# Interval-valued fuzzy soft $\beta$-covering approximation spaces

Shizhan Lu1 1School of Management, Jiangsu University, Zhengjiang 212013,
China

###### Abstract

The concept of interval-valued fuzzy soft $\beta$-covering approximation
spaces (IFS$\beta$CASs) is introduced to combine the theories of soft sets,
rough sets and interval-valued fuzzy sets, and some fundamental propositions
concerning interval-valued fuzzy soft $\beta$-neighborhoods and soft
$\beta$-neighborhoods of IFS$\beta$CASs are explored. And then four kinds of
interval-valued fuzzy soft $\beta$-coverings based fuzzy rough sets are
researched. Finally, the relationships of four kinds of interval-valued fuzzy
soft $\beta$-coverings based fuzzy rough sets are investigated.

###### keywords:

Interval-valued fuzzy soft $\beta$-covering, Interval-valued fuzzy soft
$\beta$-neighborhoods, Soft $\beta$-neighborhood.

##  1 Introduction

The fuzzy set, as proposed by Zadeh [39], stands as a renowned instrument for
handling uncertainty with diverse applications [8, 10, 17, 21, 22, 44].
Subsequently, Gorzalczany [16] introduced the notion of interval-valued fuzzy
sets, where the membership degree of set elements lies within the interval
$[0,1]$. Interval-valued fuzzy sets are adept at handling scenarios where
precise probabilities of set membership are elusive, offering instead an
interval within which such probabilities are constrained [16, 29].

Molodtsov [25] introduced soft sets as a solution to the challenge of
uncertainty. Unlike traditional mathematical approaches to uncertainty, soft
set theory offers a unique advantage in parameter handling. Soft set theory
enjoys wide-ranging applications across decision-making, rules mining, machine
learning, artificial intelligence, image processing, and beyond [4, 36, 14, 1,
24].

Pawlak [28] initially proposed rough set theory to address the challenges
posed by vagueness and granularity in information systems and data analysis.
Lower and upper approximations are central to rough set theory, offering a
mechanism to represent uncertain knowledge based on existing information. At
its essence, rough set theory constitutes an approximation space defined by a
specified universe and an equivalence relation [38]. Rough set theory has
risen to prominence as a rapidly expanding academic discipline spanning
domains such as machine learning, deep learning, data mining, decision-making,
and beyond [6, 20, 32, 9, 33, 18, 37, 41]. The amalgamation of soft sets and
rough sets frequently inspires the exploration of theories related to soft
covering-based rough sets [2, 3, 11, 13, 43], attaining substantial relevance
in specific domains. However, in fuzzy environments, rough set theory
demonstrates inherent limitations, as discussed in [42]. To overcome these
challenges, Zhang and Zhan [42] integrated fuzzy sets, soft sets, and rough
sets, expanding the notion of soft covering to fuzzy soft $\beta$-covering,
drawing insights from the works of Ma [23] and Yang [34, 35].

To combine the theories of soft sets, rough sets and interval-valued fuzzy
sets, the concept of interval-valued fuzzy soft $\beta$-covering approximation
spaces (IFS$\beta$CASs) is introduced and the propositions about neighborhoods
in IFS$\beta$CASs are researched. And then four kinds of interval-valued fuzzy
soft $\beta$-coverings based fuzzy rough sets are researched. Finally, the
relationships of four kinds of interval-valued fuzzy soft $\beta$-coverings
based fuzzy rough sets are investigated.

The following sections of this manuscript are structured as follows: Section 2
offers the Preliminaries, Section 3 introduces the notion of interval-valued
fuzzy soft $\beta$-covering approximation spaces (IFS$\beta$CASs) and
investigates the propositions regarding interval-valued fuzzy soft
$\beta$-neighborhoods and soft $\beta$-neighborhoods of IFS$\beta$CASs,
Section 4 presents four kinds of interval-valued fuzzy soft $\beta$-coverings
based fuzzy rough sets and studies their relationships.

