle [0.0,1.0],[0.0,0.3],[0.7,0.9]\rangle /u_2\}) \bigg \} \end{aligned}$$

Step 3. :

We construct ivnpivn-matrices ${\mathcal {F}}_1[9\times 2]$ and ${\mathcal {F}}_2[9\times 2]$ for ${\mathcal {F}}_1$ and ${\mathcal {F}}_2$, respectively as:

$$\begin{aligned} {\mathcal {F}}_1[9\times 2]=\left[ \begin{array}{c|cccc} \langle [0.5,0.7],[0.1,0.2],[0.7,0.8]\rangle & \langle [0.5,0.6],[0.1,0.3],[0.8,0.9]\rangle & \langle [0.4,0.6],[0.1,0.3],[0.4,0.5]\rangle \\ \langle [0.7,0.8],[0.0,0.1],[0.1,0.2]\rangle & \langle [0.6,0.7],[0.1,0.2],[0.1,0.3]\rangle & \langle [0.6,0.7],[0.3,0.4],[0.8,0.9]\rangle \\ \langle [0.3,0.6],[0.2,0.3],[0.3,0.4]\rangle & \langle [0.5,0.6],[0.2,0.3],[0.3,0.4]\rangle & \langle [0.4,0.5],[0.2,0.4],[0.7,0.9]\rangle \\ \langle [0.6,0.7],[0.1,0.2],[0.2,0.3]\rangle & \langle [0.6,0.7],[0.1,0.2],[0.2,0.3]\rangle & \langle [0.3,0.6],[0.3,0.5],[0.8,0.9]\rangle \\ \langle [0.4,0.5],[0.2,0.3],[0.3,0.4]\rangle & \langle [0.4,0.6],[0.1,0.3],[0.2,0.4]\rangle & \langle [0.7,0.9],[0.2,0.3],[0.4,0.5]\rangle \\ \langle [0.3,0.5],[0.7,0.8],[0.2,0.6]\rangle & \langle [0.5,0.7],[0.3,0.5],[0.4,0.8]\rangle & \langle [0.2,0.6],[0.5,0.6],[0.3,0.7]\rangle \\ \langle [0.3,0.8],[0.6,0.7],[0.5,0.9]\rangle & \langle [0.5,0.9],[0.5,0.8],[0.7,0.9]\rangle & \langle [0.3,0.7],[0.8,0.9],[0.4,0.5]\rangle \\ \langle [0.2,0.6],[0.3,0.4],[0.5,0.7]\rangle & \langle [0.4,0.7],[0.7,0.9],[0.3,0.6]\rangle & \langle [0.6,0.7],[0.2,0.4],[0.1,0.5]\rangle \\ \langle [0.1,0.2],[0.5,0.6],[0.1,0.6]\rangle & \langle [0.3,0.4],[0.2,0.3],[0.4,0.6]\rangle & \langle [0.6,0.9],[0.4,0.7],[0.6,0.8]\rangle \\ \end{array}\right] , \end{aligned}$$

