Inclusion degree of type-two intuitionistic fuzzy rough set

Abstract

The article provides a definition and properties of inclusion degree in type-two intuitionistic fuzzy sets and, on the basis of inclusion degree, gives the definition and attributes of type two intuitionistic fuzzy rough sets based on inclusion degree. At the same time, uncertainty measurement factors such as approximate accuracy, roughness, and importance are provided.

Keywords

type-two fuzzy set type-two intuitive fuzzy set type-two intuitive fuzzy rough set inclusion degree

Introduction

In 1975, Professor Zadeh¹ extended the membership functions in traditional fuzzy set to type-one fuzzy set. On the basis, the notion of type-two fuzzy set was proposed, which essentially fuzzies the fuzzy set again, so that the generated type-two fuzzy set can represent deeper uncertainty problems, greatly enhancing the ability to handle and characterize uncertainty problems in the objective world. Although the notion of type-two fuzzy set has been put forward for nearly 50 years, it has not attracted the attention of scholars at home and abroad for a long time due to its complexity in mathematical representation, geometric presentation, and computational efficiency. Until 2000, with the strong promotion of renowned scholar Mendel and his students^2–8 at the University of Southern California in the United States, the research on type-two fuzzy theory gradually became a hot topic in uncertain system research. Especially in the past 25 years, accompanied by the rapid development of type-two fuzzy systems and word calculation theory research, the theory of type two fuzzy decision-making has attracted increasing attention from scholars and has achieved a large number of excellent results in information fusion,^9–17 preference relationship theory,^18–24 measurement theory,^25–34 and decision-making methods.^35–42 This article provides the definition and properties of type two intuitionistic fuzzy rough sets based on inclusion degree.

Basic knowledge

Type-two fuzzy set

Zadeh¹ provided a notion of a type-n fuzzy set in 1975.

Definition 2.1

¹ A fuzzy set is an n type. If its membership function takes the value of n-1 fuzzy set, then the membership function range of one type of fuzzy set is [0,1].

Type-one fuzzy set: $H = {< (ς, u_{H} (ς) > | ς \in [0, 1], u_{H} (ς) \in [0, 1]}$ . A type-one fuzzy set is a special case when the fuzzy set domain is [0,1], which can be called a unit fuzzy set

Definition 2.2

⁸ Assuming $U$ a non-empty domain, a type-two fuzzy set $H$ can be characterized by a type-two membership function $μ_{H} (ς, ϑ)$ ,

H = {((ς, ϑ), μ_{H} (ς, ϑ)) | ς \in U, ϑ \in J_{ς} \subseteq [0, 1]}

wherein

0 \leq μ_{H} (ς, ϑ) \leq 1

If the domain of discourse U is a finite or countable set, $ψ$ can be delimited as

ψ = \sum_{ς \in U} \sum_{ϑ \in J_{ς}} μ_{ψ} (ς, ϑ) / (ς, ϑ), J_{ς} \subseteq [0, 1]

If the domain of discourse U is an infinite uncountable set, $ψ$ can be delimited as

ψ = \int_{ς \in U} \int_{ϑ \in J_{ς}} μ_{ψ} (ς, ϑ) / (ς, ϑ), J_{ς} \subseteq [0, 1]

Among them, $\iint$ represents the summation of all possible $ς$ and $ϑ$ .

Type-two intuitionistic fuzzy set

Definition 2.3

⁴³ Assuming W is a domain, is called a type-two intuitionistic fuzzy set on W. Among them,

Definition 2.4

⁴³ Suppose there are two type-two intuitionistic fuzzy sets G and H on the domain of universe U, suppose

G = {< ς, u_{G} (ς), v_{G} (ς) > | ς \in U}, H = {< ς, u_{H} (ς), v_{H} (ς) > | ς \in U}

among

\begin{array}{l} u_{H} (ς) = \int_{η \in J_{ς}^{η}} f_{ς} (η) / η, & J_{ς}^{η} \subseteq [0, 1] \end{array}

\begin{array}{l} v_{H} (ς) = \int_{ρ \in J_{ς}^{ρ}} g_{ς} (ρ) / ρ, & J_{ς}^{ρ} \subseteq [0, 1] \end{array}

\begin{array}{l} u_{H} (ς) = \int_{λ \in J_{ς}^{λ}} h_{ς} (λ) / λ, & J_{ς}^{λ} \subseteq [0, 1] \end{array}

\begin{array}{l} v_{H} (ς) = \int_{δ \in J_{ς}^{δ}} k_{ς} (δ) / δ, & J_{ς}^{δ} \subseteq [0, 1] \end{array}

The following operation can be defined:

(1) $G \cup H = {< ς, u_{G} (ς) \cup u_{H} (ς), v_{G} (ς) \cap v_{H} (ς) > | ς \in U}$

