Interval-valued pythagorean fuzzy rough approximation operators and its application

Abstract

By combining interval-valued Pythagorean fuzzy sets with rough sets, the interval-valued Pythagorean fuzzy rough set model is first constructed in this paper. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators are established subsequently. Then, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy lower and upper approximation operators. Different axiom sets of interval-valued Pythagorean fuzzy set-theoretic operators ensure the existence of different types of interval-valued Pythagorean fuzzy relations producing the same operators. Finally, we give an example to illustrate the practical application of the newly proposed model.

Keywords

Interval-valued pythagorean fuzzy sets interval-valued pythagorean fuzzy relations interval-valued pythagorean fuzzy rough sets axiomatic characterizations

1 Introduction

Rough set theory was initiated by Pawlak [15, 16]. In Pawlak rough set model, the lower and upper approximations are constructed by using an equivalence relation. It may restrict the application domain of the rough set model. On account of this reason, many researchers have generalized the pawlak rough set model [3 6 , 39].

Pythagorean fuzzy sets (PFSs) was initiated by Yager [25 –27]. Pythagorean fuzzy sets have been applied to many fields [4 , 40]. Pythagorean fuzzy rough sets was proposed as a hybrid model of PFSs and rough sets [35]. Interval-valued Pythagorean fuzzy sets (IVPFSs) [17, 37] was put forward as a generalization of PFSs. Intuitionistic fuzzy set (IFS) [1], interval-valued intuitionistic fuzzy set [2] and PFS [35] are all special cases of IVPFSs. The existing literatures on IVPFSs [5 , 32] rarely consider the combination of IVPFSs and rough sets. In this paper, by combining IVPFSs and rough sets, a new rough set model, named the interval-valued Pythagorean fuzzy rough set (IVPFRS) model will be proposed. The connections between special interval-valued Pythagorean fuzzy relations and interval-valued Pythagorean fuzzy approximation operators will also be examined.

Axiomatic characterizations of rough approximation operators are always a hot research topic. In the axiomatic approach, it treats the lower and upper approximation operators as original notions. A series of axioms are used to characterize approximation operators produced by the constructive approach. There are many literatures on the axiomatic characterizations of rough set models [14 , 40]. In this paper, the axiomatic characterizations of the IVPFRS model will be also investigated. Besides, an illustrative example is explored to illustrate the application of the IVPFRS model.

The rest of this paper is organized as follows. In Section 2, we briefly review some basic notions. The IVPFRS model is introduced and some properties of this model are examined in Section 3. Axiomatic characterizations of IVPFRSs are explored in Section 4. In Section 5, we present a decision making algorithm based on the IVPFRS model. An example is given to illustrate the effectiveness of the algorithm. This paper is summarized in the last section with some future prospects.

2 Preliminaries

In this section, we briefly recall some basic concepts and results which will be used in this paper.

2.1 Pythagorean fuzzy sets

Yager firstly proposed the concept of PFSs [25 –27] which is defined as follows.

Definition 2.1. ([25 –27]). Let U be a nonempty and finite universe. A PFS P on U can be represented as the following mathematical symbol: $P = {〈 x, P (μ_{P} (x), ν_{P} (x)) 〉 | x \in U},$ where the functions μ_P : U ⟶ [0, 1] and ν_P : U ⟶ [0, 1] define the Pythagorean membership degree and Pythagorean nonmembership degree of x ∈ U to P, respectively. For all x ∈ U, it satisfies the condition: 0≤ (μ_P (x)) ² + (ν_P (x)) ² ≤ 1. The degree of indeterminacy of x to P is $Π_{P} (x) = \sqrt{1 - (μ_{P} (x))^{2} - (ν_{P} (x))^{2}}$ .

The family of all PFSs on U is denoted as PF (U). For simplicity, Zhang and Xu [36] called P (μ_P (x) , ν_P (x)) a Pythagorean fuzzy number (PFN), denoted by β = P (μ_β, ν_β), where μ_β, ν_β ∈ [0, 1], $Π_{β} = \sqrt{1 - (μ_{β})^{2} - (ν_{β})^{2}}$ and (μ_β) ²+ (ν_β) ² ≤1.

Remark 2.1. It needs to note that a PFS will degenerate into an IFS if μ_P (x) + ν_P (x) ≤1 for all x ∈ U.

2.2 Interval-valued Pythagorean fuzzy sets

As the extension of PFSs, the concept of IVPFSs is defined as follows.

Definition 2.2. ([8 , 17]). Let U be a nonempty and finite universe. Let Int[0,1] denote the set of all closed subintervals of the unite interval [0,1]. An IVPFS $\tilde{P}$ on U can be represented as the following mathematical symbol: $\tilde{P} = {〈 x, (u_{\tilde{P}} (x), v_{\tilde{P}} (x)) 〉 | x \in U},$ where the functions

$u_{\tilde{P}} : U ⟶$ Int[0,1], $x \in U ⟶ u_{\tilde{P}} (x) (= [u_{\tilde{P}}^{-} (x), u_{\tilde{P}}^{+} (x)]) \subseteq [0, 1]$

and

$v_{\tilde{P}} : U ⟶$ Int[0,1], $x \in U ⟶ v_{\tilde{P}} (x) (= [v_{\tilde{P}}^{-} (x), v_{\tilde{P}}^{+} (x)]) \subseteq [0, 1]$

define the interval-valued Pythagorean membership degree and interval-valued Pythagorean non-membership degree of the element x ∈ U to the set $\tilde{P}$ , respectively. Moreover, $u_{\tilde{P}}$ and $v_{\tilde{P}}$ satisfy the following condition: for all x ∈ U, $0 \leq (u_{\tilde{P}}^{+} (x))^{2} + (v_{\tilde{P}}^{+} (x))^{2} \leq 1,$ where $u_{\tilde{P}}^{-} (x)$ and $u_{\tilde{P}}^{+} (x)$ are referred to the lower and upper limits of $u_{\tilde{P}} (x)$ , $v_{\tilde{P}}^{-} (x)$ and $v_{\tilde{P}}^{+} (x)$ are referred to the lower and upper limits of $v_{\tilde{P}} (x)$ . Hence, $\tilde{P}$ can also be represented as the following mathematical symbol: $\tilde{P} = {〈 x, ([u_{\tilde{P}}^{-} (x), u_{\tilde{P}}^{+} (x)], [v_{\tilde{P}}^{-} (x), v_{\tilde{P}}^{+} (x)]) 〉 | x \in U},$ where $0 \leq (u_{\tilde{P}}^{+} (x))^{2} + (v_{\tilde{P}}^{+} (x))^{2} \leq 1$ for all x ∈ U.

For convenience, the family of all IVPFSs on U is denoted by IVPF (U). $\tilde{p} = (u_{\tilde{p}} (x), v_{\tilde{p}} (x)) = ([u_{\tilde{p}}^{-} (x), u_{\tilde{p}}^{+} (x)], [v_{\tilde{p}}^{-} (x), v_{\tilde{p}}^{+} (x)])$ is called an interval-valued Pythagorean fuzzy number (IVPFN).

Definition 2.3. ([8, 17]). Let $\tilde{P}$ be an IVPFS on U. For all x ∈ U, the hesitancy degree (or the degree of indeterminacy) related to $\tilde{P}$ is defined as follows: $\begin{matrix} Π_{\tilde{P}} (x) & = [Π_{\tilde{P}}^{-} (x), Π_{\tilde{P}}^{+} (x)] \\ = [\sqrt{1 - (u_{\tilde{P}}^{+} (x))^{2} - (v_{\tilde{P}}^{+} (x))^{2}}, \\ \sqrt{1 - (u_{\tilde{P}}^{-} (x))^{2} - (v_{\tilde{P}}^{-} (x))^{2}}] . \end{matrix}$

According to Definition 2.2, we have the following remark.

Remark 2.2. (1) If $0 \leq u_{\tilde{P}}^{+} (x) + v_{\tilde{P}}^{+} (x) \leq 1$ for all x ∈ U, then the IVPFS degenerates into an interval-valued intuitionistic fuzzy set.

(2) If the upper and lower limits of the interval values $u_{\tilde{P}} (x)$ and $v_{\tilde{P}} (x)$ are identical for all x ∈ U, then the IVPFS degenerates into a PFS.

Several special IVPFSs are introduced as follows: let U be a nonempty and finite universe, and $\tilde{P} \in IVPF (U)$ ,

$\tilde{P}$ is called an empty IVPFS on U iff $u_{\tilde{P}} (x) = [0, 0]$ and $v_{\tilde{P}} (x) = [1, 1]$ for all x ∈ U. For simplicity, we denote the empty IVPFS as $\tilde{\emptyset}$ .

$\tilde{P}$ is called a full IVPFS on U iff $u_{\tilde{P}} (x) = [1, 1]$ and $v_{\tilde{P}} (x) = [0, 0]$ for all x ∈ U. For simplicity, we denote the full IVPFS as $\tilde{U}$ .

$\tilde{P}$ is called a constant IVPFS on X iff $(u_{\tilde{P}} (x), v_{\tilde{P}} (x)) = (u_{\tilde{P}} (y), v_{\tilde{P}} (y))$ for all x, y ∈ U, i.e. $(u_{\tilde{P}} (x), v_{\tilde{P}} (x)) = ([a_{1}^{-}, a_{1}^{+}], [a_{2}^{-}, a_{2}^{+}])$ for all x ∈ U, where $a_{1}^{-}$ , $a_{1}^{+}$ are referred to as the lower and upper limits of $u_{\tilde{P}} (x)$ , $a_{2}^{-}$ , $a_{2}^{+}$ are referred to as the lower and upper limits of $v_{\tilde{P}} (x)$ and $0 \leq (a_{1}^{+})^{2} + (a_{2}^{+})^{2} \leq 1$ . For simplicity, we denote the constant IVPFS as $[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ .

Meanwhile, for all y ∈ U, several interval-valued Pythagorean fuzzy singleton set [1, 1] _y, its complement [1, 1] _U-{y} and interval-valued Pythagorean fuzzy mediocre set [1, 1] _M are respectively, defined as follows: for all x ∈ U, $(u_{[1, 1]_{y}} (x), v_{[1, 1]_{y}} (x)) = {\begin{matrix} ([1, 1], [0, 0]), x = y, \\ ([0, 0], [1, 1]), x \neq y . \end{matrix}$ $(u_{[1, 1]_{U - {y}}} (x), v_{[1, 1]_{U - {y}}} (x)) = {\begin{matrix} ([0, 0], [1, 1]), x = y, \\ ([1, 1], [0, 0]), x \neq y . \end{matrix}$ $(u_{[1, 1]_{M}} (x), v_{[1, 1]_{M}} (x)) = {\begin{matrix} ([1, 1], [0, 0]), x \in M, \\ ([0, 0], [1, 1]), x \notin M . \end{matrix}$

In order to compare IVPFNs, the score function is defined as follows.

Definition 2.4. ([10, 17]). Let $\tilde{p} = (u_{\tilde{p}} (x), v_{\tilde{p}} (x)) = ([u_{\tilde{p}}^{-} (x), u_{\tilde{p}}^{+} (x)], [v_{\tilde{p}}^{-} (x), v_{\tilde{p}}^{+} (x)])$ be an IVPFN, then the score function $s (\tilde{p})$ of $\tilde{p}$ is defined as: $\begin{matrix} s (\tilde{p}) = \frac{1}{2} [(u_{\tilde{p}}^{-} (x))^{2} - (v_{\tilde{p}}^{-} (x))^{2} + (u_{\tilde{p}}^{+} (x))^{2} - (v_{\tilde{p}}^{+} (x))^{2}] . \end{matrix}$

Based on Definition 2.4, the comparison laws of IVPFNs are defined as follows.

Definition 2.5. ([10]). Let ${\tilde{p}}_{1} = (u_{{\tilde{p}}_{1}}, v_{{\tilde{p}}_{1}})$ and ${\tilde{p}}_{2} = (u_{{\tilde{p}}_{2}}, v_{{\tilde{p}}_{2}})$ be two IVPFNs, then we have:

(1) If $s ({\tilde{p}}_{1}) > s ({\tilde{p}}_{2}),$ then ${\tilde{p}}_{1}$ is bigger than ${\tilde{p}}_{2},$ denoted by ${\tilde{p}}_{1} ≻ {\tilde{p}}_{2};$

(2) If $s ({\tilde{p}}_{1}) < s ({\tilde{p}}_{2}),$ then ${\tilde{p}}_{1}$ is smaller than ${\tilde{p}}_{2},$ denoted by ${\tilde{p}}_{1} ≺ {\tilde{p}}_{2};$

(3) If $s ({\tilde{p}}_{1}) = s ({\tilde{p}}_{2})$ , then ${\tilde{p}}_{1}$ is equal to ${\tilde{p}}_{2},$ denoted by ${\tilde{p}}_{1} \sim {\tilde{p}}_{2}$ .

The operations of IVPFNs are defined as follows.