$$\begin{aligned} {\mathcal {F}}_2[9\times 2]=\left[ \begin{array}{c|cccc} \langle [0.3,0.4],[0.5,0.6],[0.4,0.5]\rangle & \langle [0.1,0.9],[0.1,0.5],[0.2,0.6]\rangle & \langle [0.0,0.9],[0.1,0.2],[0.0,0.1]\rangle \\ \langle [0.1,0.2],[0.3,0.4],[0.6,0.7]\rangle & \langle [0.0,0.1],[0.5,0.7],[0.8,0.9]\rangle & \langle [0.1,0.3],[0.0,0.2],[0.8,0.9]\rangle \\ \langle [0.2,0.4],[0.4,0.5],[0.4,0.6]\rangle & \langle [0.1,0.3],[0.4,0.6],[0.6,0.7]\rangle & \langle [0.8,0.9],[0.9,1.0],[0.6,0.7]\rangle \\ \langle [0.5,0.6],[0.6,0.7],[0.3,0.4]\rangle & \langle [0.0,0.9],[0.2,0.3],[0.0,0.1]\rangle & \langle [0.5,0.8],[0.1,0.4],[0.7,0.9]\rangle \\ \langle [0.3,0.5],[0.7,0.8],[0.2,0.3]\rangle & \langle [0.9,0.1],[0.5,0.8],[0.1,0.2]\rangle & \langle [0.8,0.9],[0.1,0.3],[0.2,0.4]\rangle \\ \langle [0.6,0.7],[0.2,0.3],[0.3,0.5]\rangle & \langle [0.1,0.3],[0.8,0.9],[0.9,1.0]\rangle & \langle [0.8,1.0],[0.8,0.9],[0.1,0.2]\rangle \\ \langle [0.7,0.8],[0.3,0.4],[0.2,0.4]\rangle & \langle [0.8,1.0],[0.7,0.8],[0.0,0.1]\rangle & \langle [0.6,0.7],[0.0,0.1],[0.6,0.9]\rangle \\ \langle [0.8,0.9],[0.2,0.6],[0.3,0.4]\rangle & \langle [0.6,0.9],[0.8,1.0],[0.3,0.4]\rangle & \langle [0.0,0.1],[0.0,0.2],[0.9,1.0]\rangle \\ \langle [0.3,0.4],[0.7,0.9],[0.1,0.2]\rangle & \langle [0.8,0.9],[0.7,0.8],[0.5,0.6]\rangle & \langle [0.0,1.0],[0.0,0.3],[0.7,0.9]\rangle \\ \end{array}\right] \end{aligned}$$

Step 4. :

We calculated the similarity between ${\mathcal {F}}_1[9\times 2]$ and ${\mathcal {F}}_2[9\times 2]$ as:

$$\begin{aligned} S({\mathcal {F}}_1[9\times 2],{\mathcal {F}}_2[9\times 2])=0.29 \langle \frac{1}{2} \end{aligned}$$

Step 5. :

The ${\mathcal {F}}_1[9\times 2]$ and ${\mathcal {F}}_2[9\times 2]$ are not significantly similar. Therefore, we conclude that the person is not possibly suffering from cancer.

Example 14

Let us consider Example 13 with different ill person.

Step 1. :

We construct an ivnpivn-soft set ${\mathcal {F}}_1$ over U for cancer with the help of a medical person as

$$\begin{aligned} {\mathcal {F}}_1&= \bigg \{(\langle x_1,[0.5,0.7],[0.1,0.2],[0.7,0.8]\rangle ,\{\langle [0.5,0.6],[0.1,0.3],[0.8,0.9]\rangle /u_1,\langle [0.4,0.6],[0.1,0.3],[0.4,0.5]\rangle /u_2\}),\\&(\langle x_2,[0.7,0.8],[0.0,0.1],[0.1,0.2]\rangle ,\{\langle [0.6,0.7],[0.1,0.2],[0.1,0.3]\rangle /u_1,\langle [0.6,0.7],[0.3,0.4],[0.8,0.9]\rangle /u_2\}),\\&(\langle x_3,[0.3,0.6],[0.2,0.3],[0.3,0.4]\rangle ,\{\langle [0.5,0.6],[0.2,0.3],[0.3,0.4]\rangle /u_1,\langle [0.4,0.5],[0.2,0.4],[0.7,0.9]\rangle /u_2\}),\\&(\langle x_4,[0.6,0.7],[0.1,0.2],[0.2,0.3]\rangle ,\{\langle [0.6,0.7],[0.1,0.2],[0.2,0.3]\rangle /u_1,\langle [0.3,0.6],[0.3,0.5],[0.8,0.9]\rangle /u_2\}),\\&(\langle x_5,[0.4,0.5],[0.2,0.3],[0.3,0.4]\rangle ,\{\langle [0.4,0.6],[0.1,0.3],[0.2,0.4]\rangle /u_1,\langle [0.7,0.9],[0.2,0.3],[0.4,0.5]\rangle /u_2\}),\\&(\langle x_6,[0.3,0.5],[0.7,0.8],[0.2,0.6]\rangle ,\{\langle [0.5,0.7],[0.3,0.5],[0.4,0.8]\rangle /u_1,\langle [0.2,0.6],[0.5,0.6],[0.3,0.7]\rangle /u_2\}),\\&(\langle x_7,[0.3,0.8],[0.6,0.7],[0.5,0.9]\rangle ,\{\langle [0.5,0.9],[0.5,0.8],[0.7,0.9]\rangle /u_1,\langle [0.3,0.7],[0.8,0.9],[0.4,0.5]\rangle /u_2\}),\\&(\langle x_8,[0.2,0.6],[0.3,0.4],[0.5,0.7]\rangle ,\{\langle [0.4,0.7],[0.7,0.9],[0.3,0.6]\rangle /u_1,\langle [0.6,0.7],[0.2,0.4],[0.1,0.5]\rangle /u_2\}),\\&(\langle x_9,[0.1,0.2],[0.5,0.6],[0.1,0.6]\rangle ,\{\langle [0.3,0.4],[0.2,0.3],[0.4,0.6]\rangle /u_1,\langle [0.6,0.9],[0.4,0.7],[0.6,0.8]\rangle /u_2\}) \bigg \} \\ \end{aligned}$$