$= {< ξ, \int_{λ \in J_{ς}^{λ}} \int_{e \in J_{ς}^{e}} f_{ς} (λ) \land h_{ς} (e) / λ \lor e, \int_{γ \in J_{ς}^{γ}} \int_{f \in J_{ς}^{f}} g_{ς} (γ) \land k_{ς} (f) / γ \land f}$

(2) $G \cap H = {< ς, u_{G} (ς) \cap u_{H} (ς), v_{G} (ς) \cup v_{H} (ς) > | ς \in U}$

$= {< ς, \int_{λ \in J_{ς}^{λ}} \int_{e \in J_{ς}^{e}} f_{ς} (λ) \land h_{ς} (e) / λ \land e, \int_{γ \in J_{ς}^{γ}} \int_{f \in J_{ς}^{f}} g_{ς} (γ) \land k_{ς} (f) / γ \lor f}$

(3) $G^{C} = {< ς, v_{G} (ς), u_{G} (ς) > | ς \in U} = {< ς, \int_{γ \in J_{ς}^{γ}} g_{ς} (γ) / γ, \int_{λ \in J_{ς}^{λ}} f_{ς} (λ) / λ | ς \in U}$

Lemma 2.1

⁴³ If S, M, and Q are three type-two intuitionistic fuzzy sets on the domain W, then the following equation holds:

(1) Law of commutation $S \cup M = M \cup S, S \cap M = M \cap S$

(2) Law of associativity: $Q \cup (M \cup S) = (Q \cup M) \cup S$ , $Q \cap (M \cap S) = (Q \cap M) \cap S$

(3) Law of idempotency: $Q \cup Q = Q, Q \cap Q = Q$

(4) Law of double complementation: ${(Q^{C})}^{C} = Q$

(5) De Morgan Law: ${(S \cup M)}^{C} = S^{C} \cap M^{C}$ , ${(S \cap M)}^{C} = S^{C} \cup M^{C}$

Definition 2.5

⁴³ Suppose W and V are two non-empty finite fields, and the type-two intuitionistic fuzzy subset defined on the direct product space $W \times V$ is said to be the type-two intuitionistic fuzzy relationship from W to V, recorded as

R = {< q, u_{R} (q, τ), v_{R} (q, τ) | q \in W, τ \in V}

\begin{array}{l} u_{R} (q, τ) = \int_{u \in J_{(q, τ)}^{u}} f_{(q, τ)} (u) / u, & J_{(q, τ)}^{u} \subseteq [0, 1] \end{array}

\begin{array}{l} v_{R} (q, τ) = \int_{v \in J_{(q, τ)}^{v}} g_{(q, τ)} (v) / v, & J_{(q, τ)}^{v} \subseteq [0, 1] \end{array}

for

\forall q, τ \in W \times V

satisfy

\max_{u \in J_{(q, τ)}^{u}} (f_{(q, τ)} (u) \times u) + \max_{v \in J_{(q, τ)}^{v}} (g_{(q, τ)} (v) \times v) \leq 1

Definition 2.6

⁴⁴ If G and H are two type-two intuitionistic fuzzy sets on the domain W,

G \subseteq H \Leftrightarrow \forall ϖ \in U, u_{G} (ϖ) ≺ u_{H} (ϖ) and v_{G} (ϖ) ≻ v_{H} (ϖ)

The order relationship $≺, ≻$ is defined as follows:

u_{G} (ϖ) ≺ u_{H} (ϖ) \Leftrightarrow u_{G} (ϖ) Δ u_{H} (ϖ) = u_{G} (ϖ) \Leftrightarrow u_{G} (ϖ) \nabla u_{H} (ϖ) = u_{H} (ϖ)

v_{G} (ϖ) ≻ v_{H} (ϖ) \Leftrightarrow v_{G} (ϖ) Δ v_{H} (ϖ) = v_{H} (ϖ) \Leftrightarrow v_{G} (ϖ) \nabla v_{H} (ϖ) = v_{G} (ϖ)

Lemma 2.2

⁴⁴ If E, F, and Q are three type two intuitionistic fuzzy sets on the domain W if $E \subseteq F$ , this is

\forall ς \in U, u_{E} (ς) ≺ u_{F} (ς) and v_{E} (ς) ≻ v_{F} (ς),

The following equation holds:

u_{E} (ς) \nabla u_{Q} (ς) ≺ u_{F} (ς) \nabla u_{Q} (ς)

v_{E} (ς) Δ v_{Q} (ς) ≻ v_{F} (ς) Δ v_{Q} (ς)

u_{E} (ς) Δ u_{Q} (ς) ≺ u_{F} (ς) Δ u_{Q} (ς)

v_{E} (ς) \nabla v_{Q} (ς) ≻ v_{F} (ς) \nabla v_{Q} (ς)