Definition 2.6. ([8 , 17]). Suppose that three IVPFNs are represented by $\tilde{p} = (u_{\tilde{p}}, v_{\tilde{p}})$ , ${\tilde{p}}_{1} = (u_{{\tilde{p}}_{1}}, v_{{\tilde{p}}_{1}})$ , and ${\tilde{p}}_{2} = (u_{{\tilde{p}}_{2}}, v_{{\tilde{p}}_{2}})$ . Some operations among them are defined as follows: $\begin{matrix} {\tilde{p}}_{1} \lor {\tilde{p}}_{2} = ([max (u_{{\tilde{p}}_{1}}^{-}, u_{{\tilde{p}}_{2}}^{-}), max (u_{{\tilde{p}}_{1}}^{+}, u_{{\tilde{p}}_{2}}^{+})], \\ [min (v_{{\tilde{p}}_{1}}^{-}, v_{{\tilde{p}}_{2}}^{-}), min (v_{{\tilde{p}}_{1}}^{+}, v_{{\tilde{p}}_{2}}^{+})]), \end{matrix}$ $\begin{matrix} = ([u_{{\tilde{p}}_{1}}^{-} \lor u_{{\tilde{p}}_{2}}^{-}, u_{{\tilde{p}}_{1}}^{+} \lor u_{{\tilde{p}}_{2}}^{+}], [v_{{\tilde{p}}_{1}}^{-} \land v_{{\tilde{p}}_{2}}^{-}, v_{{\tilde{p}}_{1}}^{+} \land v_{{\tilde{p}}_{2}}^{+}]); \end{matrix}$ $\begin{matrix} {\tilde{p}}_{1} \land {\tilde{p}}_{2} = ([\min (u_{{\tilde{p}}_{1}}^{-}, u_{{\tilde{p}}_{2}}^{-}), \min (u_{{\tilde{p}}_{1}}^{+}, u_{{\tilde{p}}_{2}}^{+})], \\ [\max (v_{{\tilde{p}}_{1}}^{-}, v_{{\tilde{p}}_{2}}^{-}), \max (v_{{\tilde{p}}_{1}}^{+}, v_{\tilde{p_{2}}}^{+})]), \end{matrix}$ $\begin{matrix} = ([u_{{\tilde{p}}_{1}}^{-} \land u_{{\tilde{p}}_{2}}^{-}, u_{{\tilde{p}}_{1}}^{+} \land u_{{\tilde{p}}_{2}}^{+}], [v_{{\tilde{p}}_{1}}^{-} \lor v_{{\tilde{p}}_{2}}^{-}, v_{{\tilde{p}}_{1}}^{+} \lor v_{\tilde{p_{2}}}^{+}]); \end{matrix}$ $\begin{matrix} {\tilde{p}}_{1} \oplus {\tilde{p}}_{2} = ([\sqrt{(u_{{\tilde{p}}_{1}}^{-})^{2} + (u_{{\tilde{p}}_{2}}^{-})^{2} - (u_{{\tilde{p}}_{1}}^{-})^{2} \cdot (u_{{\tilde{p}}_{2}}^{-})^{2}}, \\ \sqrt{(u_{{\tilde{p}}_{1}}^{+})^{2} + (u_{{\tilde{p}}_{2}}^{+})^{2} - (u_{{\tilde{p}}_{1}}^{+})^{2} \cdot (u_{{\tilde{p}}_{2}}^{+})^{2}}], \end{matrix}$ $\begin{matrix} [v_{{\tilde{p}}_{1}}^{-} \cdot v_{{\tilde{p}}_{2}}^{-}, v_{{\tilde{p}}_{1}}^{+} \cdot v_{{\tilde{p}}_{2}}^{+}]); \end{matrix}$ $\begin{matrix} {\tilde{p}}_{1} \otimes {\tilde{p}}_{2} = ([u_{{\tilde{p}}_{1}}^{-} \cdot u_{{\tilde{p}}_{2}}^{-}, u_{{\tilde{p}}_{1}}^{+} \cdot u_{{\tilde{p}}_{2}}^{+}], \\ [\sqrt{(v_{{\tilde{p}}_{1}}^{-})^{2} + (v_{{\tilde{p}}_{2}}^{-})^{2} - (v_{{\tilde{p}}_{1}}^{-})^{2} \cdot (v_{{\tilde{p}}_{2}}^{-})^{2}}, \end{matrix}$ $\begin{matrix} \sqrt{(v_{{\tilde{p}}_{1}}^{+})^{2} + (v_{{\tilde{p}}_{2}}^{+})^{2} - (v_{{\tilde{p}}_{1}}^{+})^{2} \cdot (v_{{\tilde{p}}_{2}}^{+})^{2}}]); \end{matrix}$ $\begin{matrix} {\tilde{p}}^{c} = ([v_{\tilde{p}}^{-}, v_{\tilde{p}}^{+}], [u_{\tilde{p}}^{-}, u_{\tilde{p}}^{+}]); \end{matrix}$ $\begin{matrix} α \cdot \tilde{p} = ([\sqrt{1 - (1 - (u_{\tilde{p}}^{-})^{2})^{α}}, \\ \sqrt{1 - (1 - (u_{\tilde{p}}^{+})^{2})^{α}}], [(v_{\tilde{p}}^{-})^{α}, (v_{\tilde{p}}^{+})^{α}]), α > 0; \end{matrix}$ $\begin{matrix} (\tilde{p})^{α} = ([(u_{\tilde{p}}^{-})^{α}, (u_{\tilde{p}}^{+})^{α}], [\sqrt{1 - (1 - (v_{\tilde{p}}^{-})^{2})^{α}}, \\ \sqrt{1 - (1 - (v_{\tilde{p}}^{+})^{2})^{α}}]), α > 0 . \end{matrix}$

Theorem 2.1. ([8]). All the operational results in Definition 2.6 are IVPFNs.

Definition 2.7. Let U be a nonempty and finite universe. ${\tilde{P}}_{1} = {〈 x, u_{{\tilde{P}}_{1}} (x), v_{{\tilde{P}}_{1}} (x) 〉 | x \in U}$ and ${\tilde{P}}_{2} = {〈 x, u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{2}} (x)) 〉 | x \in U}$ are two IVPFSs, the operation of ${\tilde{P}}_{1} \tilde{\oplus} {\tilde{P}}_{2}$ is defined as: $\begin{matrix} {\tilde{P}}_{1} \tilde{\oplus} {\tilde{P}}_{2} = {〈 x, (u_{{\tilde{P}}_{1}} (x), v_{{\tilde{P}}_{1}} (x)) \oplus (u_{{\tilde{P}}_{2}} (x), \\ v_{{\tilde{P}}_{2}} (x)) 〉 | x \in U} . \end{matrix}$

The operations of IVPFSs are defined as follows.

Definition 2.8. Let U be a nonempty and finite universe. Suppose $\tilde{P} = {〈 x, (u_{\tilde{P}} (x), v_{\tilde{P}} (x)$ ) 〉|x ∈ U}, ${\tilde{P}}_{1} = {〈 x, (u_{{\tilde{P}}_{1}} (x), v_{{\tilde{P}}_{1}} (x)) 〉 | x \in U}$ and ${\tilde{P}}_{2} = {〈 x, (u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{2}} (x)) 〉 | x \in U}$ are three IVPFSs on U, the union, intersection and complement are defined as follows: $\begin{matrix} (1) {\tilde{P}}_{1} ⋓ {\tilde{P}}_{2} = {〈 x, (u_{{\tilde{P}}_{1}} (x), v_{{\tilde{P}}_{1}} (x)) \lor (u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{2}} (x)) 〉 | x \in U} \\ = {〈 x, (u_{{\tilde{P}}_{1}} (x) \lor u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{1}} (x) \land v_{{\tilde{P}}_{2}} (x)) 〉 | x \in U} \\ = {〈 x, ([u_{{\tilde{P}}_{1}}^{-} (x) \lor u_{{\tilde{P}}_{2}}^{-} (x), u_{{\tilde{P}}_{1}}^{+} (x) \lor u_{{\tilde{P}}_{2}}^{+} (x)], \\ [v_{{\tilde{P}}_{1}}^{-} (x) \land v_{{\tilde{P}}_{2}}^{-} (x), v_{{\tilde{P}}_{1}}^{+} (x) \land v_{{\tilde{P}}_{2}}^{+} (x)]) 〉 | x \in U}; \\ (2) {\tilde{P}}_{1} {\tilde{P}}_{2} = {〈 x, (u_{{\tilde{P}}_{1}} (x), v_{{\tilde{P}}_{1}} (x)) \land (u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{2}} (x)) 〉 | x \in U} \\ = {〈 x, (u_{{\tilde{P}}_{1}} (x) \land u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{1}} (x) \lor v_{{\tilde{P}}_{2}} (x)) 〉 | x \in U} \\ = {〈 x, ([u_{{\tilde{P}}_{1}}^{-} (x) \land u_{{\tilde{P}}_{2}}^{-} (x), u_{{\tilde{P}}_{1}}^{+} (x) \land u_{{\tilde{P}}_{2}}^{+} (x)], \\ [v_{{\tilde{P}}_{1}}^{-} (x) \lor v_{{\tilde{P}}_{2}}^{-} (x), v_{{\tilde{P}}_{1}}^{+} (x) \lor v_{{\tilde{P}}_{2}}^{+} (x)]) 〉 | x \in U}; \\ (3) {\tilde{P}}^{c} = {〈 x, (v_{\tilde{P}} (x), u_{\tilde{P}} (x)) 〉 | x \in U} . \end{matrix}$

Theorem 2.2. Let U be a nonempty and finite universe. For all ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2} \in IVPF (U)$ , we have:

(1) $({\tilde{P}}_{1} ⋓ \tilde{B})^{c} = {\tilde{P}}_{1}^{c} ⋒ {\tilde{P}}_{2}^{c}$ ,

(2) $({\tilde{P}}_{1} ⋒ {\tilde{P}}_{2})^{c} = {\tilde{P}}_{1}^{c} ⋓ {\tilde{P}}_{2}^{c}$ .

Proof. (1) According to Definition 2.8 (1) and (2), we have $\begin{matrix} ({\tilde{P}}_{1} ⋓ {\tilde{P}}_{2})^{c} = {〈 x, (u_{{\tilde{P}}_{1}} (x) \lor u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{1}} (x) \land v_{{\tilde{P}}_{2}} (x)) 〉 | x \in U}^{c} \\ = {〈 x, (v_{{\tilde{P}}_{1}} (x) \land v_{{\tilde{P}}_{2}} (x), u_{{\tilde{P}}_{1}} (x) \lor u_{{\tilde{P}}_{2}} (x)) 〉 | x \in U} \\ = {〈 x, ([v_{{\tilde{P}}_{1}}^{-} (x) \land v_{{\tilde{P}}_{2}}^{-} (x), v_{{\tilde{P}}_{1}}^{+} (x) \land v_{{\tilde{P}}_{2}}^{+} (x)], \\ [u_{{\tilde{P}}_{1}}^{-} (x) \lor u_{{\tilde{P}}_{2}}^{-} (x), u_{{\tilde{P}}_{1}}^{+} (x) \lor u_{{\tilde{P}}_{2}}^{+} (x)]) 〉 | x \in U} \\ = {〈 x, (v_{{\tilde{P}}_{1}} (x), u_{{\tilde{P}}_{1}} (x)) 〉 | x \in X} ⋒ {〈 x, (v_{{\tilde{P}}_{2}} (x), u_{{\tilde{P}}_{2}} (x)) 〉 | x \in U} \\ = {\tilde{P}}_{1}^{c} ⋒ {\tilde{P}}_{2}^{c} . \end{matrix}$

Therefore, we have (1) holds.

(2) It is similar to the proof of (1).

Yager [25] introduced the concept of Pythagorean fuzzy subsets. We give the concept of interval-valued Pythagorean fuzzy subsets.

Definition 2.9. Let U be a nonempty and finite universe. For all ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2} \in IVPF (U),$ ${\tilde{P}}_{1}$ is called an interval-valued Pythagorean fuzzy subset of ${\tilde{P}}_{2}$ , if $u_{{\tilde{P}}_{1}} (x) ⪯ u_{{\tilde{P}}_{2}} (x)$ and $v_{{\tilde{P}}_{1}} (x) ⪰ v_{{\tilde{P}}_{2}} (x)$ hold for all x ∈ U, i.e. $\begin{matrix} u_{{\tilde{P}}_{1}}^{-} (x) \leq u_{{\tilde{P}}_{2}}^{-} (x), u_{{\tilde{P}}_{1}}^{+} (x) \leq u_{{\tilde{P}}_{2}}^{+} (x), v_{{\tilde{P}}_{1}}^{-} (x) \geq v_{{\tilde{P}}_{2}}^{-} (x), \\ v_{{\tilde{P}}_{1}}^{+} (x) \geq v_{{\tilde{P}}_{2}}^{+} (x) \end{matrix}$ for all x ∈ U. In such a circumstance, we denote it by ${\tilde{P}}_{1} ⊑ {\tilde{P}}_{2}$ .

Theorem 2.3. Let U be a nonempty and finite universe. For all ${\tilde{P}}_{1}$ , ${\tilde{P}}_{2}$ , ${\tilde{P}}_{3} \in IVPF (U)$ , we have:

(1) ${\tilde{P}}_{1} ⋓ {\tilde{P}}_{2} ⊒ {\tilde{P}}_{1}$ , ${\tilde{P}}_{1} ⋓ {\tilde{P}}_{2} ⊒ {\tilde{P}}_{2}$ ,

(2) ${\tilde{P}}_{1} ⋒ {\tilde{P}}_{2} ⊑ {\tilde{P}}_{1}$ , ${\tilde{P}}_{1} ⋒ {\tilde{P}}_{2} ⊑ {\tilde{P}}_{2}$ ,

(3) ${\tilde{P}}_{1} ⊑ {\tilde{P}}_{2}, {\tilde{P}}_{1} ⊑ {\tilde{P}}_{3}$ $\Rightarrow {\tilde{P}}_{1} ⊑ {\tilde{P}}_{2} ⋒ {\tilde{P}}_{3}$ ,

${\tilde{P}}_{1} ⊒ {\tilde{P}}_{2}, {\tilde{P}}_{1} ⊒ {\tilde{P}}_{3}$ $\Rightarrow {\tilde{P}}_{1} ⊒ {\tilde{P}}_{2} ⋓ {\tilde{P}}_{3}$ .

Proof. (1) According to Definitions 2.8 and 2.9, we have $\begin{matrix} {\tilde{P}}_{1} ⋓ {\tilde{P}}_{2} = {〈 x, (u_{{\tilde{P}}_{1}} (x) \lor u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{1}} (x) \land v_{{\tilde{P}}_{2}} (x)) 〉 \\ | x \in U} = {〈 x, ([u_{{\tilde{P}}_{1}}^{-} (x) \lor u_{{\tilde{P}}_{2}}^{-} (x), u_{{\tilde{P}}_{1}}^{+} (x) \lor u_{{\tilde{P}}_{2}}^{+} (x)], \\ [v_{{\tilde{P}}_{1}}^{-} (x) \land v_{{\tilde{P}}_{2}}^{-} (x), v_{{\tilde{P}}_{1}}^{+} (x) \land v_{{\tilde{P}}_{2}}^{+} (x)]) 〉 | x \in U} ⊒ \\ {〈 x, ([u_{{\tilde{P}}_{1}}^{-} (x), u_{{\tilde{P}}_{1}}^{+} (x)], [v_{{\tilde{P}}_{1}}^{-} (x), v_{{\tilde{P}}_{1}}^{+} (x)]) 〉 | x \in U} \\ = {\tilde{P}}_{1}, \end{matrix}$ and $\begin{matrix} {\tilde{P}}_{1} ⋓ {\tilde{P}}_{2} = {〈 x, (u_{{\tilde{P}}_{1}} (x) \lor u_{{\tilde{P}}_{2}} (x), v_{{\tilde{P}}_{1}} (x) \land v_{{\tilde{P}}_{2}} (x)) 〉 \\ | x \in U} = {〈 x, ([u_{{\tilde{P}}_{1}}^{-} (x) \lor u_{{\tilde{P}}_{2}}^{-} (x), u_{{\tilde{P}}_{1}}^{+} (x) \lor u_{{\tilde{P}}_{2}}^{+} (x)], \\ [v_{{\tilde{P}}_{1}}^{-} (x) \land v_{{\tilde{P}}_{2}}^{-} (x), v_{{\tilde{P}}_{1}}^{+} (x) \land v_{{\tilde{P}}_{2}}^{+} (x)]) 〉 | x \in U} ⊒ \\ {〈 x, ([u_{{\tilde{P}}_{2}}^{-} (x), u_{{\tilde{P}}_{2}}^{+} (x)], [v_{{\tilde{P}}_{2}}^{-} (x), v_{{\tilde{P}}_{2}}^{+} (x)]) 〉 | x \in U} \\ = {\tilde{P}}_{2} . \end{matrix}$

Therefore, we have ${\tilde{P}}_{1} ⋓ {\tilde{P}}_{2} ⊒ {\tilde{P}}_{1}$ , ${\tilde{P}}_{1} ⋓ {\tilde{P}}_{2} ⊒ {\tilde{P}}_{2}$ .