Step 2. :

We construct an ivnpivn-soft sets ${\mathcal {F}}_3$ based on data of ill person as

$$\begin{aligned} {\mathcal {F}}_3&= \bigg \{(\langle x_1,[0.3,0.4],[0.5,0.6],[0.4,0.5]\rangle ,\{\langle [0.1,0.2],[0.7,0.8],[0.0,0.1]\rangle /u_1,\langle [0.0,0.1],[0.9,1.0],[0.9,1.0]\rangle /u_2\}),\\&(\langle x_2,[0.1,0.2],[0.3,0.4],[0.6,0.7]\rangle ,\{\langle [0.0,0.1],[0.5,0.6],[0.8,0.9]\rangle /u_1,\langle [0.1,0.2],[0.0,0.1],[0.3,0.4]\rangle /u_2\}),\\&(\langle x_3,[0.2,0.3],[0.4,0.5],[0.6,0.7]\rangle ,\{\langle [0.1,0.2],[0.0,0.1],[0.6,0.7]\rangle /u_1,\langle [0.8,0.9],[0.9,1.0],[0.0,0.1]\rangle /u_2\}),\\&(\langle x_4,[0.5,0.6],[0.6,0.7],[0.3,0.4]\rangle ,\{\langle [0.0,0.1],[0.6,0.7],[0.1,0.2]\rangle /u_1,\langle [0.4,0.5],[0.7,0.8],[0.2,0.3]\rangle /u_2\}),\\&(\langle x_5,[0.3,0.4],[0.7,0.8],[0.9,1.0]\rangle ,\{\langle [0.9,1.0],[0.6,0.7],[0.7,0.8]\rangle /u_1,\langle [0.8,0.9],[0.4,0.5],[0.2,0.3]\rangle /u_2\}),\\&(\langle x_6,[0.6,0.7],[0.4,0.5],[0.5,0.6]\rangle ,\{\langle [0.1,0.2],[0.8,0.9],[0.9,1.0]\rangle /u_1,\langle [0.3,0.4],[0.4,0.5],[0.1,0.2]\rangle /u_2\}),\\&(\langle x_7,[0.7,0.8],[0.9,1.0],[0.1,0.2]\rangle ,\{\langle [0.1,0,2],[0.7,0.8],[0.0,0.1]\rangle /u_1,\langle [0.9,1.0],[0.0,0.1],[0.6,0.7]\rangle /u_2\}),\\&(\langle x_8,[0.8,0.9],[0.2,0.3],[0.3,0.4]\rangle ,\{\langle [0.6,0.7],[0.2,0.3],[0.3,0.4]\rangle /u_1,\langle [0.0,0.1],[0.1,0.2],[0.9,1.0]\rangle /u_2\}),\\&(\langle x_9,[0.0,0.1],[0.7,0.8],[0.8,0.9]\rangle ,\{\langle [0.8,0.9],[0.7,0.8],[0.2,0.3]\rangle /u_1,\langle [0.0,0.1],[0.4,0.5],[0.2,0.3]\rangle /u_2\}) \bigg \} \\ \end{aligned}$$

Step 3. :