Type-two fuzzy rough set

Definition 2.7

⁴⁵ Suppose R is a type-two fuzzy relationship on $W \times W$ , where $(W, R)$ is a type-two fuzzy approximation space, E is a type-two fuzzy set on the domain W, E’s upper and lower approximations of the approximation space $(W, R)$ are the type-two fuzzy sets defined on W, as follows

\bar{A P P} (E) = \int_{ς \in W} u_{\bar{A P P} (E)} (ς) / ς, u_{\bar{R} (E)} (ς) = \underset{ϖ \in U}{Δ} [u_{R (ς, ϖ)} \nabla u_{E} (ϖ)]

\underline{A P P} (E) = \int_{ς \in W} u_{\underline{A P P} (E)} (ς) / ς, u_{\underline{R} (E)} (ς) = \underset{ϖ \in U}{\nabla} [u_{R^{C} (ς, ϖ)} Δ u_{A} (ϖ)]

$(\underline{A P P} (E), \bar{A P P} (E))$ is called a type-two fuzzy rough set of E with respect to the approximate space $(W, R)$ .

Lemma 2.3

⁴⁵ Suppose $\bar{A P P}$ and $\underline{A P P}$ are the upper and lower approximation operators in definition 2.7, for any $E, V, W \in F_{2} (U)$ and the following properties hold:

(1) ${\underline{A P P}}^{C} (E^{C}) = \bar{A P P} (E)$ , ${\bar{A P P}}^{C} (E^{C}) = \underline{A P P} (E)$

(2) if for $\forall ς \in U$ , thus $u_{E} (ς) = \frac{1}{0}, u_{Q} (ς) = \frac{1}{1}$ , then for $\forall ς \in U$ ,

thus $u_{\bar{A P P} (E)} (ς) = \frac{1}{0}$ , and $u_{\underline{A P P} (Q)} (ς) = \frac{1}{1}$

(3) $V \subseteq W \Rightarrow \bar{A P P} (V) \subseteq \bar{A P P} (W), \underline{A P P} (V) \subseteq \underline{A P P} (W)$

(4) $\bar{A P P} (V \cup W) = \bar{A P P} (V) \cup \bar{A P P} (W)$ , $\underline{A P P} (V \cap W) = \underline{A P P} (V) \cap \underline{A P P} (W)$

(5) $\bar{A P P} (V \cap W) \subseteq \bar{A P P} (V) \cap \bar{A P P} (W)$ , $\underline{A P P} (V \cup W) \supseteq \underline{A P P} (V) \cup \underline{A P P} (W)$

(6) $R_{1} \subseteq R_{2} \Rightarrow {\bar{A P P}}_{1} (E) \subseteq {\bar{A P P}}_{2} (E), {\underline{A P P}}_{2} (E) \subseteq {\underline{A P P}}_{1} (E)$ .

Type-two intuitionistic fuzzy rough set

Definition 2.8

⁴⁵ Suppose R is a type-two intuitionistic fuzzy relationship on $U \times U$ , where $(U, R)$ is a type-two intuitionistic fuzzy approximation space, S is a type-two intuitionistic fuzzy set on the domain U, and S’s upper and lower approximations to the approximation space $(U, R)$ are the type two intuitionistic fuzzy sets defined on U, as follows

\bar{A P P} (S) = {< ς, u_{\bar{A P P} (S)} (ς), v_{\bar{A P P} (S)} (ς) > | ς \in U}

\underline{A P P} (S) = {< ς, u_{\underline{A P P} (S)} (ς), v_{\underline{A P P} (S)} (ς) > | ς \in U}

among

u_{\bar{A P P} (S)} (ς) = \nabla_{m \in U} [u_{S} (m) Δ u_{R} (ς, m)]

v_{\bar{A P P} (S)} (ς) = Δ_{m \in U} [v_{S} (m) \nabla v_{R} (ς, m)]

u_{\underline{A P P} (S)} (ς) = Δ_{m \in U} [u_{S} (m) \nabla u_{R} (ς, m)]

v_{\underline{A P P} (S)} (ς) = \nabla_{m \in U} [v_{S} (m) Δ v_{R} (ς, m)]

(\underline{A P P} (S), \bar{A P P} (S))

is called a type-two intuitionistic fuzzy rough set of S with respect to the approximate space

(U, R)

Definition 2.9

⁴⁵ Suppose $E, Q \subseteq I F_{2} (U)$ , if

E \subseteq Q \Leftrightarrow \forall ς \in U, u_{E} (ς) ≺ u_{Q} (ς) 且 v_{E} (ς) ≻ v_{Q} (ς)

The order relationship $≺, ≻$ is defined as follows:

u_{E} (ς) ≺ u_{Q} (ς) \Leftrightarrow u_{E} (ς) Δ u_{Q} (ς) = u_{E} (ς) \Leftrightarrow u_{E} (ς) \nabla u_{Q} (ς) = u_{Q} (ς)

v_{E} (ς) ≻ v_{Q} (ς) \Leftrightarrow v_{E} (ς) Δ v_{Q} (ς) = v_{Q} (ς) \Leftrightarrow v_{E} (ς) \nabla v_{Q} (ς) = v_{E} (ς)