(2) It is similar to the proof of (1).

(3) According to Definitions 2.8 and 2.9. It can be easily checked without any difficulties.

3 Interval-valued pythagorean fuzzy rough sets

In this section, we first give the concept of interval-valued Pythagorean fuzzy relations on two universes. Then we propose the IVPFRS model by combining IVPFSs and rough sets. Some properties of this model will be also examined.

3.1 Interval-valued Pythagorean fuzzy relations

We first introduce the notion of interval-valued Pythagorean fuzzy relations on two universes as follows.

Definition 3.1. Let U and V be two nonempty and finite universes. An interval-valued Pythagorean fuzzy subset $\tilde{R}$ of U × V is called an interval-valued Pythagorean fuzzy relation on U × V, i.e., $\tilde{R}$ is given by $\tilde{R} = {〈 (x, y), u_{\tilde{R}} (x, y), v_{\tilde{R}} (x, y) 〉 | (x, y) \in U \times V},$ where $u_{\tilde{R}} : U \times V ⟶$ Int[0,1], $(x, y) \in U \times V ⟶ u_{\tilde{R}} (x, y)$ $(= [u_{\tilde{R}}^{-} (x, y), u_{\tilde{R}}^{+} (x, y)])$ ,

$v_{\tilde{R}} : U \times V ⟶$ Int[0,1], $(x, y) \in U \times V ⟶ v_{\tilde{R}} (x, y)$ $(= [v_{\tilde{R}}^{-} (x, y), v_{\tilde{R}}^{+} (x, y)])$ , i.e. $u_{\tilde{R}} (x, y)$ and $v_{\tilde{R}} (x, y)$ are two interval numbers in Int[0, 1], denoting the interval-valued Pythagorean membership degree and interval-valued Pythagorean nonmembership degree of the relationship between x and y, respectively, with the condition: $0 \leq (u_{\tilde{R}}^{+} (x, y))^{2} + (v_{\tilde{R}}^{+} (x, y))^{2} \leq 1$ for all (x, y) ∈ U × V .

The family of all interval-valued Pythagorean fuzzy relations on U × V is denoted by IVPFR (U × V). Particularly, if U = V, then we say that $\tilde{R}$ is an interval-valued Pythagorean fuzzy relation on U. For simplicity, the family of all interval-valued Pythagorean fuzzy relations on U is denoted by IVPFR (U).

Definition 3.2. Let $\tilde{R} \in IVPFR (U)$ ,

(1) $\tilde{R}$ is called serial, if for all x ∈ U, there exists a y ∈ U such that $u_{\tilde{R}} (x, y) = [1, 1]$ and $v_{\tilde{R}} (x, y) = [0, 0];$

(2) $\tilde{R}$ is called reflexive, if for all x ∈ U, $u_{\tilde{R}} (x, x) = [1, 1]$ and $v_{\tilde{R}} (x, x) = [0, 0];$

(3) $\tilde{R}$ is called symmetric, if for all (x, y) ∈ U × U, $u_{\tilde{R}} (x, y) = u_{\tilde{R}} (y, x)$ and $v_{\tilde{R}} (x, y) = v_{\tilde{R}} (y, x);$

(4) $\tilde{R}$ is called transitive, if $u_{\tilde{R}} (x, y) \land u_{\tilde{R}} (y, z) \leq u_{\tilde{R}} (x, z)$ and $v_{\tilde{R}} (x, y) \lor v_{\tilde{R}} (y, z) \geq v_{\tilde{R}} (x, z)$ for all (x, y) ∈ U × U, (x, z) ∈ U × U.

Alternatively, $\tilde{R}$ is transitive if the following conditions are satisfied:

$u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{R}}^{-} (y, z) \leq u_{\tilde{R}}^{-} (x, z),$

$u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{R}}^{+} (y, z) \leq u_{\tilde{R}}^{+} (x, z),$

$v_{\tilde{R}}^{-} (x, y) \lor v_{\tilde{R}}^{-} (y, z) \geq v_{\tilde{R}}^{-} (x, z),$

$v_{\tilde{R}}^{+} (x, y) \lor v_{\tilde{R}}^{+} (y, z) \geq v_{\tilde{R}}^{+} (x, z) .$

3.2 The interval-valued Pythagorean fuzzy rough sets

The interval-valued Pythagorean fuzzy lower and upper approximation operators are defined as follows.

Definition 3.3. Let U and V be two nonempty and finite universes. $\tilde{R}$ is an interval-valued Pythagorean fuzzy relation on U × V. The triple ( $U, V, \tilde{R}$ ) is called an interval-valued Pythagorean fuzzy approximation space. For all $\tilde{A} \in IVPF (V)$ , the lower and upper approximations of $\tilde{A}$ w.r.t. ( $U, V, \tilde{R}$ ), denoted as $\underline{\tilde{R}} (\tilde{A})$ and $\bar{\tilde{R}} (\tilde{A})$ , respectively, are two IVPFSs on U and are defined as follows: $\underline{\tilde{R}} (\tilde{A}) = {〈 x, (u_{\underline{\tilde{R}} (\tilde{A})} (x), v_{\underline{\tilde{R}} (\tilde{A})} (x)) 〉 | x \in U},$ $\bar{\tilde{R}} (\tilde{A}) = {〈 x, (u_{\bar{\tilde{R}} (\tilde{A})} (x), v_{\bar{\tilde{R}} (\tilde{A})} (x)) 〉 | x \in U},$ where $u_{\underline{\tilde{R}} (\tilde{A})} (x) = ⋀_{y \in V} {v_{\tilde{R}} (x, y) \lor u_{\tilde{A}} (y)}, v_{\underline{\tilde{R}} (\tilde{A})} (x)$ $= ⋁_{y \in V} {u_{\tilde{R}} (x, y) \land v_{\tilde{A}} (y)}$ ,

$u_{\bar{\tilde{R}} (\tilde{A})} (x) = ⋁_{y \in V} {u_{\tilde{R}} (x, y) \land u_{\tilde{A}} (y)}, v_{\bar{\tilde{R}} (\tilde{A})} (x)$ $= ⋀_{y \in V} {v_{\tilde{R}} (x, y) \lor v_{\tilde{A}} (y)}$ . $\underline{\tilde{R}} (\tilde{A})$ and $\bar{\tilde{R}} (\tilde{A})$ are referred to as the interval-valued Pythagorean fuzzy lower and upper approximations of $\tilde{A}$ w.r.t. $(U, V, \tilde{R})$ , respectively. The pair $(\underline{\tilde{R}} (\tilde{A}), \bar{\tilde{R}} (\tilde{A}))$ is referred to as an IVPFRS w.r.t. $(U, V, \tilde{R})$ . $\underline{\tilde{R}}, \bar{\tilde{R}} : IVPF (V) ⟶ IVPF (U)$ are called interval-valued Pythagorean fuzzy lower and upper approximation operators, respectively. If $\underline{\tilde{R}} (\tilde{A}) = \bar{\tilde{R}} (\tilde{A})$ , then we say that $\tilde{A}$ is definable, otherwise, we say that $\tilde{A}$ is undefinable.

Particularly, if U = V, then we denote $(U, V, \tilde{R})$ as $(U, \tilde{R})$ .

Table 1
The interval-valued Pythagorean fuzzy relation $\tilde{R}$

$\tilde{R}$ x ₁ x ₂ x ₃

x ₁ 〈[1, 1] , [0, 0] 〉〈[0.6, 0.8] , [0.1, 0.2] 〉〈[0.4, 0.7] , [0.2, 0.4] 〉

x ₂ 〈{0.5, 0.6} , {0.4, 0.7} 〉〈[1, 1] , [0, 0] 〉〈[0.7, 0.8] , [0.3, 0.4] 〉

x ₃ 〈[0.4, 0.8] , [0.1, 0.3] 〉〈[0.4, 0.7] , [0.5, 0.6] 〉〈[1, 1] , [0, 0] 〉

$\tilde{R}$	x ₁	x ₂	x ₃
x ₁	〈[1, 1] , [0, 0] 〉	〈[0.6, 0.8] , [0.1, 0.2] 〉	〈[0.4, 0.7] , [0.2, 0.4] 〉
x ₂	〈{0.5, 0.6} , {0.4, 0.7} 〉	〈[1, 1] , [0, 0] 〉	〈[0.7, 0.8] , [0.3, 0.4] 〉
x ₃	〈[0.4, 0.8] , [0.1, 0.3] 〉	〈[0.4, 0.7] , [0.5, 0.6] 〉	〈[1, 1] , [0, 0] 〉

Obviously, the interval-valued Pythagorean fuzzy lower and upper approximations in Definition 3.3 are referred to as the following equivalent forms:

$u_{\underline{\tilde{R}} (\tilde{A})} (x) = [⋀_{y \in V} v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{A}}^{-} (y), ⋀_{y \in V} v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{A}}^{+} (y)]$ ,

$v_{\underline{\tilde{R}} (\tilde{A})} (x) = [⋁_{y \in V} u_{\tilde{R}}^{-} (x, y) \land v_{A}^{-} (y), ⋁_{y \in V} u_{\tilde{R}}^{+} (x, y) \land v_{A}^{+} (y)]$ ,

$u_{\bar{\tilde{R}} (\tilde{A})} (x) = [⋁_{y \in V} u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{A}}^{-} (y), ⋁_{y \in V} u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{A}}^{+} (y)]$ ,

$v_{\bar{\tilde{R}} (\tilde{A})} (x) = [⋀_{y \in V} v_{\tilde{R}}^{-} (x, y) \lor v_{\tilde{A}}^{-} (y), ⋀_{y \in V} v_{\tilde{R}}^{+} (x, y) \lor v_{\tilde{A}}^{+} (y)]$ .

Example 3.1. Let U = {x₁, x₂, x₃} and $\tilde{R}$ be an interval-valued Pythagorean fuzzy relation on U, see Table 1.

Let $\tilde{A} = {〈 x_{1}, [0.4, 0.7], [0.3, 0.4] 〉, 〈 x_{2}, [0.5, 0.6],$ [0.5, 0.7] 〉, 〈x₃, [0.7, 0.8] , [0.1, 0.3] 〉}.

From Definition 3.3, the interval-valued Pythagorean fuzzy lower and upper approximations of $\tilde{A}$ associated with interval-valued Pythagorean fuzzy relation $\tilde{R}$ can be calculated as follows. $\begin{matrix} \underline{\tilde{R}} (\tilde{A}) = {〈 x_{1}, [0.4, 0.7], [0.5, 0.7] 〉, 〈 x_{2}, [0.4, 0.6], \\ [0.5, 0.7] 〉, 〈 x_{3}, [0.4, 0.7], [0.4, 0.7] 〉}, \\ \bar{\tilde{R}} (\tilde{A}) = {〈 x_{1}, [0.5, 0.7], [0.2, 0.4] 〉, 〈 x_{2}, [0.7, 0.8], \\ [0.3, 0.4] 〉, 〈 x_{3}, [0.7, 0.8], [0.1, 0.3] 〉} . \end{matrix}$

Remark 3.1. If $\tilde{R}$ in Definition 3.3 is degenerated to a Pythagorean fuzzy relation and $\tilde{A}$ is degenerated to a Pythagorean fuzzy set, then interval-valued Pythagorean fuzzy lower and upper approximation operators will be degenerated to Pythagorean fuzzy lower and upper approximation operators in [35].

In the following, some properties of IVPFRSs will be discussed. We only discuss the case of U = V.

Theorem 3.1. Let ( $U, \tilde{R}$ ) be an interval-valued Pythagorean fuzzy approximation space. For all $\tilde{A} \in IVPF (U)$ , the interval-valued Pythagorean fuzzy lower and upper approximations of $\tilde{A}$ w.r.t. $(U, \tilde{R})$ satisfy the following properties:

(IVPFL1) $\underline{\tilde{R}} ({\tilde{A}}^{c}) = (\bar{\tilde{R}} (\tilde{A}))^{c}$ ,

(IVPFU1) $\bar{\tilde{R}} ({\tilde{A}}^{c}) = (\underline{\tilde{R}} (\tilde{A}))^{c}$ ;

(IVPFL2) $\underline{\tilde{R}} (\tilde{A} ⋒ \tilde{B}) = \underline{R} (\tilde{A}) ⋒ \underline{R} (\tilde{B})$ ,

(IVPFU2) $\bar{\tilde{R}} (\tilde{A} ⋓ \tilde{B}) = \bar{\tilde{R}} (\tilde{A}) ⋓ \bar{\tilde{R}} (\tilde{B})$ ;

(IVPFL3) $\tilde{A} ⊑ \tilde{B} \Rightarrow \underline{\tilde{R}} (\tilde{A}) ⊑ \underline{\tilde{R}} (\tilde{B})$ ,

(IVPFU3) $\tilde{A} ⊑ \tilde{B} \Rightarrow \bar{\tilde{R}} (\tilde{A}) ⊑ \bar{\tilde{R}} (\tilde{B})$ ;

(IVPFL4) $\underline{\tilde{R}} (\tilde{A}) ⋓ \underline{\tilde{R}} (\tilde{B}) \subseteq \underline{\tilde{R}} (\tilde{A} ⋓ \tilde{B})$ ,

(IVPFU4) $\bar{\tilde{R}} (\tilde{A} ⋒ \tilde{B}) \subseteq \bar{\tilde{R}} (\tilde{A}) ⋒ \bar{\tilde{R}} (\tilde{B})$ ;

(IVPFL5) $\underline{\tilde{R}} (\tilde{A} ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]) = \underline{R} (A) ⋓$ $[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ ,

(IVPFU5) $\bar{\tilde{R}} (\tilde{A} ⋒ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]) = \bar{\tilde{R}} (\tilde{A}) ⋒ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ ;

(IVPFL6) $\underline{\tilde{R}} (\tilde{U}) = \tilde{U}$ ,

(IVPFU6) $\bar{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ .

Proof. (IVPFL1) According to Theorem 2.2, Definitions 3.3 and 2.9, for all x ∈ U, we have

$- 2.3 {ptu}_{\underline{\tilde{R}} ({\tilde{A}}^{c})} (x) = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{{\tilde{A}}^{c}} (y)$

$= ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor v_{\tilde{A}} (y)$

$= {{⋀_{y \in U} v_{\tilde{R}} (x, y) \lor v_{\tilde{A}} (y)}^{c}}^{c}$

$= {⋁_{y \in U} (v_{\tilde{R}} (x, y) \lor v_{\tilde{A}} (y))^{c}}^{c}$

$= {⋁_{y \in U} (v_{\tilde{R}} (x, y))^{c} \land (v_{\tilde{A}} (y))^{c}}^{c}$

$= {⋁_{y \in U} u_{\tilde{R}} (x, y) \land u_{\tilde{A}} (y)}^{c}$

$= u_{(\bar{\tilde{R}} (\tilde{A}))^{c}} (x),$ and

$- 3.9 {ptv}_{\underline{\tilde{R}} ({\tilde{A}}^{c})} (x) = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{{\tilde{A}}^{c}} (y)$

$= ⋁_{y \in U} u_{\tilde{R}} (x, y) \land u_{\tilde{A}} (y)$

$= {{⋁_{y \in U} u_{\tilde{R}} (x, y) \land u_{\tilde{A}} (y)}^{c}}^{c}$

$= {⋀_{y \in U} {u_{\tilde{R}} (x, y) \land u_{\tilde{A}} (y)}^{c}}^{c}$

$= {⋀_{y \in U} (u_{\tilde{R}} (x, y))^{c} \lor (u_{\tilde{A}} (y))^{c}}^{c}$

$= {⋀_{y \in U} v_{\tilde{R}} (x, y) \lor v_{\tilde{A}} (y)}^{c}$

$= v_{(\bar{\tilde{R}} (\tilde{A}))^{c}} (x) .$

Therefore, we have $\underline{\tilde{R}} ({\tilde{A}}^{c}) = (\bar{\tilde{R}} (\tilde{A}))^{c}$ .

(IVPFU1) According to Definitions 3.3 and 2.9, for all x ∈ U, we have

$- . 9 pt (\bar{\tilde{R}} ({\tilde{A}}^{c}))^{c} = {〈 x, u_{\bar{\tilde{R}} ({\tilde{A}}^{c})} (x), v_{\bar{\tilde{R}} ({\tilde{A}}^{c})} (x) 〉 | x \in U}^{c}$

$= {〈 x, v_{\bar{\tilde{R}} ({\tilde{A}}^{c})} (x), u_{\bar{\tilde{R}} ({\tilde{A}}^{c})} (x) 〉 | x \in U}$

$= {〈 x, v_{(\underline{\tilde{R}} (\tilde{A}))^{c}} (x), u_{(\underline{\tilde{R}} (\tilde{A}))^{c}} (x) 〉 | x \in U}$

$= {〈 x, u_{\underline{\tilde{R}} (\tilde{A})} (x), v_{\underline{\tilde{R}} (\tilde{A})} (x) 〉 | x \in U}$

$= \underline{\tilde{R}} (\tilde{A})$ .

Therefore, we have $\bar{\tilde{R}} ({\tilde{A}}^{c}) = (\underline{\tilde{R}} (\tilde{A}))^{c} .$

From above conclusions (IVPFL1) and (IVPFU1), we know that $\underline{\tilde{R}}$ , $\bar{\tilde{R}}$ have the property of the duality, we only discuss the case of $\underline{\tilde{R}}$ .

(IVPFL2) According to Definition 3.3, for all x ∈ U, we have $\begin{matrix} u_{\underline{\tilde{R}} (\tilde{A} ⋒ \tilde{B})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{\tilde{A} ⋒ \tilde{B}} (y) \\ = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor (u_{\tilde{A}} (y) \land u_{\tilde{B}} (y)) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor (u_{\tilde{A}}^{-} (y) \land u_{\tilde{B}}^{-} (y)), \\ ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor (u_{\tilde{A}} (y)^{+} \land u_{\tilde{B}}^{+} (y))] \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{A}}^{-} (y) \land \\ ⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{B}}^{-} (y), \\ ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{A}}^{+} (y) \land \\ ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{B}}^{+} (y)] \\ = u_{\underline{\tilde{R}} (\tilde{A})} (x) \land u_{\underline{\tilde{R}} (\tilde{B})} (x) = u_{\underline{\tilde{R}} (\tilde{A}) ⋒ \underline{\tilde{R}} (\tilde{B})} (x), \end{matrix}$ and $\begin{matrix} v_{\underline{\tilde{R}} (\tilde{A} ⋒ \tilde{B})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{\tilde{A} ⋒ \tilde{B}} (y) \\ = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land (v_{\tilde{A}} (y) \lor v_{\tilde{B}} (y)) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land (v_{\tilde{A}}^{-} (y) \lor v_{\tilde{B}}^{-} (y)), \\ ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land (v_{\tilde{A}}^{+} (y) \lor v_{\tilde{B}}^{+} (y))] \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land v_{\tilde{A}}^{-} (y) \lor \\ ⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land v_{\tilde{B}}^{-} (y), \\ ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land v_{\tilde{A}}^{+} (y) \lor \\ ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land v_{\tilde{B}}^{+} (y)] \end{matrix}$ $\begin{matrix} = v_{\underline{\tilde{R}} (\tilde{A})} (x) \lor v_{\underline{\tilde{R}} (\tilde{B})} (x) = v_{\underline{\tilde{R}} (\tilde{A}) ⋒ \underline{\tilde{R}} (\tilde{B})} (x) . \end{matrix}$

Therefore, we have $\underline{\tilde{R}} (\tilde{A} ⋒ \tilde{B}) = \underline{\tilde{R}} (\tilde{A}) ⋒ \underline{\tilde{R}} (\tilde{B})$ .

(IVPFL3) Since $\tilde{A} ⊑ \tilde{B}$ , then ∀y ∈ U, we have $u_{\tilde{A}}^{-} (y) \leq u_{\tilde{B}}^{-} (y)$ , $u_{A}^{+} (y) \leq u_{\tilde{B}}^{+} (y)$ , $v_{\tilde{A}}^{-} (y) \geq v_{\tilde{B}}^{-} (y)$ and $v_{\tilde{A}}^{+} (y) \geq v_{\tilde{B}}^{+} (y)$ . Consequently, for all x ∈ U, we have $\begin{matrix} u_{\underline{\tilde{R}} (\tilde{A})} (x) & = [⋀_{y \in U} u_{\tilde{A}}^{-} (y) \lor v_{\tilde{R}}^{-} (x, y), ⋀_{y \in U} u_{\tilde{A}}^{+} (y) \lor v_{\tilde{R}}^{+} (x, y)] \\ ⪯ [⋀_{y \in U} u_{\tilde{B}}^{-} (y) \lor v_{\tilde{R}}^{-} (x, y), ⋀_{y \in U} u_{\tilde{B}}^{+} (y) \lor v_{\tilde{R}}^{+} (x, y)] \\ = u_{\underline{\tilde{R}} (\tilde{B})} (x), \end{matrix}$ and $\begin{matrix} v_{\underline{\tilde{R}} (\tilde{A})} (x) & = [⋁_{y \in U} v_{\tilde{A}}^{-} (y) \land u_{\tilde{R}}^{-} (x, y), ⋁_{y \in U} v_{\tilde{A}}^{+} (y) \land u_{\tilde{R}}^{+} (x, y)] \\ ⪰ [⋁_{y \in U} v_{\tilde{B}}^{-} (y) \land u_{\tilde{R}}^{-} (x, y), ⋁_{y \in U} v_{\tilde{B}}^{-} (y) \land u_{\tilde{R}}^{-} (x, y)] \\ = v_{\underline{\tilde{R}} (\tilde{B})} (x) . \end{matrix}$

Therefore, $u_{\underline{\tilde{R}} (\tilde{A})} (x) ⪯ u_{\underline{\tilde{R}} (\tilde{B})} (x)$ and $v_{\underline{\tilde{R}} (\tilde{A})} (x) ⪰ v_{\underline{\tilde{R}} (\tilde{B})} (x)$ for all x ∈ U, we have $\underline{\tilde{R}} (\tilde{A}) ⊑ \underline{\tilde{R}} (\tilde{B})$ .

(IVPFL4) From above result (IVPFL3) and Theorem 2.3, it can be easily checked without any difficulties.

(IVPFL5) For all x ∈ U, we have $\begin{matrix} u_{\underline{\tilde{R}} (\tilde{A} ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}])} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{\tilde{A} ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (y) \\ = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor (u_{\tilde{A}} (y) \lor u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (y)) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor (u_{\tilde{A}}^{-} (y) \lor u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]}^{-} (y)), \\ ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor (u_{\tilde{A}}^{+} (y) \lor u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]}^{+} (y))] \\ = [⋀_{y \in U} (v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{A}}^{-} (y)) \lor a_{1}^{-}, \\ ⋀_{y \in U} (v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{A}}^{+} (y)) \lor a_{1}^{+}] \\ = u_{\underline{\tilde{R}} (\tilde{A})} (x) \lor u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (x) \\ = u_{\underline{\tilde{R}} (\tilde{A}) ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (x), \end{matrix}$ and{{ $\begin{matrix} v_{\underline{\tilde{R}} (\tilde{A} ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}])} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{\tilde{A} ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (y) \\ = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land (v_{\tilde{A}} (y) \land v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (y)) \\ = [⋁_{y \in U} u_{R}^{-} (x, y) \land (v_{A}^{-} (y) \land v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{-}}]}^{-} (y)), \\ ⋁_{y \in U} u_{R}^{+} (x, y) \land (v_{A}^{+} (y) \land v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]}^{+} (y))] \\ = [⋁_{y \in U} (u_{\tilde{R}}^{-} (x, y) \land v_{\tilde{A}}^{-} (y)) \land a_{2}^{-}, \\ ⋁_{y \in U} (u_{\tilde{R}}^{+} (x, y) \land v_{\tilde{A}}^{+} (y)) \land a_{2}^{+}] \\ = v_{\underline{\tilde{R}} (\tilde{A})} (x) \land v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (x) \\ = v_{\underline{\tilde{R}} (\tilde{A}) ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (x) . \end{matrix}$

Therefore, we conclude that (IVPFL5) holds.

(IVPFL6) According to Definition 3.3, for all x ∈ U, we have $\begin{matrix} u_{\underline{\tilde{R}} (\tilde{U})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{\tilde{U}} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{U}}^{-} (y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{U}}^{+} (y)] \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor 1, ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor 1] \\ = [1, 1] = u_{\tilde{U}} (x), \end{matrix}$ and $\begin{matrix} v_{\underline{R} (\tilde{U})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{\tilde{U}} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land v_{\tilde{U}}^{-} (y), ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land v_{\tilde{U}}^{+} (y)] \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land 0, ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land 0] \\ = [0, 0] = v_{\tilde{U}} (x) . \end{matrix}$

Therefore, we have (IVPFL6) holds.

Theorem 3.2. Let ( $U, \tilde{R}$ ) be an interval-valued Pythagorean fuzzy approximation space. $\tilde{R}$ is reflexive. For all $\tilde{A} \in IVPF (U)$ , the interval-valued Pythagorean fuzzy lower and upper approximations of $\tilde{A}$ w.r.t. $(U, \tilde{R})$ satisfy the following properties:

(1) $\underline{\tilde{R}} (\tilde{U}) = \bar{\tilde{R}} (\tilde{U}) = \tilde{U}$ ,

(2) $\underline{\tilde{R}} (\tilde{\emptyset}) = \bar{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ ,

(3) $\underline{\tilde{R}} (\tilde{A}) ⊑ \tilde{A} ⊑ \bar{\tilde{R}} (\tilde{A})$ .

Proof. (1) Since $\tilde{R}$ is reflexive, then for all x ∈ U, $u_{\tilde{R}} (x, x) = [1, 1], v_{\tilde{R}} (x, x) = [0, 0]$ , $\begin{matrix} u_{\bar{\tilde{R}} (\tilde{U})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land u_{\tilde{U}} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{U}}^{-} (y), ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{U}}^{+} (y)] \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land 1, ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land 1] \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y), ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y)] \\ = [u_{\tilde{R}}^{-} (x, x) \lor ⋁_{y \neq x} u_{\tilde{R}}^{-} (x, y), u_{\tilde{R}}^{+} (x, x) \lor \\ ⋁_{y \neq x} u_{\tilde{R}}^{+} (x, y)] = [1, 1] = u_{\tilde{U}} (x), \end{matrix}$ and $\begin{matrix} v_{\bar{\tilde{R}} (\tilde{U})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor v_{\tilde{U}} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor v_{\tilde{U}}^{-} (y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor v_{\tilde{U}}^{+} (y)] \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor 0, ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor 0] \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y)] \\ = [v_{\tilde{R}}^{-} (x, x) \land ⋀_{y \neq x} v_{\tilde{R}}^{-} (x, y), v_{\tilde{R}}^{+} (x, x) \land \\ ⋀_{y \neq x} v_{\tilde{R}}^{+} (x, y)] = [0, 0] = v_{\tilde{U}} (x), \end{matrix}$ which implies that $\bar{\tilde{R}} (\tilde{U}) = \tilde{U}$ . By (IVPFL6), we can obtain $\underline{\tilde{R}} (\tilde{U}) = \tilde{U}$ . Therefore, we have $\bar{\tilde{R}} (\tilde{U}) = \underline{R} (\tilde{U}) = \tilde{U}$ .

(2) Since $\tilde{R}$ is reflexive, then for all x ∈ U, $u_{\tilde{R}} (x, x) = [1, 1], v_{\tilde{R}} (x, x) = [0, 0]$ , $\begin{matrix} u_{\underline{\tilde{R}} (\tilde{\emptyset})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{\tilde{\emptyset}} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{\emptyset}}^{-} (y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{\emptyset}}^{+} (y)] \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor 0, ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor 0] \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y)] \end{matrix}$ $\begin{matrix} = [v_{\tilde{R}}^{-} (x, x) \land ⋀_{y \neq x} v_{\tilde{R}}^{-} (x, y), v_{\tilde{R}}^{+} (x, x) \land \\ ⋀_{y \neq x} v_{\tilde{R}}^{+} (x, y)] = [0, 0] = u_{\tilde{\emptyset}} (x), \end{matrix}$ and $\begin{matrix} v_{\underline{\tilde{R}} (\tilde{\emptyset})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{\tilde{\emptyset}} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land v_{\tilde{\emptyset}}^{-} (y), ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land v_{\tilde{\emptyset}}^{+} (y)] \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land 1, ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land 1] \\ = [u_{\tilde{R}}^{-} (x, x) \lor ⋁_{y \neq x} u_{\tilde{R}}^{-} (x, y), u_{\tilde{R}}^{+} (x, x) \lor \\ ⋁_{y \neq x} u_{\tilde{R}}^{+} (x, y)] = [1, 1] = v_{\tilde{\emptyset}} (x), \end{matrix}$ which implies that $\underline{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ . By (IVPFU6), we can obtain $\bar{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ . Therefore, we have $\bar{\tilde{R}} (\tilde{\emptyset}) = \underline{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ .

(3) According to Definition 3.3, for all x ∈ U, we have $\begin{matrix} u_{\underline{\tilde{R}} (\tilde{A})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{\tilde{A}} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{A}}^{-} (y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{A}}^{+} (y)] \\ = [(v_{\tilde{R}}^{-} (x, x) \lor u_{\tilde{A}}^{-} (x)) \land ⋀_{y \neq x} v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{A}}^{-} (y), \\ (v_{\tilde{R}}^{+} (x, x) \lor u_{\tilde{A}}^{+} (x)) \land ⋀_{y \neq x} v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{A}}^{+} (y)] \\ = [u_{\tilde{A}}^{-} (x) \land ⋀_{y \neq x} v_{\tilde{R}}^{-} (x, y) \lor u_{\tilde{A}}^{-} (y), u_{\tilde{A}}^{+} (x) \land \\ ⋀_{y \neq x} v_{\tilde{R}}^{+} (x, y) \lor u_{\tilde{A}}^{+} (y)] ⪯ [u_{\tilde{A}}^{-} (x)), u_{\tilde{A}}^{+} (x))] \\ = u_{\tilde{A}} (x), \end{matrix}$ and $\begin{matrix} v_{\underline{\tilde{R}} (\tilde{A})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{\tilde{A}} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land v_{\tilde{A}}^{-} (y), ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land v_{\tilde{A}}^{+} (y)] \\ = [(u_{\tilde{R}}^{-} (x, x) \land v_{\tilde{A}}^{-} (x)) \lor ⋁_{y \neq x} u_{\tilde{R}}^{-} (x, y) \land v_{\tilde{A}}^{-} (y), \\ (u_{\tilde{R}}^{+} (x, x) \land v_{\tilde{A}}^{+} (x)) \lor ⋁_{y \neq x} u_{\tilde{R}}^{+} (x, y) \land v_{\tilde{A}}^{+} (y)] \end{matrix}$ $\begin{matrix} = [v_{\tilde{A}}^{-} (x) \lor ⋁_{y \neq x} u_{\tilde{R}}^{-} (x, y) \land v_{\tilde{A}}^{-} (y), v_{\tilde{A}}^{+} (x)) \lor \\ ⋁_{y \neq x} u_{\tilde{R}}^{+} (x, y) \land v_{\tilde{A}}^{+} (y)] ⪰ [v_{\tilde{A}}^{-} (x), v_{\tilde{A}}^{+} (x))] \\ = v_{\tilde{A}} (x) . \end{matrix}$ On the other hand, we have $\begin{matrix} u_{\bar{\tilde{R}} (\tilde{A})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land u_{\tilde{A}} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{A}}^{-} (y), ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{A}}^{+} (y)] \\ = [(u_{\tilde{R}}^{-} (x, x) \land u_{\tilde{A}}^{-} (x)) \lor ⋁_{y \neq x} u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{A}}^{-} (y), \\ (u_{\tilde{R}}^{+} (x, x) \land u_{\tilde{A}}^{+} (x)) \lor ⋁_{y \neq x} u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{A}}^{+} (y)] \\ = [u_{\tilde{A}}^{-} (x) \lor ⋁_{y \neq x} u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{A}}^{-} (y), u_{\tilde{A}}^{+} (x) \lor \\ ⋁_{y \neq x} u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{A}}^{+} (y)] ⪰ [u_{\tilde{A}}^{-} (x), u_{\tilde{A}}^{+} (x)] \\ = u_{\tilde{A}} (x), \end{matrix}$ and $\begin{matrix} v_{\bar{\tilde{R}} (\tilde{A})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor v_{\tilde{A}} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor v_{\tilde{A}}^{-} (y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor v_{\tilde{A}}^{+} (y)] \\ = [(v_{\tilde{R}}^{-} (x, x) \lor v_{\tilde{A}}^{-} (x)) \land ⋀_{y \neq x} v_{\tilde{R}}^{-} (x, y) \lor v_{\tilde{A}}^{-} (y), \\ (v_{\tilde{R}}^{+} (x, x) \lor v_{\tilde{A}}^{+} (x)) \land ⋀_{y \neq x} v_{\tilde{R}}^{+} (x, y) \lor v_{\tilde{A}}^{+} (y)] \\ = [v_{\tilde{A}}^{-} (x) \land ⋀_{y \neq x} v_{\tilde{R}}^{-} (x, y) \lor v_{\tilde{A}}^{-} (y), v_{\tilde{A}}^{+} (x) \land \\ ⋀_{y \neq x} v_{\tilde{R}}^{+} (x, y) \lor v_{\tilde{A}}^{+} (y)] ⪯ [v_{\tilde{A}}^{-} (x), v_{\tilde{A}}^{+} (x)] \\ = v_{\tilde{A}} (x), \end{matrix}$ which implies that $\underline{\tilde{R}} (\tilde{A}) ⊑ \tilde{A} ⊑ \bar{\tilde{R}} (\tilde{A})$ .

Corollary 3.1. Let ( $U, \tilde{R}$ ) be an interval-valued Pythagorean fuzzy approximation space. $\tilde{R}$ is reflexive, for all $\tilde{A} \in IVPF (U)$ , we have the following properties:

(1) $\underline{\tilde{R}} (\underline{\tilde{R}} (\tilde{A})) ⊑ \underline{\tilde{R}} (\tilde{A}),$

(2) $\bar{\tilde{R}} (\underline{\tilde{R}} (\tilde{A})) ⊑ \bar{\tilde{R}} (\tilde{A}),$

(3) $\bar{\tilde{R}} (\tilde{A}) ⊑ \bar{\tilde{R}} (\bar{\tilde{R}} (\tilde{A})),$

(4) $\underline{\tilde{R}} (\tilde{A}) ⊑ \underline{\tilde{R}} (\bar{\tilde{R}} (\tilde{A})) .$

Proof. It follows immediately from Theorems 3.2 and 3.1.

Theorem 3.3. Let U be a nonempty and finite universe. Assume that ${\tilde{R}}_{1}, {\tilde{R}}_{2} \in IVPFR (U)$ are two interval-valued Pythagorean fuzzy relations on U. If ${\tilde{R}}_{1} ⊑ {\tilde{R}}_{2}$ , for all $\tilde{A} \in IVPF (U)$ , we have the following properties:

(1) $\underline{{\tilde{R}}_{1}} (\tilde{A}) ⊒ \underline{{\tilde{R}}_{2}} (\tilde{A})$ ,

(2) $\bar{{\tilde{R}}_{1}} (\tilde{A}) ⊑ \bar{{\tilde{R}}_{2}} (\tilde{A}) .$

Proof. (1) Since ${\tilde{R}}_{1} ⊑ {\tilde{R}}_{2}$ , then for all (x, y) ∈ U × U, we have $u_{{\tilde{R}}_{1}}^{-} (x, y) \leq u_{{\tilde{R}}_{2}}^{-} (x, y)$ , $u_{{\tilde{R}}_{1}}^{+} (x, y) \leq u_{{\tilde{R}}_{2}}^{+} (x, y)$ ; $v_{{\tilde{R}}_{1}}^{-} (x, y)$ $\geq v_{{\tilde{R}}_{2}}^{-} (x, y)$ , $v_{{\tilde{R}}_{1}}^{+} (x, y)$ $\geq v_{{\tilde{R}}_{2}}^{+} (x, y)$ .

Consequently, we have $\begin{matrix} u_{\underline{{\tilde{R}}_{1}} (\tilde{A})} (x) & = ⋀_{y \in U} v_{{\tilde{R}}_{1}} (x, y) \lor u_{\tilde{A}} (y) \\ = [⋀_{y \in U} v_{{\tilde{R}}_{1}}^{-} (x, y) \lor u_{\tilde{A}}^{-} (y), ⋀_{y \in U} v_{{\tilde{R}}_{1}}^{+} (x, y) \lor u_{\tilde{A}}^{+} (y)] \\ ⪰ [⋀_{y \in U} v_{{\tilde{R}}_{2}}^{-} (x, y) \lor u_{\tilde{A}}^{-} (y), ⋀_{y \in U} v_{{\tilde{R}}_{2}}^{+} (x, y) \lor u_{\tilde{A}}^{+} (y)] \\ = u_{\underline{{\tilde{R}}_{2}} (\tilde{A})} (x), \end{matrix}$ and $\begin{matrix} v_{\underline{{\tilde{R}}_{1}} (\tilde{A})} (x) & = ⋁_{y \in U} u_{{\tilde{R}}_{1}} (x, y) \land v_{\tilde{A}} (y) \\ = [⋁_{y \in U} u_{{\tilde{R}}_{1}}^{-} (x, y) \land v_{\tilde{A}}^{-} (y), ⋁_{y \in U} u_{{\tilde{R}}_{1}}^{+} (x, y) \land v_{\tilde{A}}^{+} (y)] \\ ⪯ [⋁_{y \in U} u_{{\tilde{R}}_{2}}^{-} (x, y) \land v_{\tilde{A}}^{-} (y), ⋁_{y \in U} u_{{\tilde{R}}_{2}}^{+} (x, y) \land v_{\tilde{A}}^{+} (y)] \\ = v_{\underline{{\tilde{R}}_{2}} (\tilde{A})} (x) . \end{matrix}$ Therefore, we have $u_{\underline{{\tilde{R}}_{1}} (\tilde{A})} (x) ⪰ u_{\underline{{\tilde{R}}_{2}} (\tilde{A})} (x), v_{\underline{{\tilde{R}}_{1}} (\tilde{A})} (x) ⪯ v_{\underline{{\tilde{R}}_{2}} (\tilde{A})} (x),$ consequently, $\underline{{\tilde{R}}_{1}} (\tilde{A}) ⊒ \underline{{\tilde{R}}_{2}} (\tilde{A})$ .

(2) Similarly, it can be easily checked without any difficulties.

Corollary 3.2. Let U be a nonempty and finite universe. Assume that ${\tilde{R}}_{1}, {\tilde{R}}_{2} \in IVPFR (U)$ are two interval-valued Pythagorean fuzzy relations on U. For all $\tilde{A} \in IVPF (U)$ , we have the following properties:

(1) $(\underline{{\tilde{R}}_{1} ⋒ {\tilde{R}}_{2}}) (\tilde{A}) ⊒ \underline{{\tilde{R}}_{1}} (\tilde{A}) ⋓ \underline{{\tilde{R}}_{2}} (\tilde{A})$ , $(\bar{{\tilde{R}}_{1} ⋒ {\tilde{R}}_{2}}) (\tilde{A}) ⊑ \bar{{\tilde{R}}_{1}} (\tilde{A})$

$⋒ \bar{{\tilde{R}}_{2}} (\tilde{A})$ ,

(2) $(\underline{{\tilde{R}}_{1} ⋓ {\tilde{R}}_{2}}) (\tilde{A}) ⊑ \underline{{\tilde{R}}_{1}} (\tilde{A}) ⋒ \underline{{\tilde{R}}_{2}} (\tilde{A})$ , $(\bar{{\tilde{R}}_{1} ⋓ {\tilde{R}}_{2}}) (\tilde{A}) ⊒ \bar{{\tilde{R}}_{1}} (\tilde{A})$

$⋓ \bar{{\tilde{R}}_{2}} (\tilde{A})$ .

Proof. It follows immediately from Theorems 2.3 and 3.3.

3.3 Connections between special interval-valued Pythagorean fuzzy relations and approximation operators

In this subsection, we discuss the connections between special interval-valued Pythagorean fuzzy relations and approximation operators.

Theorem 3.4. Let U be a nonempty and finite universe and $\tilde{R} \in IVPFR (U)$ . ∀ (x, y) ∈ U × U, M ⊑ U, we have:

(1) $(u_{\underline{\tilde{R}} ([1, 1]_{M})} (x), v_{\underline{\tilde{R}} ([1, 1]_{M})} (x)) = (⋀_{y \notin M} v_{\tilde{R}} (x, y),$

$⋀_{y \notin M} u_{\tilde{R}} (x, y)),$

(2) $(u_{\bar{\tilde{R}} ([1, 1]_{M})} (x), v_{\bar{\tilde{R}} ([1, 1]_{M})} (x)) = (⋁_{y \in M} u_{\tilde{R}} (x, y),$

$⋀_{y \in M} v_{\tilde{R}} (x, y)),$

(3) $(u_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x), v_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x)) = (v_{\tilde{R}} (x, y), u_{\tilde{R}} (x, y))$

$= (u_{\tilde{R}} (x, y), v_{\tilde{R}} (x, y))^{c},$

(4) $(u_{\bar{\tilde{R}} ([1, 1]_{y})} (x), v_{\bar{\tilde{R}} ([1, 1]_{y})} (x)) = (u_{\tilde{R}} (x, y), v_{\tilde{R}} (x, y)) .$

Proof. (1) For all x ∈ U, we have $\begin{matrix} u_{\underline{\tilde{R}} ([1, 1]_{M})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{[1, 1]_{M}} (y) \\ = (⋀_{y \in M} v_{\tilde{R}} (x, y) \lor u_{[1, 1]_{M}} (y)) \land \\ (⋀_{y \notin M} v_{\tilde{R}} (x, y) \lor u_{[1, 1]_{M}} (y)) \\ = [1, 1] \land (⋀_{y \notin M} v_{\tilde{R}} (x, y) \lor [0, 0]) \\ = ⋀_{y \notin M} v_{\tilde{R}} (x, y), \end{matrix}$ and $\begin{matrix} v_{\underline{\tilde{R}} ([1, 1]_{M})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{[1, 1]_{M}} (y) \\ = (⋁_{y \in M} u_{\tilde{R}} (x, y) \land v_{[1, 1]_{M}} (y)) \lor \\ (⋁_{y \notin M} u_{\tilde{R}} (x, y) \land v_{[1, 1]_{M}} (y)) \\ = [0, 0] \lor (⋁_{y \notin M} u_{\tilde{R}} (x, y) \land [1, 1]) \\ = ⋁_{y \notin M} u_{\tilde{R}} (x, y) . \end{matrix}$ Therefore, (1) holds.