We construct ivnpivn-matrices ${\mathcal {F}}_1[9\times 2]$ and ${\mathcal {F}}_3[9\times 2]$ for ${\mathcal {F}}_1$ and ${\mathcal {F}}_3$, respectively as

$$\begin{aligned} {\mathcal {F}}_1[9\times 2]=\left[ \begin{array}{c|cccc} \langle [0.5,0.7],[0.1,0.2],[0.7,0.8]\rangle & \langle [0.5,0.6],[0.1,0.3],[0.8,0.9]\rangle & \langle [0.4,0.6],[0.1,0.3],[0.4,0.5]\rangle \\ \langle [0.7,0.8],[0.0,0.1],[0.1,0.2]\rangle & \langle [0.6,0.7],[0.1,0.2],[0.1,0.3]\rangle & \langle [0.6,0.7],[0.3,0.4],[0.8,0.9]\rangle \\ \langle [0.3,0.6],[0.2,0.3],[0.3,0.4]\rangle & \langle [0.5,0.6],[0.2,0.3],[0.3,0.4]\rangle & \langle [0.4,0.5],[0.2,0.4],[0.7,0.9]\rangle \\ \langle [0.6,0.7],[0.1,0.2],[0.2,0.3]\rangle & \langle [0.6,0.7],[0.1,0.2],[0.2,0.3]\rangle & \langle [0.3,0.6],[0.3,0.5],[0.8,0.9]\rangle \\ \langle [0.4,0.5],[0.2,0.3],[0.3,0.4]\rangle & \langle [0.4,0.6],[0.1,0.3],[0.2,0.4]\rangle & \langle [0.7,0.9],[0.2,0.3],[0.4,0.5]\rangle \\ \langle [0.3,0.5],[0.7,0.8],[0.2,0.6]\rangle & \langle [0.5,0.7],[0.3,0.5],[0.4,0.8]\rangle & \langle [0.2,0.6],[0.5,0.6],[0.3,0.7]\rangle \\ \langle [0.3,0.8],[0.6,0.7],[0.5,0.9]\rangle & \langle [0.5,0.9],[0.5,0.8],[0.7,0.9]\rangle & \langle [0.3,0.7],[0.8,0.9],[0.4,0.5]\rangle \\ \langle [0.2,0.6],[0.3,0.4],[0.5,0.7]\rangle & \langle [0.4,0.7],[0.7,0.9],[0.3,0.6]\rangle & \langle [0.6,0.7],[0.2,0.4],[0.1,0.5]\rangle \\ \langle [0.1,0.2],[0.5,0.6],[0.1,0.6]\rangle & \langle [0.3,0.4],[0.2,0.3],[0.4,0.6]\rangle & \langle [0.6,0.9],[0.4,0.7],[0.6,0.8]\rangle \\ \end{array}\right] , \\ {\mathcal {F}}_3[9\times 2]=\left[ \begin{array}{c|cccc} \langle [0.3,0.4],[0.5,0.6],[0.4,0.5]\rangle & \langle [0.1,0.2],[0.7,0.8],[0.0,0.1]\rangle & \langle [0.0,0.1],[0.9,1.0],[0.9,1.0]\rangle \\ \langle [0.1,0.2],[0.3,0.4],[0.6,0.7]\rangle & \langle [0.0,0.1],[0.5,0.6],[0.8,0.9]\rangle & \langle [0.1,0.2],[0.0,0.1],[0.3,0.4]\rangle \\ \langle [0.2,0.3],[0.4,0.5],[0.6,0.7]\rangle & \langle [0.1,0.2],[0.0,0.1],[0.6,0.7]\rangle & \langle [0.8,0.9],[0.9,1.0],[0.0,0.1]\rangle \\ \langle [0.5,0.6],[0.6,0.7],[0.3,0.4]\rangle & \langle [0.0,0.1],[0.6,0.7],[0.1,0.2]\rangle & \langle [0.4,0.5],[0.7,0.8],[0.2,0.3]\rangle \\ \langle [0.3,0.4],[0.7,0.8],[0.9,1.0]\rangle & \langle [0.9,1.0],[0.6,0.7],[0.7,0.8]\rangle & \langle [0.8,0.9],[0.4,0.5],[0.2,0.3]\rangle \\ \langle [0.6,0.7],[0.4,0.5],[0.5,0.6]\rangle & \langle [0.1,0.2],[0.8,0.9],[0.9,1.0]\rangle & \langle [0.3,0.4],[0.4,0.5],[0.1,0.2]\rangle \\ \langle [0.7,0.8],[0.9,1.0],[0.1,0.2]\rangle & \langle [0.1,0.2],[0.7,0.8],[0.0,0.1]\rangle & \langle [0.9,1.0],[0.0,0.1],[0.6,0.7]\rangle \\ \langle [0.8,0.9],[0.2,0.3],[0.3,0.4]\rangle & \langle [0.6,0.7],[0.2,0.3],[0.3,0.4]\rangle & \langle [0.0,0.1],[0.1,0.2],[0.9,1.0]\rangle \\ \langle [0.0,0.1],[0.7,0.8],[0.8,0.9]\rangle & \langle [0.8,0.9],[0.7,0.8],[0.2,0.3]\rangle & \langle [0.0,1.0],[0.4,0.5],[0.2,0.3]\rangle \\ \end{array}\right] \end{aligned}$$