Lemma 2.4

⁴⁵ Suppose $M, N, P \subseteq I F_{2} (U)$ , if $M \subseteq N$ , this is $\forall ς \in U, u_{M} (ς) ≺ u_{N} (ς)$

and $v_{M} (ς) ≻ v_{N} (ς)$ , then

u_{M} (ς) \nabla u_{P} (ς) ≺ u_{N} (ς) \nabla u_{P} (ς)

u_{M} (ς) Δ u_{P} (ς) ≺ u_{N} (ς) Δ u_{P} (ς)

v_{M} (ς) Δ v_{P} (ς) ≻ v_{N} (ς) Δ v_{P} (ς)

v_{M} (ς) \nabla v_{P} (ς) ≻ v_{N} (ς) \nabla v_{P} (ς)

Lemma 2.5

⁴⁵ Suppose $\underline{A P P}, \bar{A P P}$ be the lower and upper approximations in Definition 1.8, for any $R, F \subseteq I F_{2} (U)$ , then:

(1) $\bar{A P P} (\sim R) = {\underline{A P P}}^{\sim} (R)$ , $\underline{A P P} (\sim R) = {\bar{A P P}}^{\sim} (R)$ ,

(2) $R \subseteq F \Rightarrow \underline{A P P} (R) \subseteq \underline{A P P} (F), \bar{A P P} (R) \subseteq \bar{A P P} (F)$ ,

(3) $A P P_{1} \subseteq A P P_{2} \Rightarrow \bar{A P P_{1}} \subseteq \bar{A P P_{2}}, \underline{A P P_{1}} \supseteq \underline{A P P_{2}}$ ,

(4) $\bar{A P P} (R \cup F) = \bar{A P P} (R) \cup \bar{A P P} (F)$ , $\underline{A P P} (R \cap F) = \underline{A P P} (R) \cap \underline{A P P} (F)$ , and

(5) $\bar{A P P} (R \cap F) \subseteq \bar{A P P} (R) \cap \bar{A P P} (F)$ , $\underline{A P P} (R \cup F) \supseteq \underline{A P P} (R) \cup \underline{A P P} (F)$ .

Type-two intuitionistic fuzzy rough set based on inclusion degree

Definition 3.1

Suppose U is a non-empty finite set, where $P (U)$ represents the entire set of classical sets in U, and $I F S_{2} (U)$ represents the entire set of type-two intuitionistic fuzzy sets in U设 $I F {S_{2}}^{'} (U) \subseteq I F S_{2} (U)$ , if for any $H, B \in I F {S_{2}}^{'} (U)$ , there is a corresponding value of $D (B / H)$ , and it satisfies:

(1) $0 \leq D (B / H) \leq 1$ ,

(2) $\forall H, B \in I F {S_{2}}^{'} (U)$ , $H \subseteq B \Rightarrow D (B / H) = 1$ , and

(3) $\forall H, S, Q \in I F {S_{2}}^{'} (U)$ , $Q \subseteq H \subseteq S \Rightarrow D (Q / S) \geq D (Q / H)$ .

Then D is called the inclusion degree on $I F {S_{2}}^{'} (U)$ .

On the basis of the above definition and the Euclidean distance formula between interval numbers, a new formula for calculating inclusion is given:

Theorem 3.1

Suppose W is a non-empty finite set, for any $M, N \in I F S_{2} (W)$ ,

D (N / M) = \frac{1}{2} = [\int_{δ \in W} \sqrt{{(\min (1, - u_{M} (δ) + u_{N} (δ) + 1) - 0)}^{2}} d δ

+ \int_{δ \in W} \sqrt{{(\min (1, v_{M} (δ) - v_{N} (δ) + 1) - 0)}^{2}} d δ]

Then D is called the inclusion degree.

The following proves that this formula satisfies the above three conditions.

Proof: (1) $D (M / A) \geq 0$ Clearly established.