(2) For all x ∈ U, we have $\begin{matrix} u_{\bar{\tilde{R}} ([1, 1]_{M})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land u_{[1, 1]_{M}} (y) \end{matrix}$ $\begin{matrix} = (⋁_{y \in M} u_{\tilde{R}} (x, y) \land u_{[1, 1]_{M}} (y)) \lor \\ (⋁_{y \notin M} u_{\tilde{R}} (x, y) \land u_{[1, 1]_{M}} (y)) \\ = (⋁_{y \in M} u_{\tilde{R}} (x, y) \land [1, 1]) \lor [0, 0] \\ = ⋁_{y \in M} u_{\tilde{R}} (x, y), \end{matrix}$ and $\begin{matrix} v_{\bar{\tilde{R}} ([1, 1]_{M})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor v_{[1, 1]_{M}} (y) \\ = (⋀_{y \in M} v_{\tilde{R}} (x, y) \lor v_{[1, 1]_{M}} (y)) \land \\ (⋀_{y \notin M} v_{\tilde{R}} (x, y) \lor v_{[1, 1]_{M}} (y)) \\ = (⋀_{y \in M} v_{\tilde{R}} (x, y) \lor [0, 0]) \land [1, 1] \\ = ⋀_{y \in M} v_{\tilde{R}} (x, y) . \end{matrix}$

Therefore, (2) holds.

(3) For all x ∈ U, we have $\begin{matrix} u_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x) & = ⋀_{z \in U} v_{\tilde{R}} (x, z) \lor u_{[1, 1]_{U - {y}}} (z) \\ = (v_{\tilde{R}} (x, y) \lor [0, 0]) \land \\ (⋀_{z \neq y} v_{\tilde{R}} (x, z) \lor [1, 1]) \\ = v_{\tilde{R}} (x, y) \land [1, 1] = v_{\tilde{R}} (x, y), \end{matrix}$ and $\begin{matrix} v_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x) & = ⋁_{z \in U} u_{\tilde{R}} (x, z) \land v_{[1, 1]_{U - {y}}} (z) \\ = (u_{\tilde{R}} (x, y) \land [1, 1]) \lor \\ (⋁_{z \neq y} u_{\tilde{R}} (x, z) \land [0, 0]) \\ = u_{\tilde{R}} (x, y) \lor [0, 0] = u_{\tilde{R}} (x, y) . \end{matrix}$

Therefore, (3) holds.

(4) For all x ∈ U, we have $\begin{matrix} u_{\bar{\tilde{R}} ([1, 1]_{y})} (x) & = ⋁_{z \in U} u_{\tilde{R}} (x, z) \land u_{[1, 1]_{y}} (z) \end{matrix}$ $\begin{matrix} = (u_{\tilde{R}} (x, y) \land u_{[1, 1]_{y}} (y)) \lor \\ (⋁_{z \neq y} u_{\tilde{R}} (x, z) \land u_{[1, 1]_{y}} (z)) \\ = u_{\tilde{R}} (x, y) \lor [0, 0] = u_{\tilde{R}} (x, y), \end{matrix}$ and $\begin{matrix} v_{\bar{\tilde{R}} ([1, 1]_{y})} (x) & = ⋀_{z \in U} v_{\tilde{R}} (x, z) \lor v_{[1, 1]_{y}} (z) \\ = (v_{\tilde{R}} (x, y) \lor v_{[1, 1]_{y}} (y)) \land \\ (⋀_{z \neq y} v_{\tilde{R}} (x, z) \lor v_{[1, 1]_{y}} (z)) \\ = v_{\tilde{R}} (x, y) \land [1, 1] = v_{\tilde{R}} (x, y) . \end{matrix}$ Therefore, (4) holds.

Theorem 3.5. Let $(U, \tilde{R})$ be an interval-valued Pythagorean fuzzy approximation space. Let $\tilde{R} \in IVPFR (U)$ , $\underline{\tilde{R}}$ and $\bar{\tilde{R}}$ are referred to as interval-valued Pythagorean fuzzy lower and upper approximation operators given in Definition 3.3, then $\tilde{R}$ is serial iff one of the following properties hold:

(IVPFL0) $\underline{\tilde{R}} (\tilde{\emptyset}) = \tilde{\emptyset}$ ,

(IVPFU0) $\bar{\tilde{R}} (\tilde{U}) = \tilde{U}$ ,

(IVPFL0) $\underline{\tilde{R}} ([\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]) = [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ ,

(IVPFU0) $\bar{\tilde{R}} ([\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]) = [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ .

Proof. From Theorem 3.1, we know that $\underline{\tilde{R}}$ and $\bar{\tilde{R}}$ have the property of the duality. Consequently, we can conclude that (IVPFL0) and (IVPFU0) are equivalent. Similarly, (IVPFL0) and (IVPFU0) are also equivalent. We only need to proof (IVPFU0) and (IVPFL0).

Firstly, we prove that $\tilde{R}$ is serial ⇔ (IVPFU0).

Since $\tilde{R}$ is serial, then for all x ∈ U, there exists a z ∈ U such that $u_{\tilde{R}} (x, z) = [1, 1]$ , $v_{\tilde{R}} (x, z) = [0, 0]$ . According to Definition 3.3, we have $\begin{matrix} u_{\bar{\tilde{R}} (\tilde{U})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land u_{\tilde{U}} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{U}}^{-} (y), ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{U}}^{+} (y)] \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land 1, ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land 1] \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y), ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y)] \\ = [u_{\tilde{R}}^{-} (x, z) \lor ⋁_{y \neq z} u_{\tilde{R}}^{-} (x, y), u_{\tilde{R}}^{+} (x, z) \lor \\ ⋁_{y \neq z} u_{\tilde{R}}^{+} (x, y)] = [1, 1] = u_{\tilde{U}} (x), \end{matrix}$ and $\begin{matrix} v_{\bar{\tilde{R}} (\tilde{U})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor v_{\tilde{U}} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor v_{\tilde{U}}^{-} (y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor v_{\tilde{U}}^{+} (y)] \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor 0, ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor 0] \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y), ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y)] \\ = [v_{\tilde{R}}^{-} (x, z) \land ⋀_{y \neq z} v_{\tilde{R}}^{-} (x, y), v_{\tilde{R}}^{+} (x, z) \land \\ ⋀_{y \neq z} v_{\tilde{R}}^{+} (x, y)] = [0, 0] = v_{\tilde{U}} (x), \end{matrix}$ which implies that $\bar{\tilde{R}} (\tilde{U}) = \tilde{U}$ .

Conversely, assume that (IVPFU0) holds, then ∀x ∈ U, $u_{\bar{\tilde{R}} (\tilde{U})} (x) = [1, 1]$ and $v_{\bar{\tilde{R}} (\tilde{U})} (x) = [0, 0]$ . If $\tilde{R}$ is not serial, then there exists x ∈ U, ∀y ∈ U, such that $u_{\tilde{R}} (x, y) \neq [1, 1]$ or $v_{\tilde{R}} (x, y) \neq [0, 0]$ . Since $u_{\tilde{U}} (y) = [1, 1]$ and $v_{\tilde{U}} (y) = [0, 0]$ for all y ∈ U, then $u_{\tilde{R}} (x, y) \land u_{\tilde{U}} (y) = u_{\tilde{R}} (x, y) \neq [1, 1]$ , or $v_{\tilde{R}} (x, y) \lor v_{\tilde{U}} (y) = v_{\tilde{R}} (x, y) \neq [0, 0]$ .

From above discussions, it follows that there exists x ∈ U, such that $u_{\bar{\tilde{R}} (\tilde{U})} (x) \neq [1, 1]$ or $v_{\bar{\tilde{R}} (\tilde{U})} (x) \neq [0, 0]$ , which contradicts $u_{\bar{\tilde{R}} (\tilde{U})} (x) = [1, 1]$ and $v_{\bar{\tilde{R}} (\tilde{U})} (x) = [0, 0]$ (∀x ∈ U). Consequently, $\tilde{R}$ is serial.

Finally, we are to prove that $\tilde{R}$ is serial ⇔ (IVPFL0).

Assume that $\tilde{R}$ is serial. For all x ∈ U, there exists a z ∈ U such that $u_{\tilde{R}} (x, z) = [1, 1]$ and $v_{\tilde{R}} (x, z) = [0, 0]$ , we have $\begin{matrix} u_{\underline{\tilde{R}} ([\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}])} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]}^{-} (y), \\ ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]}^{+} (y)] \\ = [(v_{\tilde{R}}^{-} (x, z) \lor a_{1}^{-}) \land (⋀_{y \neq z} v_{\tilde{R}}^{-} (x, y) \lor a_{1}^{-}), \\ (v_{\tilde{R}}^{+} (x, z) \lor a_{1}^{+}) \land (⋀_{y \neq z} v_{\tilde{R}}^{+} (x, y) \lor a_{1}^{+})] \\ = [a_{1}^{-} \land (⋀_{y \neq z} v_{\tilde{R}}^{-} (x, y) \lor a_{1}^{-}), \\ a_{1}^{+} \land (⋀_{y \neq z} v_{\tilde{R}}^{+} (x, y) \lor a_{1}^{+})] \\ = [a_{1}^{-}, a_{1}^{+}] = u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (x), \end{matrix}$ and $\begin{matrix} v_{\underline{\tilde{R}} ([\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}])} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]}^{-} (y), \\ ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]}^{+} (y)] \\ = [(u_{\tilde{R}}^{-} (x, z) \land a_{2}^{-}) \lor (⋁_{y \neq z} u_{\tilde{R}}^{-} (x, y) \land a_{2}^{-}), \\ (u_{\tilde{R}}^{+} (x, z) \land a_{2}^{+}) \lor (⋁_{y \neq z} u_{\tilde{R}}^{+} (x, y) \land a_{2}^{+})] \\ = [a_{2}^{-} \lor (⋁_{y \neq z} u_{\tilde{R}}^{-} (x, y) \land a_{2}^{-}), \\ a_{2}^{+} \lor (⋁_{y \neq z} u_{\tilde{R}}^{+} (x, y) \land a_{2}^{+})] \\ = [a_{2}^{-}, a_{2}^{+}] = v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (x) . \end{matrix}$

Conversely, if (IVPFL0) holds, then for all x ∈ U, we have $u_{\underline{\tilde{R}} ([\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}])} (x) = u_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (x)$ and $v_{\underline{\tilde{R}} ([\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}])}$ $(x) = v_{[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]} (x)$ , which implies that $⋀_{y \in U} v_{R}^{-} (x, y) \leq a_{1}^{-}$ , $⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \leq a_{1}^{+}$ ; $⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \geq a_{2}^{-}$ and ⋁_y∈U $u_{\tilde{R}}^{+} (x, y) \geq a_{2}^{+}$ . If we let $[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}] = ([0, 0], [1, 1])$ i.e. $a_{1}^{-} = a_{1}^{+} = 0$ and $a_{2}^{-} = a_{2}^{+} = 1$ then for all x ∈ U, there exists a y ∈ U such that $u_{\tilde{R}} (x, y) = [1, 1]$ and $v_{\tilde{R}} (x, y) = [0, 0]$ . Hence, $\tilde{R}$ is serial.

Therefore, Theorem 3.5 holds.

Theorem 3.6. Let $(U, \tilde{R})$ be an interval-valued Pythagorean fuzzy approximation space. Let $\tilde{R} \in IVPFR (U)$ , $\underline{\tilde{R}}$ and $\bar{\tilde{R}}$ are referred to as the interval-valued Pythagorean fuzzy lower and upper approximation operators induced from $(U, \tilde{R})$ , then we have:

(1) $\tilde{R}$ is reflexive ⇔ (IVPFLR) $\underline{\tilde{R}} (\tilde{A}) ⊑ \tilde{A}$ , $\forall \tilde{A} \in IVPF (U)$ ,

⇔ (IVPFUR) $\tilde{A} ⊑ \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in IVPF (U)$ .

(2) $\tilde{R}$ is symmetric ⇔ (IVPFLS) $u_{\underline{\tilde{R}} ([1, 1]_{U - {x}})} (y) = u_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x)$ , $v_{\underline{\tilde{R}} ([1, 1]_{U - {x}})} (y) = v_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x)$ , ∀x, y ∈ U.

⇔ (IVPFUS) $u_{\bar{\tilde{R}} ([1, 1]_{x})} (y) = u_{\bar{\tilde{R}} ([1, 1]_{y})} (x),$

$v_{\bar{\tilde{R}} ([1, 1]_{x})} (y) = v_{\bar{\tilde{R}} ([1, 1]_{y})} (x),$ ∀x, y ∈ U.

(3) $\tilde{R}$ is transitive ⇔ (IVPFLT) $\underline{\tilde{R}} (\tilde{A}) ⊑ \underline{\tilde{R}} (\underline{\tilde{R}} (\tilde{A}))$ , $\forall \tilde{A} \in IVPF (U)$ ,

⇔ (IVPFUT) $\bar{\tilde{R}} (\bar{\tilde{R}} (\tilde{A})) ⊑ \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in IVPF (U)$ .

Proof. (1) Since the dual properties of interval-valued Pythagorean fuzzy approximation operators, it is only to prove that $\tilde{R}$ is reflexive ⇔ (IVPFLR) $\underline{\tilde{R}} (\tilde{A}) ⊑ \tilde{A}, \forall \tilde{A} \in IVPF (U)$ .

If $\tilde{R}$ is reflexive, then according to Theorem 3.2, we have (IVPFLR) holds.

Conversely, if (IVPFLR) holds, then for all x ∈ U, we have $u_{\underline{\tilde{R}} (\tilde{A})}^{-} (x) \leq u_{\tilde{A}}^{-} (x)$ , $u_{\underline{\tilde{R}} (\tilde{A})}^{+} (x) \leq u_{\tilde{A}}^{+} (x)$ ; $v_{\underline{\tilde{R}} (\tilde{A})}^{-} (x) \geq v_{\tilde{A}}^{-} (x)$ and $v_{\underline{\tilde{R}} (\tilde{A})}^{+} (x) \geq v_{\tilde{A}}^{+} (x)$ .

Consequently, $u_{\underline{\tilde{R}} ([1, 1]_{U - {x}})}^{-} (x) \leq u_{[1, 1]_{U - {x}}}^{-} (x) = 0$ , $u_{\underline{\tilde{R}} ([1, 1]_{U - {x}})}^{+} (x) \leq u_{[1, 1]_{U - {x}}}^{+} (x) = 0$ ; $v_{\underline{\tilde{R}} ([1, 1]_{U - {x}})}^{-} (x) \geq v_{[1, 1]_{U - {x}}}^{-} (x) = 1$ and $v_{\underline{\tilde{R}} ([1, 1]_{U - {x}})}^{+} (x) \geq v_{[1, 1]_{U - {x}}}^{+} (x) = 1$ , from which we conclude that $u_{\underline{\tilde{R}} ([1, 1]_{U - {x}})} (x) = [0, 0]$ and $v_{\underline{\tilde{R}} ([1, 1]_{U - {x}})} (x) = [1, 1]$ .