Step 4. :

We calculated the similarity between ${\mathcal {F}}_1[9\times 2]$ and ${\mathcal {F}}_3[9\times 2]$ as

$$\begin{aligned} S({\mathcal {F}}_1[9\times 2],{\mathcal {F}}_3[9\times 2])=0.94 \end{aligned}$$

Step 5. :

Here the ${\mathcal {F}}_1[9\times 2]$ and ${\mathcal {F}}_3[9\times 2]$ are significantly similar. Therefore, we conclude that the person is possibly suffering from cancer.

7 Comparison analysis and discussion

In this section, we present a comparative analysis aiming to certify the feasibility of the proposed method based on similarity measures. The comparative analysis compares the proposed method with four other methods which use similarity measure based on distance measures under ivnpivn-soft set environments.

Firstly, proposed method, the method 1 based on Hamming distance measure, method 2 based on normalized Hamming distance measure, method 3 based on Euclidean distance measure and method 4 based on normalized Euclidean distance measure between two ivnpivn-soft set are compared. The results from the different methods used to resolve the decision-making problem in Example 13 are shown in Table 1.

Table 1 Results for different similarity measures

From Table 1, similarity measure between two ivnpivn-soft set are significantly similar in method 2 and 4. Therefore, we conclude that the person is possibly suffering from cancer in the methods. Similarity measure between two ivnpivn-soft set are not significantly similar in the method 1 and method 3. Therefore, we conclude that the person is not possibly suffering from cancer in the methods.

Secondly, the method 1 based on Hamming distance measure, method 2 based on normalized Hamming distance measure, method 3 based on Euclidean distance measure and method 4 based on normalized Euclidean distance measure between two ivnpivn-soft set are compared. The results from the different methods used to resolve the decision-making problem in Example 14 are shown in Table 2.

Table 2 Results for different similarity measures

From Table 2, similarity measure between two ivnpivn-soft set are significantly similar in proposed method, method 2 and 4. Therefore, we conclude that the person is possibly suffering from cancer in the methods. Similarity measure between two ivnpivn-soft set are not significantly similar in the method 1 and method 3. Therefore, we conclude that the person is not possibly suffering from cancer in the methods.

The reasons why the differences exist is given as. The some methods does not consider the normalized values of distance measures while the other methods does. In addition, the methods use both distance measure and similarity measure while the proposed method used the only similarity measure and the results of these methods may be different with the change in distance measures. Also the distance measures cannot take into account the included angle between two ivnpivn-soft set while the proposed similarity measure can. Consequently, the methods and the proposed method may have different results. Generally speaking, the proposed method can effectively tackle the decision-making problems under ivnpivn-soft set environments including medical diagnosis.

8 Conclusion

In this paper, we define the notion of interval-valued neutrosophic parameterized interval-valued neutrosophic soft set, called ivnpivn-soft set, in a new way by using interval-valued neutrosophic set and soft set. Furthermore, we proposed some definitions and operations on ivnpivn-soft set and constructed ivnpivn-soft matrix which are more functional to make theoretical studies in the ivnpivn-soft set theory. Also, ivnpivn-soft set can be expanding with new research subjects such as algebraic structures, graph, soft computing techniques and game theory.