∵ A \subseteq A,

∴ D (A / A) = \frac{1}{2} = [\int_{δ \in W} \sqrt{{(\min (1, - u_{A} (δ) + u_{A} (δ) + 1) - 0)}^{2}} d δ

+ \int_{δ \in W} \sqrt{{(\min (1, v_{A} (δ) - v_{A} (δ) + 1) - 0)}^{2}} d δ] = 1

(2) Necessity: if $M \subseteq N$ , then $\forall δ \in W, u_{M} (δ) ≺ u_{N} (δ)$ and $v_{M} (δ) ≻ v_{N} (δ)$ , so

D (N / M) = \frac{1}{2} = [\int_{δ \in W} \sqrt{{(\min (1, - u_{M} (δ) + u_{N} (δ) + 1) - 0)}^{2}} d δ

+ \int_{δ \in W} \sqrt{{(\min (1, v_{M} (δ) - v_{N} (δ) + 1) - 0)}^{2}} d δ] = 1

Sufficiency: if $D (N / M) = \frac{1}{2} = [\int_{δ \in W} \sqrt{{(\min (1, - u_{M} (δ) + u_{N} (δ) + 1) - 0)}^{2}} d δ$

+ \int_{δ \in W} \sqrt{{(\min (1, v_{M} (δ) - v_{N} (δ) + 1) - 0)}^{2}} d δ] = 1

Then $\min (1, - u_{M} (δ) + u_{N} (δ) + 1) = 1$ , $\min (1, v_{M} (δ) - v_{N} (δ) + 1) = 1$

Thus $1 - u_{M} (δ) + u_{N} (δ) \leq 1$ , $1 + v_{M} (δ) - v_{N} (δ) \leq 1$

So $\forall δ \in W, u_{M} (δ) ≺ u_{N} (δ)$ and $v_{M} (δ) ≻ v_{N} (δ)$ , then $M \subseteq N$ .

(3) If $M \subseteq N \subseteq C$ , then $\forall δ \in W, u_{M} (δ) ≺ u_{N} (δ) ≺ u_{C} (δ)$ , $v_{M} (δ) ≻ v_{N} (δ) ≻ v_{C} (δ)$ , so

\min (1, - u_{M} (δ) + u_{C} (δ) + 1) \geq \min (1, - u_{M} (δ) + u_{N} (δ) + 1)

\min (1, v_{M} (δ) - v_{C} (δ) + 1) \geq \min (1, v_{M} (δ) - v_{N} (δ) + 1)

Because $D (M / C) = \frac{1}{2} = [\int_{δ \in W} \sqrt{{(\min (1, - u_{M} (δ) + u_{C} (δ) + 1) - 0)}^{2}} d δ$

+ \int_{δ \in W} \sqrt{{(\min (1, v_{M} (δ) - v_{C} (δ) + 1) - 0)}^{2}} d δ]

D (M / N) = \frac{1}{2} = [\int_{δ \in W} \sqrt{{(\min (1, - u_{M} (δ) + u_{N} (δ) + 1) - 0)}^{2}} d δ

+ \int_{δ \in W} \sqrt{{(\min (1, v_{M} (δ) - v_{N} (δ) + 1) - 0)}^{2}} d δ]

Thus $D (M / C) \geq D (M / N)$ .

Theorem 3.2

For any $E, Q, N \in I F {S_{2}}^{'} (W)$ , if $M \subseteq Q$ , then

(1) $D (N / M) \leq D (N / Q)$ .

(2) $D (N / M) \leq D (Q / M)$ .

Proof: (1) if $M \subseteq Q$ , then $\forall ς \in W, u_{M} (ς) ≺ u_{Q} (ς)$ and $v_{M} (ς) ≻ v_{Q} (ς)$ , because

\min (1, - u_{N} (ς) + u_{M} (ς) + 1) \leq \min (1, - u_{N} (ς) + u_{Q} (ς) + 1)

\min (1, v_{Q} (ς) - v_{M} (ς) + 1) \leq \min (1, v_{N} (ς) - v_{Q} (ς) + 1)

Also because $D (N / M) = \frac{1}{2} = [\int_{ς \in W} \sqrt{{(\min (1, - u_{N} (ς) + u_{M} (ς) + 1) - 0)}^{2}} d ς$

+ \int_{ς \in W} \sqrt{{(\min (1, 1 + v_{N} (ς) - v_{M} (ς)) - 0)}^{2}} d ς]

D (N / Q) = \frac{1}{2} = [\int_{ς \in W} \sqrt{{(\min (1, 1 - u_{N} (ς) + u_{Q} (ς)) - 0)}^{2}} d ς

+ \int_{ς \in W} \sqrt{{(\min (1, 1 + v_{N} (ς) - v_{Q} (ς)) - 0)}^{2}} d ς]

D (N / M) \leq D (Q / M)

(2) Similarly, it can be proven.