On the other hand, $\begin{matrix} u_{\underline{\tilde{R}} ([1, 1]_{U - {x}})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{[1, 1]_{U - {x}}} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor u_{[1, 1]_{U - {x}}}^{-} (y), \\ ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor u_{[1, 1]_{U - {x}}}^{+} (y)] \\ = [(v_{\tilde{R}}^{-} (x, x) \lor u_{[1, 1]_{U - {x}}}^{-} (x)) \land \\ (⋀_{y \neq x} v_{\tilde{R}}^{-} (x, y) \lor u_{[1, 1]_{U - {x}}}^{-} (y)), \\ (v_{\tilde{R}}^{+} (x, x) \lor u_{[1, 1]_{U - {x}}}^{+} (x)) \land \\ (⋀_{y \neq x} v_{\tilde{R}}^{+} (x, y) \lor u_{[1, 1]_{U - {x}}}^{+} (y))] \\ = [v_{\tilde{R}}^{-} (x, x) \land (⋀_{y \neq x} v_{\tilde{R}}^{-} (x, y) \lor 1), \\ v_{\tilde{R}}^{+} (x, x) \land (⋀_{y \neq x} v_{\tilde{R}}^{+} (x, y) \lor 1)] \\ = [v_{\tilde{R}}^{-} (x, x), v_{\tilde{R}}^{+} (x, x)] = v_{\tilde{R}} (x, x) = [0, 0], \end{matrix}$ $\begin{matrix} v_{\underline{\tilde{R}} ([1, 1]_{U - {x}})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{[1, 1]_{U - {x}}} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land v_{[1, 1]_{U - {x}}}^{-} (y), \\ ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land v_{[1, 1]_{U - {x}}}^{+} (y)] \\ = [(u_{\tilde{R}}^{-} (x, x) \land v_{[1, 1]_{U - {x}}}^{-} (x)) \lor \\ (⋁_{y \neq x} u_{\tilde{R}}^{-} (x, y) \land v_{[1, 1]_{U - {x}}}^{-} (y)), \end{matrix}$ $\begin{matrix} (u_{\tilde{R}}^{+} (x, x) \land v_{[1, 1]_{U - {x}}}^{+} (x)) \lor \\ (⋁_{y \neq x} u_{\tilde{R}}^{+} (x, y) \land v_{[1, 1]_{U - {x}}}^{+} (y))] \\ = [u_{\tilde{R}}^{-} (x, x) \lor (⋁_{y \neq x} u_{\tilde{R}}^{-} (x, y) \land 0), \\ u_{\tilde{R}}^{+} (x, x) \lor (⋁_{y \neq x} u_{\tilde{R}}^{+} (x, y) \land 0)] \\ = [u_{\tilde{R}}^{-} (x, x), u_{\tilde{R}}^{+} (x, x)] = u_{\tilde{R}} (x, x) = [1, 1] . \end{matrix}$

Therefore, we have $u_{\tilde{R}} (x, x) = [1, 1]$ and $v_{\tilde{R}} (x, x) = [0, 0]$ , which implies that $\tilde{R}$ is reflexive.

(2) It follows immediately from Theorem 3.4.

(3) By the dual properties of interval-valued Pythagorean fuzzy approximation operators, it is only to prove that $\tilde{R}$ is transitive ⇔ (IVPFLT).

If $\tilde{R}$ is transitive, then we have $u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{R}}^{-} (y, z) \leq u_{\tilde{R}}^{-} (x, z)$ , $u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{R}}^{-} (y, z) \leq u_{\tilde{R}}^{+} (x, z)$ ; $v_{\tilde{R}}^{-} (x, y) \land$ $v_{\tilde{R}}^{-} (y, z) \geq v_{\tilde{R}}^{-} (x, z)$ and $v_{\tilde{R}}^{+} (x, y) \land v_{\tilde{R}}^{-} (y, z) \geq v_{\tilde{R}}^{+} (x, z)$ . Moreover, we have $\begin{matrix} u_{\underline{R} (\underline{R} (\tilde{A}))} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor u_{\underline{\tilde{R}} (\tilde{A})} (y) \\ = [⋀_{y \in U} v_{\tilde{R}}^{-} (x, y) \lor u_{\underline{\tilde{R}} (\tilde{A})}^{-} (y), \\ ⋀_{y \in U} v_{\tilde{R}}^{+} (x, y) \lor u_{\underline{\tilde{R}} (\tilde{A})}^{+} (y)] \\ = [⋀_{y \in U} (v_{\tilde{R}}^{-} (x, y) \lor (⋀_{z \in U} v_{\tilde{R}}^{-} (y, z) \lor u_{\tilde{A}}^{-} (z))), \\ ⋀_{y \in U} (v_{\tilde{R}}^{+} (x, y) \lor (⋀_{z \in U} v_{\tilde{R}}^{+} (y, z) \lor u_{\tilde{A}}^{+} (z)))] \\ = [⋀_{y \in U} ⋀_{z \in U} (v_{\tilde{R}}^{-} (x, y) \lor v_{\tilde{R}}^{-} (y, z) \lor u_{\tilde{A}}^{-} (z)), \\ ⋀_{y \in U} ⋀_{z \in U} (v_{\tilde{R}}^{+} (x, y) \lor v_{\tilde{R}}^{+} (y, z) \lor u_{\tilde{A}}^{+} (z))] \\ ⪰ [⋀_{z \in U} v_{\tilde{R}}^{-} (x, z) \lor u_{\tilde{A}}^{-} (z), ⋀_{z \in U} v_{\tilde{R}}^{+} (x, z) \lor u_{\tilde{A}}^{+} (z)] \\ = u_{\underline{\tilde{R}} (\tilde{A})} (x), \end{matrix}$ and $\begin{matrix} v_{\underline{\tilde{R}} (\underline{\tilde{R}} (\tilde{A}))} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land v_{\underline{\tilde{R}} (\tilde{A})} (y) \\ = [⋁_{y \in U} u_{\tilde{R}}^{-} (x, y) \land v_{\underline{\tilde{R}} (\tilde{A})}^{-} (y), \end{matrix}$ $\begin{matrix} ⋁_{y \in U} u_{\tilde{R}}^{+} (x, y) \land v_{\underline{\tilde{R}} (\tilde{A})}^{+} (y)] \\ = [⋁_{y \in U} (u_{\tilde{R}}^{-} (x, y) \land (⋁_{z \in U} u_{\tilde{R}}^{-} (y, z) \land v_{\tilde{A}}^{-} (z))), \\ ⋁_{y \in U} (u_{\tilde{R}}^{+} (x, y) \land (⋁_{z \in U} u_{\tilde{R}}^{+} (y, z) \land v_{\tilde{A}}^{+} (z)))] \\ = [⋁_{y \in U} ⋁_{z \in U} (u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{R}}^{-} (y, z) \land v_{\tilde{A}}^{-} (z)), \\ ⋁_{y \in U} ⋁_{z \in U} (u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{R}}^{+} (y, z) \land v_{\tilde{A}}^{+} (z))] \\ ⪯ [⋁_{z \in U} u_{\tilde{R}}^{-} (x, z) \land v_{\tilde{A}}^{-} (z), ⋁_{z \in U} u_{\tilde{R}}^{+} (x, z) \land v_{\tilde{A}}^{+} (z)] \\ = v_{\underline{\tilde{R}} (\tilde{A})} (x) . \end{matrix}$ Therefore, (IVPFLT) holds.

Conversely, if $\underline{\tilde{R}} (\tilde{A}) ⊑ \underline{\tilde{R}} (\underline{\tilde{R}} (\tilde{A}))$ for all $\tilde{A} \in IVPF (U)$ , then for all x ∈ U, y ∈ U, $u_{\underline{\tilde{R}} (\underline{\tilde{R}} ([1, 1]_{U - {y}}))} (x) ⪰ u_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x)$ , $v_{\underline{\tilde{R}} (\underline{\tilde{R}} ([1, 1]_{U - {y}}))} (x) ⪯ v_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x)$ . On the other hand, according to Theorem 3.4, we have $\begin{matrix} u_{\underline{\tilde{R}} (\underline{\tilde{R}} ([1, 1]_{U - {y}}))} (x) & = ⋀_{z \in U} v_{\tilde{R}} (x, z) \lor u_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (z) \\ = ⋀_{z \in U} v_{\tilde{R}} (x, z) \lor v_{\tilde{R}} (z, y); \end{matrix}$ $\begin{matrix} v_{\underline{\tilde{R}} (\underline{\tilde{R}} ([1, 1]_{U - {y}}))} (x) & = ⋁_{z \in U} u_{\tilde{R}} (x, z) \land v_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (z) \\ = ⋁_{z \in U} u_{\tilde{R}} (x, z) \land u_{\tilde{R}} (z, y); \end{matrix}$ and $u_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x) = v_{\tilde{R}} (x, y)$ , $v_{\underline{\tilde{R}} ([1, 1]_{U - {y}})} (x) = u_{\tilde{R}} (x, y)$ , from which we can conclude that $⋀_{z \in U} v_{\tilde{R}}^{-} (x, z) \lor v_{\tilde{R}}^{-} (z, y) ⪰ v_{\tilde{R}}^{-} (x, y)$ , $⋀_{z \in U} v_{\tilde{R}}^{+} (x, z) \lor v_{\tilde{R}}^{+} (z, y) ⪰ v_{\tilde{R}}^{+} (x, y)$ ; $⋁_{z \in U} u_{\tilde{R}}^{-} (x, z) \land u_{\tilde{R}}^{-} (z, y) ⪯ u_{\tilde{R}}^{-} (x, y)$ and $⋁_{z \in U} u_{\tilde{R}}^{+} (x, z) \land u_{\tilde{R}}^{+} (z, y) ⪯ u_{\tilde{R}}^{+} (x, y)$ . Consequently, for all z ∈ U, we have $v_{\tilde{R}}^{-} (x, y) \land v_{\tilde{R}}^{-} (y, z) \geq v_{\tilde{R}}^{-} (x, z)$ , $v_{\tilde{R}}^{+} (x, y) \land v_{\tilde{R}}^{-} (y, z) \geq v_{\tilde{R}}^{+} (x, z)$ ; $u_{\tilde{R}}^{-} (x, y) \land u_{\tilde{R}}^{-} (y, z) \leq u_{\tilde{R}}^{-} (x, z)$ , and $u_{\tilde{R}}^{+} (x, y) \land u_{\tilde{R}}^{+} (y, z) \leq u_{\tilde{R}}^{+} (x, z)$ . By the definition of transitivity, we know that $\tilde{R}$ is transitive.

Corollary 3.3. Let ( $U, \tilde{R}$ ) be an interval-valued Pythagorean fuzzy approximation space. $\tilde{R}$ is reflexive and transitive, for all $\tilde{A} \in IVPF (U)$ , we have the following properties:

(1) $\underline{\tilde{R}} (\underline{\tilde{R}} (\tilde{A})) = \underline{\tilde{R}} (\tilde{A}),$

(2) $\bar{\tilde{R}} (\tilde{A}) = \bar{\tilde{R}} (\bar{\tilde{R}} (\tilde{A}))$ .

Proof. It follows immediately from Theorems 3.1, 3.2 and 3.6.

4 Axiomatic characterizations of interval-valued Pytha-gorean fuzzy approximation operators

In this section, we study the axiomatic characterizations of interval-valued Pythagorean fuzzy approximation operators.

Definition 4.1. Let L, H: IVPF (U) ⟶ IVPF (U) be two interval-valued Pythagorean fuzzy set operators. L and H are called dual operators if the following axioms are satisfied: $\forall \tilde{A} \in IVPF (U)$ ,

(AL1) $L (\tilde{A}) = (H ({\tilde{A}}^{c}))^{c}$ ,

(AU1) $H (\tilde{A}) = (L ({\tilde{A}}^{c}))^{c}$ .

Theorem 4.1. Let L, H: IVPF (U) ⟶ IVPF (U) be two dual operators, then there exists an interval-valued Pythagorean fuzzy relation $\tilde{R}$ on U such that $L (\tilde{A}) = \underline{\tilde{R}} (\tilde{A})$ and $H (\tilde{A}) = \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in IVPF (U)$ iff L satisfies axioms (AL2) and (AL3), or equivalently, H satisfies axioms (AU2) and (AU3): $\forall \tilde{A}, \tilde{B} \in IVPF (U)$ , $[\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}] \in IVPF (U)$ ,

(AL2) $L (\tilde{A} ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]) = L (\tilde{A}) ⋓ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ ,

(AL3) $L (\tilde{A} ⋒ \tilde{B}) = L (\tilde{A}) ⋒ L (\tilde{B})$ ,

(AU2) $H (\tilde{A} ⋒ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]) = H (\tilde{A}) ⋒ [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ ,

(AU3) $H (\tilde{A} ⋓ \tilde{B}) = H (\tilde{A}) ⋓ H (\tilde{B})$ .

Proof. (⇒) It follows immediately from Theorem 3.1.

(⟸) Suppose that the operator H satisfies axioms (AU2) and (AU3), then we can define an interval-valued Pythagorean fuzzy relation $\tilde{R} = {〈 (x, y)$ , $u_{\tilde{R}} (x, y)$ , $v_{\tilde{R}} (x, y) 〉 | (x, y)$ ∈U × U} on U by H as follows:

$u_{\tilde{R}} (x, y) = u_{H ([1, 1]_{y})} (x)$ , (x, y) ∈ U × U, $v_{\tilde{R}} (x, y) = v_{H ([1, 1]_{y})} (x)$ , (x, y) ∈ U × U.

Moreover, we can prove that $\forall \tilde{A} \in IVPF (U)$ ,

$\tilde{A} = ⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})$ .