##  2 Preliminaries

In this section we briefly review some foundations. If two sets $A$ and $B$
are classical sets (set theory established by Cantor [27]), $A\sqcap B$ and
$A\sqcup B$ represent the intersection and union of $A$ and $B$, respectively.
Furthermore, $A\sqsubset B$ represents that $A$ is a subset of the classical
set $B$. Let $U$ be a universal set and $E$ be a set of parameters. Let
$IF(U)$ be the set of all interval-valued fuzzy sets defined over $U$.

###  2.1 Fuzzy sets, soft sets, rough sets and approximation spaces

######  Definition 2.1.

[39] Let $U$ be an initial universe set. A fuzzy set $F$ on $U$ is a mapping
$F:U\rightarrow[0,1]$.

######  Definition 2.2.

[25] Let $U$ be an initial universe set and $E$ be a set of parameters. $P(U)$
is denoted the power set of $U$. A pair $(F,A)$ is called a soft set over $U$,
where $A\subset E$ and $F$ is a mapping given by $F:A\rightarrow P(U)$.

######  Definition 2.3.

[28] Let $U$ be an initial universe set and $R$ be an equivalence relation on
$U$. A pair $(U,R)$ is called a Pawlak approximation space. Two rough
approximations $R_{-}(X)=\\{x\in U:[x]_{R}\sqsubset X\\}$ and
$R_{+}(X)=\\{x\in U:[x]_{R}\sqcap X\neq\emptyset\\}$ are called the lower and
upper approximation of $X$, respectively, where $X\sqsubset U$.

######  Definition 2.4.

[23] Let $U$ be an initial universe set and $F(U)$ be the set of all fuzzy
subsets of $U$. For $\beta\in(0,1]$,
$\mathcal{C}=\\{C_{1},C_{2},\cdots,C_{m}\\}$ is called a fuzzy
$\beta$-covering of $U$ if $\bigcup_{i=1}^{m}C(x)\geqslant\beta$ for all $x\in
U$, where $\mathcal{C}\sqsubset F(U)$. And $(U,\mathcal{C})$ is called a fuzzy
$\beta$-covering approximation space.

If $\beta=1$, then $(U,\mathcal{C})$ is called a fuzzy covering approximation
space.

###  2.2 Interval-valued fuzzy sets and interval-valued fuzzy soft sets

######  Definition 2.5.

[7, 31] Assuming that $I_{1}=[I_{1}^{-},I_{1}^{+}]$ and
$I_{2}=[I_{2}^{-},I_{2}^{+}]$ are two bounded closed interval on the real
number field, then $I_{1}\leqslant I_{2}$ if and only if $I_{1}^{+}\leqslant
I_{2}^{+}$ and $I_{1}^{-}\leqslant I_{2}^{-}$.

######  Definition 2.6.

[12, 29, 40] Assuming that $I_{1}=[I_{1}^{-},I_{1}^{+}]$ and
$I_{2}=[I_{2}^{-},I_{2}^{+}]$ are two bounded closed interval on the real
number field, then we call $I_{1}$ and $I_{2}$ as two interval values. For any
two interval values $I_{1}$ and $I_{2}$, the following operations and product
order relation are valid:

(1) $I_{1}\wedge I_{2}=[I_{1}^{-}\wedge I_{2}^{-},I_{1}^{+}\wedge I_{2}^{+}]$,

(2) $I_{1}\vee I_{2}=[I_{1}^{-}\vee I_{2}^{-},I_{1}^{+}\vee I_{2}^{+}]$,

(3) $I_{1}^{c}=[1-I_{1}^{+},1-I_{1}^{-}]$,

where $x\wedge y=\min\\{x,y\\}$ and $x\vee y=\max\\{x,y\\}$ for all
$x,y\in[0,1]$.

######  Definition 2.7.

[12, 16, 29] An interval-valued fuzzy set $F$ on an universe $U$ is a mapping
such that $F:x\rightarrow I$, where $x\in U$ and $I$ is a closed sub-intervals
of $[0,1]$.