Definition 3.2

Suppose $(W, B, D)$ is a fuzzy inclusion approximation space, where $B = {Q_{1}, Q_{2}, \dots, Q_{k}}$ is a weak fuzzy partition on W, and D is the inclusion degree on $I F S_{2} (W)$ . For any $G \in I F S_{2} (W)$ , upper approximations and lower approximations of M to $B$ bases on the parameter $0 \leq β < α \leq 1$ for a type-two intuitionistic fuzzy rough set stem from inclusion degree are defined as follows:

{\bar{A P P}}_{β} (G) = \cup {Q_{i} \in B | D (G / Q_{i}) > β}

{\underline{A P P}}_{α} (G) = \cup {Q_{i} \in B | D (G / Q_{i}) \geq α}

We can define positive domain, negative domain, and boundary domain :

$P o s (G) = {\underline{A P P}}_{α} (G) = \cup {Q_{i} \in B | D (G / Q_{i}) \geq α}$

N e g (G) = \cup {Q_{i} \in B | D (G / Q_{i}) \leq β}

B n (G) = \cup {Q_{i} \in B | β < D (G / Q_{i}) < α}

For parameters $α, β$ , rough sets in this definition can be divided into the following four categories:

(1) If ${\underline{A P P}}_{α} (R) \neq ϕ$ , ${\bar{A P P}}_{β} (R) \neq W$ , then $R$ is said to be partially definable.

(2) If ${\underline{A P P}}_{α} (R) \neq ϕ$ , ${\bar{A P P}}_{β} (R) = W$ , then $R$ is said to be internally definable.

(3) If ${\underline{A P P}}_{α} (R) = ϕ$ , ${\bar{A P P}}_{β} (R) \neq W$ , then $R$ is said to be externally definable.

(4) If ${\underline{A P P}}_{α} (R) = ϕ$ , ${\bar{A P P}}_{β} (R) = W$ , then $R$ is said to be completely undefinable.

Then, we can obtain the following properties.

Theorem 3.3

Suppose $0 \leq β < α \leq 1$ , if D is the inclusion degree on $I F S_{2} (U)$ , then the approximation operator based on inclusion degree satisfies the following properties:

(1) For any $M \in I F S_{2} (U)$ , then ${\underline{A P P}}_{α} (M) \subseteq {\bar{A P P}}_{β} (M)$ .

(2) For any $S, R \in I F S_{2} (U)$ and $S \subseteq R$ , then ${\underline{A P P}}_{α} (S) \subseteq {\underline{A P P}}_{α} (R)$ ,

${\bar{A P P}}_{β} (S) \subseteq {\bar{A P P}}_{β} (R)$ .

(3) For any $S, R \in I F S_{2} (U)$ , ${\underline{A P P}}_{α} (S \cap R) \subseteq {\underline{A P P}}_{α} (S) \cap {\underline{A P P}}_{α} (R)$ ,

${\underline{A P P}}_{α} (S \cup R) \supseteq {\underline{A P P}}_{α} (S) \cup {\underline{A P P}}_{α} (R)$ .

(4) For any $S, R \in I F S_{2} (U)$ , ${\bar{A P P}}_{β} (S \cap R) \subseteq {\bar{A P P}}_{β} (S) \cap {\bar{A P P}}_{β} (R)$ ,

${\bar{A P P}}_{β} (S \cup R) \supseteq {\bar{A P P}}_{β} (S) \cup {\bar{A P P}}_{β} (R)$ .

(5) For any $R \in I F S_{2} (U)$ , if $α_{1} \leq α_{2}$ , $β_{1} \leq β_{2}$ , then ${\underline{A P P}}_{α_{2}} (R) \subseteq {\underline{A P P}}_{α_{1}} (R)$ ,

${\bar{A P P}}_{β_{2}} (R) \subseteq {\bar{A P P}}_{β_{1}} (R)$ .

Proof: (1) According to Definition 3.1, it can be inferred that

(2) If $S \subseteq R$ , then $\forall ς \in U, u_{S} (ς) ≺ u_{R} (ς)$ and $v_{S} (ς) ≻ v_{R} (ς)$ , so

\min (1, 1 - u_{X} (ς) + u_{S} (ς)) \leq \min (1, 1 - u_{X} (ς) + u_{R} (ς))

\min (1, 1 + v_{X} (ς) - v_{S} (ς)) \leq \min (1, 1 + v_{X} (ς) - v_{R} (ς))

This is $D (X / S) \leq D (X / R)$ , if $D (X / S) \geq α$ , then $D (X / R) \geq α$ ,

so ${\underline{A P P}}_{α} (S) \subseteq {\underline{A P P}}_{α} (R)$ .

The same as ${\bar{A P P}}_{β} (S) \subseteq {\bar{A P P}}_{β} (R)$ .

(3) Because $S \cap R \subseteq S$ , $S \cap R \subseteq R$ , by (2) we hold ${\underline{A P P}}_{α} (S \cap R) \subseteq {\underline{A P P}}_{α} (S)$ ,

${\underline{A P P}}_{α} (S \cap R) \subseteq {\underline{A P P}}_{α} (R)$ , so ${\underline{A P P}}_{α} (S \cap R) \subseteq {\underline{A P P}}_{α} (S) \cap {\underline{A P P}}_{α} (R)$ .