In fact, for all x ∈ U, we have $\begin{matrix} u_{⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})} (x) & = ⋁_{y \in U} u_{([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})} (x) \\ = ⋁_{y \in U} (u_{[1, 1]_{y}} (x) \land u_{\hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]}} (x)) \\ = ([1, 1] \land u_{\hat{[u_{\tilde{A}} (x), v_{\tilde{A}} (x)]}} (x)) \lor [0, 0] \\ = ([1, 1] \land u_{\tilde{A}} (x)) \lor [0, 0] = u_{\tilde{A}} (x), \end{matrix}$ and $\begin{matrix} v_{⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})} (x) & = ⋀_{y \in U} v_{([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})} (x) \end{matrix}$ $\begin{matrix} = ⋀_{y \in U} (v_{[1, 1]_{y}} (x) \lor v_{\hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]}} (x)) \\ = ([0, 0] \lor v_{\hat{[u_{\tilde{A}} (x), v_{\tilde{A}} (x)]}} (x)) \land [1, 1] \\ = ([0, 0] \lor v_{\tilde{A}} (x)) \land [1, 1] = v_{\tilde{A}} (x) . \end{matrix}$

Consequently, we have $\tilde{A} = ⋓_{y \in U} ([1, 1]_{y} ⋒$

$\hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})$ . $\begin{matrix} u_{\bar{\tilde{R}} (\tilde{A})} (x) & = ⋁_{y \in U} u_{\tilde{R}} (x, y) \land u_{\tilde{A}} (y) \\ = ⋁_{y \in U} u_{H ([1, 1]_{y})} (x) \land u_{\hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]}} (x) \\ = ⋁_{y \in U} u_{H ([1, 1]_{y}) ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]}} (x) \\ = ⋁_{y \in U} u_{H ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})} (x) \\ = u_{⋓_{y \in U} H ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})} (x) \\ = u_{H (⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]}))} (x) \\ = u_{H (\tilde{A})} (x), \end{matrix}$ and $\begin{matrix} v_{\bar{\tilde{R}} (\tilde{A})} (x) & = ⋀_{y \in U} v_{\tilde{R}} (x, y) \lor v_{\tilde{A}} (y) \\ = ⋀_{y \in U} v_{H ([1, 1]_{y})} (x) \lor v_{\hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]}} (x) \\ = ⋀_{y \in U} v_{H ([1, 1]_{y}) ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]}} (x) \\ = ⋀_{y \in U} v_{H ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})} (x) \\ = v_{⋓_{y \in U} H ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]})} (x) \\ = v_{H (⋓_{y \in U} ([1, 1]_{y} ⋒ \hat{[u_{\tilde{A}} (y), v_{\tilde{A}} (y)]}))} (x) \\ = v_{H (\tilde{A})} (x) . \end{matrix}$

Therefore, we have $H (\tilde{A}) = \bar{\tilde{R}} (\tilde{A})$ . According to Definition 4.1 and $H (\tilde{A}) = \bar{\tilde{R}} (\tilde{A})$ , we have $L (\tilde{A}) = \underline{\tilde{R}} (\tilde{A})$ .

Theorem 4.2. Let L, H: IVPF (U) ⟶ IVPF (U) be a pair of dual operators, i.e., L satisfies axioms (AL1), (AL2) and (AL3), and H satisfies axioms (AU1), (AU2) and (AU3). Then there exists a serial interval-valued Pythagorean fuzzy relation $\tilde{R}$ on U such that $L (\tilde{A}) = \underline{\tilde{R}} (\tilde{A})$ and $H (\tilde{A}) = \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in IVPF (U)$ iff L satisfies the axiom (ALS), or equivalently, H satisfies the axiom (AUS):

(ALS) $L ([\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]) = [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ ,

(AUS) $H ([\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]) = [\hat{a_{1, 2}^{-}, a_{1, 2}^{+}}]$ .

Proof. (⇒) It follows immediately from Theorem 3.5.

(⟸) It follows immediately from Theorems 4.1 and 3.5.

By Theorem 4.2, we have the following corollary:

Corollary 4.1. Let L, H: IVPF (U) ⟶ IVPF (U) be a pair of dual operators. By Theorem 3.5, it can be easily seen that axioms (ALS) and (AUS) can be replaced by one of the following axioms:

(ALS1) $L (\tilde{\emptyset}) = \tilde{\emptyset}$ ,

(AUS1) $H (\tilde{U}) = \tilde{U}$ ,

(ALUS) $L (\tilde{A}) ⊑ H (\tilde{A})$ .

Theorem 4.3. Let L, H: IVPF (U) ⟶ IVPF (U) be a pair of dual operators, i.e., L satisfies axioms (AL1), (AL2) and (AL3), and H satisfies axioms (AU1), (AU2) and (AU3). Then there exists a reflexive interval-valued Pythagorean fuzzy relation $\tilde{R}$ on U such that $L (\tilde{A}) = \underline{\tilde{R}} (\tilde{A})$ and $H (\tilde{A}) = \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in IVPF (U)$ iff L satisfies the axiom (ALR), or equivalently, H satisfies the axiom (AUR):

(ALR) $L (\tilde{A}) ⊑ \tilde{A}$ ,

$\forall \tilde{A} \in IVPF (U)$ ,

(AUR) $\tilde{A} ⊑ H (\tilde{A})$ ,

$\forall \tilde{A} \in IVPF (U)$ .

Proof. (⇒) It follows immediately from Theorem 3.6.

Table 2
The interval-valued Pythagorean fuzzy relation $\tilde{R}$

$\tilde{R}$ y ₁ y ₂ y ₃ y ₄

x ₁ 〈[1, 1] , [0, 0] 〉〈[0.4, 0.8] , [0.1, 0.4] 〉〈[0.3, 0.4] , [0.6, 0.7] 〉〈[0.6, 0.8] , [0.3, 0.4] 〉

x ₂ 〈[0.3, 0.4] , [0.5, 0.6] 〉〈[1, 1] , [0, 0] 〉〈[0.4, 0.5] , [0.1, 0.6] 〉〈[0.2, 0.3] , [0.5, 0.9] 〉

x ₃ 〈[0.4, 0.6] , [0.3, 0.5] 〉〈[0.7, 0.9] , [0.2, 0.3] 〉〈[1, 1] , [0, 0] 〉〈[0.7, 0.8] , [0.1, 0.2] 〉

x ₄ 〈[0.4, 0.9] , [0.2, 0.3] 〉〈[0.4, 0.5] , [0.7, 0.8] 〉〈[0.5, 0.6] , [0.3, 0.6] 〉〈[1, 1] , [0, 0] 〉

$\tilde{R}$	y ₁	y ₂	y ₃	y ₄
x ₁	〈[1, 1] , [0, 0] 〉	〈[0.4, 0.8] , [0.1, 0.4] 〉	〈[0.3, 0.4] , [0.6, 0.7] 〉	〈[0.6, 0.8] , [0.3, 0.4] 〉
x ₂	〈[0.3, 0.4] , [0.5, 0.6] 〉	〈[1, 1] , [0, 0] 〉	〈[0.4, 0.5] , [0.1, 0.6] 〉	〈[0.2, 0.3] , [0.5, 0.9] 〉
x ₃	〈[0.4, 0.6] , [0.3, 0.5] 〉	〈[0.7, 0.9] , [0.2, 0.3] 〉	〈[1, 1] , [0, 0] 〉	〈[0.7, 0.8] , [0.1, 0.2] 〉
x ₄	〈[0.4, 0.9] , [0.2, 0.3] 〉	〈[0.4, 0.5] , [0.7, 0.8] 〉	〈[0.5, 0.6] , [0.3, 0.6] 〉	〈[1, 1] , [0, 0] 〉

(⟸) It follows immediately from Theorems 4.1 and 3.6.

Theorem 4.4. Let L, H: IVPF (U) ⟶ IVPF (U) be a pair of dual operators, i.e., L satisfies axioms (AL1), (AL2) and (AL3), and H satisfies axioms (AU1), (AU2) and (AU3). Then there exists a symmetric interval-valued Pythagorean fuzzy relation $\tilde{R}$ on U such that $L (\tilde{A}) = \underline{\tilde{R}} (\tilde{A})$ and $H (\tilde{A}) = \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in IVPF (U)$ iff L satisfies the axiom (ALSY), or equivalently, H satisfies the axiom (AUSY): ∀ (x, y) ∈ U × U,

(ALSY) u_{L([1,1]_U-{x})} (y) = u_{L([1,1]_U-{y})} (x) ∧ v_{L([1,1]_U-{x})}(y) = v_{L([1,1]_U-{y})} (x),

(AUSY) u_{H([1,1]_x)} (y) = u_{H([1,1]_y)} (x) ∧ v_{H([1,1]_x)}(y) = v_{H([1,1]_y)} (x).

Proof. (⇒) It follows immediately from Theorem 3.6.

(⟸) It follows immediately from Theorems 4.1 and 3.6.

Theorem 4.5. Let L, H: IVPF (U) ⟶ IVPF (U) be a pair of dual operators, i.e., L satisfies axioms (AL1), (AL2) and (AL3), and H satisfies axioms (AU1), (AU2) and (AU3). Then there exists a transitive interval-valued Pythagorean fuzzy relation $\tilde{R}$ on U such that $L (\tilde{A}) = \underline{\tilde{R}} (\tilde{A})$ and $H (\tilde{A}) = \bar{\tilde{R}} (\tilde{A})$ , $\forall \tilde{A} \in IVPF (U)$ iff L satisfies the axiom (ALT), or equivalently, H satisfies the axiom (AUT):

(ALT) $L (\tilde{A}) ⊑ L (L (\tilde{A}))$ ,

$\forall \tilde{A} \in IVPF (U)$ ,

(AUT) $H (H (\tilde{A})) ⊑ H (\tilde{A})$ ,

$\forall \tilde{A} \in IVPF (U)$ .

Proof. (⇒) It follows immediately from Theorem 3.6.

(⟸) It follows immediately from Theorems 4.1 and 3.6.

5 Application of interval-valued Pythagorean fuzzy rough sets in medical diagnosis

In this section, we explore the application of interval-valued Pythagorean fuzzy rough sets.

5.1 An application model

Assume that U = {x₁, x₂, . . . , x_n} is a disease set in clinic and V = {y₁, y₂, . . . , y_j} is a symptom set. Let $\tilde{R}$ be an interval-valued Pythagorean fuzzy relation from U to V, which represents the related degrees between these diseases and symptoms. Let $\tilde{A} \in IVPF (V)$ be an interval-valued Pythagorean fuzzy set which represents some symptoms of the patient. Our assignment is to decide which disease the patient is suffered from most probably based on the information table. We can use IVPFRSs proposed in this paper to involve the problem. The concrete process is presented by an algorithm as follows:

Algorithm 1. The selection of optimal decision in medical diagnosis by using IVPFRSs.

Step 1. Calculate the interval-valued Pythagorean fuzzy lower approximation $\underline{\tilde{R}} (\tilde{A})$ and upper approximation $\bar{\tilde{R}} (\tilde{A})$ of $\tilde{A}$ .

Step 2. Compute $\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A})$ .

Step 3. Compute the score values $s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A})) (x_{i}))$ , i = 1, 2, ·· · , n.

Step 4. Rank the score values $s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A})) (x_{i}))$ (i = 1, 2, ·· · , n). The optimal decision is x_k if the score value $s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A})) (x_{k}))$ is the highest score value.

5.2 An illustrative example

Example 5.1. Assume that U = {x₁, x₂, x₃, x₄} is a disease set, where x₁ stands for “viral fever”, x₂ stands for “malaria”, x₃ stands for “typhoid” and x₄ stands for “stomach ulcer”. Assume V = {y₁, y₂, y₃, y₄} is a symptoms set, where y₁ stands for “temperature”, y₂ stands for “headache”, y₃ stands for “cough” and y₄ stands for “stomach pain”. $\tilde{R}$ is given in Table 2. The symptoms of the patient $\tilde{A}$ are described by an interval-valued Pythagorean fuzzy set as follows: $\begin{matrix} \tilde{A} = {〈 y_{1}, ([0.4, 0.8], [0.1, 0.3]) 〉, 〈 y_{2}, ([0.3, 0.5], \\ [0.6, 0.7]) 〉, 〈 y_{3}, ([0.2, 0.4], [0.3, 0.7]) 〉, \\ 〈 y_{4}, ([0.4, 0.6], [0.2, 0.5]) 〉} . \end{matrix}$

Next, we will diagnose which kind of disease the patient $\tilde{A}$ is suffering from according to Algorithm 1.

Step 1. From Definition 3.3, the interval-valued Pythagorean fuzzy lower and upper approximations of $\tilde{A}$ can be calculated as follows, respectively: $\begin{matrix} \underline{\tilde{R}} (\tilde{A}) = {〈 x_{1}, [0.3, 0.5], [0.4, 0.7] 〉, 〈 x_{2}, [0.2, 0.5], \\ [0.6, 0.7] 〉, 〈 x_{3}, [0.2, 0.4], [0.6, 0.7] 〉, \\ 〈 x_{4}, [0.3, 0.6], [0.4, 0.6] 〉}, \end{matrix}$ $\begin{matrix} \bar{\tilde{R}} (\tilde{A}) = {〈 x_{1}, [0.4, 0.8], [0.1, 0.3] 〉, 〈 x_{2}, [0.3, 0.5], \\ [0.3, 0.6] 〉, 〈 x_{3}, [0.4, 0.6], [0.2, 0.5] 〉, \\ 〈 x_{4}, [0.4, 0.8], [0.2, 0.3] 〉} . \end{matrix}$

Step 2. By Definition 2.7, we get $\begin{matrix} \underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) = {〈 x_{1}, [0.4854, 0.8544], [0.04, 0.21] 〉, \\ 〈 x_{2}, [0.3555, 0.6614], [0.18, 0.42] 〉, \\ 〈 x_{3}, [0.4400, 0.6800], [0.12, 0.35] 〉, \\ 〈 x_{4}, [0.4854, 0.8773], [0.08, 0.18] 〉}, \end{matrix}$

Step 3. By Definition 2.4, we have

$s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) (x_{1})) = 0.4600$ ,

$s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) (x_{2})) = 0.1775$ ,

$s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) (x_{3})) = 0.2596$ ,

$s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) (x_{4})) = 0.4832$ .

Step 4. According to the above results, we have $\begin{matrix} s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) (x_{4})) > \\ s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) (x_{1})) > \\ s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) (x_{3})) > \\ s ((\underline{\tilde{R}} (\tilde{A}) \tilde{\oplus} \bar{\tilde{R}} (\tilde{A}) (x_{2})), \end{matrix}$ which means that the optimal decision is x₄. So the patient $\tilde{A}$ is suffering from the stomach ulcer.

6 Conclusion

In this paper, we propose the IVPFRS model by integrating the IVPFSs and rough sets. Some basic properties of IVPFRSs are discussed. Furthermore, we explore the axiomatic characterizations of IVPFRSs. A decision making algorithm based on IVPFRS model is constructed. An illustrative example is presented to show the validity of the algorithm. In the forthcoming research, we will study the hesitant interval-valued Pythagorean fuzzy rough sets and its application.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (No. 61473181).

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