For any two interval-valued fuzzy sets $F_{1}$ and $F_{2}$ on $U$, the
following basic operations are valid for all $x\in U$:

(1) $(F_{1}\cap F_{2})(x)=F_{1}(x)\wedge F_{2}(x)$,

(2) $(F_{1}\cup F_{2})(x)=F_{1}(x)\vee F_{2}(x)$,

(3) $F_{1}^{c}(x)=[1-F_{1}^{+}(x),1-F_{1}^{-}(x)]$,

(4) $F_{1}\subset F_{2}\Leftrightarrow F_{1}(x)\leqslant F_{2}(x)$.

######  Definition 2.8.

[15, 19, 30] Let $U$ be an initial universe set and $E$ be a set of
parameters. A pair $(F,E)$ is an interval-valued fuzzy soft sets if $F(e)\in
IF(U)$ for every $e\in E$.

For two interval-valued fuzzy soft sets $(F,A)$ and $(G,B)$ over a common
universe $U$, $A\sqsubset B\sqsubset E$. We denote $(F,A)\subset(G,B)$ if
$F(e)\subset G(e)$ for every $e\in A$.

######  Theorem 2.9.

[5, 26] Let $A,B$ and $C$ be three interval-valued fuzzy soft sets, the
following statements hold.

(1) $A\cap B=B\cap A$, $A\cup B=B\cup A$.

(2) $(A\cup B)\cap C=(A\cap C)\cup(B\cap C)$, $(A\cap B)\cup C=(A\cup
C)\cap(B\cup C)$.

(3) $(A\cap B)\cap C=A\cap(B\cap C)$, $(A\cup B)\cup C=A\cup(B\cup C)$.

(4) $(A^{c})^{c}=A$.

(5) $(A\cap B)^{c}=A^{c}\cup B^{c}$, $(A\cup B)^{c}=A^{c}\cap B^{c}$.

##  3 Neighborhoods in IFS$\beta$CASs

###  3.1 Interval-valued fuzzy soft $\beta$-neighborhoods in IFS$\beta$CASs

Let the interval-valued number $\beta$ be an indicator to survey other
interval-valued fuzzy sets in the processes of decision-making.
$I^{U}(x)=[1,1]$ and $I^{\emptyset}(x)=[0,0]$ for all $x\in U$.

######  Definition 3.10.

Suppose $U$ and $E$ are the sets of objects and parameters, respectively. Let
$A\sqsubset E$, $F:E\rightarrow IF(U)$. If $I^{U}\subset\bigcup\limits_{e\in
A}F(e)$, then we called $(F,A)$ an interval-valued fuzzy soft covering over
$U$.

Here, $(U,F,A)$ is called an interval-valued fuzzy soft covering approximation
space.

######  Definition 3.11.

Let $\beta$ be an interval-valued number, $F:E\rightarrow IF(U)$. If
$\beta\leqslant(\bigcup\limits_{e\in A}F(e))(x)$ for all $x\in U$, then
$(F,A)$ is called an interval-valued fuzzy soft $\beta$-covering over $U$, the
triple $T=(U,F,A)_{\beta}$ is called an interval-valued fuzzy soft
$\beta$-covering approximation space.

######  Definition 3.12.

Let $\beta$ be an interval-valued number, and $U$ and $E$ be the sets of
objects and parameters, respectively. Let $e\in A\sqsubset E$, $F:E\rightarrow
IF(U)$. $\widetilde{SN}^{\beta}_{x}=\bigcap\\{F(e)\in F(A):\beta\leqslant
F(e)(x)\\}$ is called an interval-valued fuzzy soft $\beta$-neighborhood of
$x$ in $(U,F,A)_{\beta}$.

######  Lemma 3.13.

Let $I_{*}$ be an interval-valued number, $\mathcal{I}$ and $\mathcal{I}_{1}$
be two families of interval-valued numbers, $|\mathcal{I}|=\Lambda$,
$|\mathcal{I}_{1}|=\Gamma$, and $\mathcal{I}\sqsubset\mathcal{I}_{1}$, where
$\Lambda$ and $\Gamma$ are two random sets of indicators.

(1) $I_{*}\leqslant I$ for all $I\in\mathcal{I}$ if and only if
$I_{*}\leqslant\bigwedge\\{I:I\in\mathcal{I}\\}$.