Because $S \subseteq S \cup R$ , $R \subseteq S \cup R$ , then ${\underline{A P P}}_{α} (S \cup R) \supseteq {\underline{A P P}}_{α} (S)$ ,

${\underline{A P P}}_{α} (S \cup R) \supseteq {\underline{A P P}}_{α} (R)$ , thus ${\underline{A P P}}_{α} (S \cup R) \supseteq {\underline{A P P}}_{α} (S) \cup {\underline{A P P}}_{α} (R)$ .

(4) Similar to (3) evidence method.

(5) If $α_{1} \leq α_{2}$ , when $D (X / R) \geq α_{2}$ and $D (X / R) \geq α_{1}$ , so ${\underline{A P P}}_{α_{2}} (R) \subseteq {\underline{A P P}}_{α_{1}} (R)$ .

The same as ${\bar{A P P}}_{β_{2}} (R) \subseteq {\bar{A P P}}_{β_{1}} (R)$ .

From (5) of Theorem 3.2, it can be seen that the positive field increases with the decrease of $α$ , the negative field increases with the increase of $β$ , and the boundary shrinks. Therefore, the following theorem holds:

Theorem 3.3. Suppose $0 < ε < 1$ , then for any $N \subseteq W$ , we hold

(1) $\lim_{α ↓ ε} {\underline{A P P}}_{α} (N) = \underset{α > ε}{\cup} {\underline{A P P}}_{α} (N) = {\bar{A P P}}_{ε} (N)$ .

(2) $\lim_{β ↑ ε} {\bar{A P P}}_{β} (N) = \underset{β < ε}{\cap} {\bar{A P P}}_{β} (N) = {\underline{A P P}}_{ε} (N)$ .

Proof: (1) If $α > ε$ , from the definitions of lower approximations, it can be inferred:

{\underline{A P P}}_{α} (N) = \cup {Q_{i} \in B | D (N / Q_{i}) \geq α}

\subseteq \cup {Q_{i} \in B | D (N / Q_{i}) > ε} {\bar{A P P}}_{ε} (K)

Because ${\underline{A P P}}_{α} (N)$ increases as $α$ decreases, so

\lim_{α ↓ ε} {\underline{A P P}}_{α} (N) = \underset{α > ε}{\cup} {\underline{A P P}}_{α} (N) \subseteq {\bar{A P P}}_{ε} (N)

If there exists $g \in {\bar{A P P}}_{ε} (N) \ \underset{α > ε}{\cup} {\underline{A P P}}_{α} (N)$ , from the definitions of upper approximations, it can be inferred $D (N / Q_{i}) > ε$ .

If $g \notin {\underline{A P P}}_{α} (N)$ , then $D (N / Q_{i}) < α$ , because $α > ε$ , so $D (N / Q_{i}) \leq ε$ , contradiction between $D (N / Q_{i}) > ε$ and $D (N / Q_{i}) \leq ε$ , so ${\bar{A P P}}_{ε} (N) = \underset{α > ε}{\cup} {\underline{A P P}}_{α} (N)$ .

Then (1) holds.

(2) If $β < ε$ , by the definitions of upper approximations, we hold

{\bar{A P P}}_{β} (N) = \cup {Q_{i} \in B | D (N / Q_{i}) > β}

\supseteq \cup {Q_{i} \in B | D (N / Q_{i}) \geq ε} = {\underline{A P P}}_{ε} (N)

According to Theorem 3.2 of (5), ${\bar{A P P}}_{β} (N)$ decreases as $β$ increases, therefore

\lim_{β ↑ ε} {\bar{A P P}}_{β} (N) = \underset{β < ε}{\cap} {\bar{A P P}}_{β} (N) \supseteq {\underline{A P P}}_{ε} (N)

If there exists $g \in \underset{β < ε}{\cap} {\bar{A P P}}_{β} (N) \ {\underline{A P P}}_{ε} (N)$ , so $g \in \underset{β < ε}{\cap} {\bar{A P P}}_{β} (N)$ and $g \notin {\underline{A P P}}_{ε} (N)$ .

Thus $D (N / Q_{i}) > β$ , for any $β < ε$ hold, so $D (N / Q_{i}) \geq ε$ . But contradictory to $D (N / Q_{i}) < ε$ , so $\underset{β < ε}{\cap} {\bar{A P P}}_{β} (N) = {\underline{A P P}}_{ε} (N)$

Thus (2) holds.

Theorem 3.4

Suppose $0 < ε < 1$ , then

\lim_{β ↑ ε, α ↓ ε} B n (N) = \underset{β < ε < α}{\cap} ({\bar{A P P}}_{β} (N) \ {\underline{A P P}}_{α} (N))

= {\bar{A P P}}_{ε} (N) \ {\underline{A P P}}_{ε} (N) = \cup {Q_{i} \in B | D (N / Q_{i}) = ε}

Proof: obviously $B n (N) = {\bar{A P P}}_{ε} (N) \ {\underline{A P P}}_{ε} (N) = \cup {Q_{i} \in B | D (N / Q_{i}) = ε}$ .

Because $B n (N) = \cup {Q_{i} \in B | β < D (N / Q_{i}) < α}$ , so when $α$ monotonically decreases towards $ε$ and $β$ monotonically increases towards $ε$ , the boundary domain monotonically decreases, so

\lim_{β ↑ ε, α ↓ ε} B n (N) = \underset{β < ε < α}{\cap} ({\bar{A P P}}_{β} (N) \ {\underline{A P P}}_{α} (N))

\supseteq {\bar{A P P}}_{ε} (N) \ {\underline{A P P}}_{ε} (N) = \cup {Q_{i} \in B | D (N / Q_{i}) = ε}

if there exists

g \in \underset{β < ε < α}{\cap} ({\bar{A P P}}_{β} (N) \ {\underline{A P P}}_{α} (N)) \ \cup {Q_{i} \in B | D (N / Q_{i}) = ε}

then $g \in {\bar{A P P}}_{β} (N) \ {\underline{A P P}}_{α} (N)$ , but $D (N / Q_{i}) \neq ε$ .

When $g \in {\bar{A P P}}_{β} (N)$ , then $D (N / Q_{i}) > β$ . For any $β < ε$ , thus $D (N / Q_{i}) \geq ε$ .

When $g \notin {\underline{A P P}}_{α} (N)$ , then $D (N / Q_{i}) < α$ . For any $α > ε$ , thus $D (N / Q_{i}) \leq ε$ .

So $D (N / Q_{i}) = ε$ . This contradicts with $D (N / Q_{i}) \neq ε$ ; therefore, the theorem is proven.

From the above theorem, it can be seen that as $α$ monotonically decreases towards $ε$ and $β$ monotonically increases towards $ε$ , the boundary domain gradually shrinks to:

B n (N) = \cup {Q_{i} \in B | D (N / Q_{i}) = ε} .

Uncertainty metrics

Definition 4.1

If $α_{R} (T)$ is the approximate precision of the set $T (T \subseteq U)$ , and

α_{R} (T) = \frac{| {\underline{A P P}}_{α} (T) |}{| {\bar{A P P}}_{β} (T) |}

Among them $T \neq ϕ$ , $| T |$ represents the cardinality of set $T$ .

Obviously $0 \leq α_{R} (T) \leq 1$ .

Definition 4.2

If $ρ_{R} (M)$ is the roughness degree of the set $M (M \subseteq U)$ , and

ρ_{R} (M) = 1 - α_{R} (M)

Definition 4.3

If $B = {B_{1}, B_{2}, \dots, B_{k}}$ is a weak fuzzy partition on U, $α_{R} (B)$ is called the approximate classification accuracy of $B$ . If

α_{R} (B) = \frac{\sum_{i = 1}^{n} | {\underline{A P P}}_{α} (B_{i}) |}{\sum_{i = 1}^{n} | {\bar{A P P}}_{β} (B_{i}) |}

Definition 4.4

If $B = {B_{1}, B_{2}, \dots, B_{k}}$ is a weak fuzzy partition on U, $γ_{R} (B)$ is called the approximate quality of classification of $B$ . If

γ_{R} (B) = \frac{\sum_{i = 1}^{n} | {\underline{A P P}}_{α} (B_{i}) |}{| U |}

Definition 4.5

If $B = {B_{1}, B_{2}, \dots, B_{k}}$ is a weak fuzzy partition on U, $G D (B)$ is called the granularity of $B$ , if

G D (B) = \sum_{i = 1}^{n} {| B_{i} |}^{2} / {| U |}^{2}

Definition 4.6

If $B = {B_{1}, B_{2}, \dots, B_{k}}$ is a weak fuzzy partition on U, $D I S (B)$ is called the resolution $B$ , if

D I S (B) = 1 - G D (B) = 1 - \sum_{i = 1}^{n} {| B_{i} |}^{2} / {| U |}^{2}

Definition 4.7

If A is the set of all attributes, $P \subseteq A$ , $m \in A$ , then the importance of m to $P$ is denoted as $S i g_{P} (m)$ , if

S i g_{P} (m) = 1 - | P \cup {m} | / | P |

Definition 4.8

Suppose $A = {< τ, u_{A} (τ), v_{A} (τ) > | τ \in U}$ be a type-two intuitionistic fuzzy set on U, where $\nabla_{A} (τ)$ is the degree of ambiguity of $τ$ with respect to $A$ . If

\nabla_{A} (τ) = | u_{A} (τ) - v_{A} (τ) |

Conclusion

This article provides the definition of inclusion degree in type-two intuitionistic fuzzy sets, a new calculation formula for inclusion degree, and verifies the feasibility of the formula. Furthermore, the definition and attributes of type two intuitionistic fuzzy rough sets based on inclusion degree are given, and at the same time, the type two fuzzy set was expanded.

Statements and declarations

Footnotes

Conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

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