(2) If $I_{*}\leqslant I_{\alpha}$ for an $I_{\alpha}\in\mathcal{I}$, then
$I_{*}\leqslant\bigvee\\{I:I\in\mathcal{I}\\}$.

(3) If $\mathcal{I}\sqsubset\mathcal{I}_{1}$, then
$\bigwedge\\{I:I\in\mathcal{I}_{1}\\}\leqslant\bigwedge\\{I:I\in\mathcal{I}\\}$.

Proof (1) $I_{*}\leqslant I$ for all $I\in\mathcal{I}$, i.e.,
$I_{*}^{-}\leqslant I^{-}$ and $I_{*}^{+}\leqslant I^{+}$ for all
$I\in\mathcal{I}$.

$I_{*}^{-}\leqslant inf\\{I^{-}:I\in\mathcal{I}\\}$ if and only if
$I_{*}^{-}\leqslant I^{-}$ for all $I\in\mathcal{I}$. $I_{*}^{+}\leqslant
inf\\{I^{+}:I\in\mathcal{I}\\}$ if and only if $I_{*}^{+}\leqslant I^{+}$ for
all $I\in\mathcal{I}$. That is,
$I_{*}\leqslant\bigwedge\\{I:I\in\mathcal{I}\\}$ if and only if
$I_{*}\leqslant I$ for all $I\in\mathcal{I}$.

(2) $I_{*}\leqslant I_{\alpha}$ for an $I_{\alpha}\in\mathcal{I}$, i.e.,
$I_{*}^{-}\leqslant I_{\alpha}^{-}$ and $I_{*}^{+}\leqslant I_{\alpha}^{+}$.
Then, $I_{*}^{-}\leqslant I_{\alpha}^{-}\leqslant
sup\\{I^{-}:I\in\mathcal{I}\\}$ and $I_{*}^{+}\leqslant
I_{\alpha}^{+}\leqslant sup\\{I^{+}:I\in\mathcal{I}\\}$, it means that
$I_{*}\leqslant\bigvee\\{I:I\in\mathcal{I}\\}$.

(3) $\mathcal{I}\sqsubset\mathcal{I}_{1}$, then
$inf\\{I^{-}:I\in\mathcal{I}_{1}\\}\leqslant inf\\{I^{-}:I\in\mathcal{I}\\}$
and $inf\\{I^{+}:I\in\mathcal{I}_{1}\\}\leqslant
inf\\{I^{+}:I\in\mathcal{I}\\}$. It means that
$\bigwedge\\{I:I\in\mathcal{I}_{1}\\}\leqslant\bigwedge\\{I:I\in\mathcal{I}\\}$.$\blacksquare$

######  Theorem 3.14.

Let $\beta$ be an interval-valued number,
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(x)$ for all $x\in U$ in
$(U,F,A)_{\beta}$.

Proof For each $x\in U$, $\widetilde{SN}^{\beta}_{x}=\bigcap\\{F(e)\in
F(A):\beta\leqslant F(e)(x)\\}$. By Lemma 3.13 (1), if $\beta\leqslant
F(e)(x)$ for all $e\in A^{*}\sqsubset A$, then
$\beta\leqslant(\bigcap\limits_{e\in A^{*}}F(e))(x)$, i.e.,
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(x)$.$\blacksquare$

######  Theorem 3.15.

Let $\beta$ be an interval-valued number. If
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$ and
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)$, then
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(z)$ for all $x,y,z\in U$.

Proof For each $e\in A\in(U,F,A)_{\beta}$,
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)=(\bigcap\limits_{\beta\leqslant
F(e)(x)}F(e))(y)$. By the sufficiency of Lemma 3.13 (1), we can obtain that
$\beta\leqslant F(e)(y)$ for each $F(e)\in\\{F(e):\beta\leqslant F(e)(x)\\}$.
It means that if $\beta\leqslant F(e)(x)$ then $\beta\leqslant F(e)(y)$ for
each $e\in A$.

In the similar way above, $\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)$
implies that if $\beta\leqslant F(e)(y)$ then $\beta\leqslant F(e)(z)$ for
each $e\in A$.

To sum up, $\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$ and
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)$ imply that if $\beta\leqslant
F(e)(x)$ then $\beta\leqslant F(e)(z)$ for each $e\in A$. By the necessity of
Lemma 3.13 (1), $\beta\leqslant(\bigcap\\{F(e):\beta\leqslant
F(e)(x)\\})(z)=\widetilde{SN}^{\beta}_{x}(z)$.$\blacksquare$

######  Theorem 3.16.

For two interval-valued numbers $\beta_{1}$ and $\beta_{2}$, if
$\beta_{1}\leqslant\beta_{2}$, then
$\widetilde{SN}^{\beta_{1}}_{x}\subset\widetilde{SN}^{\beta_{2}}_{x}$ for all
$x\in U$.

Proof Since $\beta_{1}\leqslant\beta_{2}$, if $\beta_{2}\leqslant F(e)(x)$
then $\beta_{1}(x)\leqslant F(e)(x)$ for $e\in A\in(U,F,A)_{\beta}$, i.e.,
$\\{F(e):\beta_{2}\leqslant F(e)(x),e\in
A\\}\sqsubset\\{F(e):\beta_{1}\leqslant F(e)(x),e\in A\\}$.

By Lemma 3.13 (3), $\\{F(e):\beta_{2}\leqslant F(e)(x),e\in
A\\}\sqsubset\\{F(e):\beta_{1}\leqslant F(e)(x),e\in A\\}$ implies
$\widetilde{SN}^{\beta_{1}}_{x}=\bigcap\\{F(e):\beta_{1}\leqslant F(e)(x),e\in
A\\}\subset\bigcap\\{F(e):\beta_{2}\leqslant F(e)(x),e\in
A\\}=\widetilde{SN}^{\beta_{2}}_{x}$.$\blacksquare$

######  Theorem 3.17.

Let $\beta$ be an interval-valued number, $x,y\in U$.

(1) $\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$ if and only if
$\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}$.

(2) $\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$ and
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(x)$ if and only if
$\widetilde{SN}^{\beta}_{x}=\widetilde{SN}^{\beta}_{y}$.

Proof (1) For $e\in A\in(U,F,A)_{\beta}$, if
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)=(\bigcap\limits_{\beta\leqslant
F(e)(x)}F(e))(y)$, by the sufficiency of Lemma 3.13 (1), $\beta\leqslant
F(e)(y)$ holds for each $F(e)\in\\{F(e):\beta\leqslant F(e)(x)\\}$, then
$\\{F(e):\beta\leqslant F(e)(x)\\}\sqsubset\\{F(e):\beta\leqslant F(e)(y)\\}$
is obtained.

By Lemma 3.13 (1), $\\{F(e):\beta\leqslant
F(e)(x)\\}\sqsubset\\{F(e):\beta\leqslant F(e)(y)\\}$ implies
$\bigcap\\{F(e):\beta\leqslant F(e)(y)\\}\subset\bigcap\\{F(e):\beta\leqslant
F(e)(x)\\}$, i.e.,
$\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}$.

On the other hand, by Theorem 3.14 (1),
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(y)$ holds. If
$\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}$, then
$\widetilde{SN}^{\beta}_{y}(y)\leqslant\widetilde{SN}^{\beta}_{x}(y)$, then
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(y)\leqslant\widetilde{SN}^{\beta}_{x}(y)$.

(2) By the results of (1), $\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$ and
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(x)$ if and only if
$\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}$ and
$\widetilde{SN}^{\beta}_{x}\subset\widetilde{SN}^{\beta}_{y}$, i.e.,
$\widetilde{SN}^{\beta}_{y}=\widetilde{SN}^{\beta}_{x}$.$\blacksquare$

###  3.2 Soft $\beta$-neighborhoods in IFS$\beta$CASs

######  Definition 3.18.

Let $\beta$ be an interval-valued number, $F:E\rightarrow IF(U)$, $A\sqsubset
E$. $\overline{SN}^{\beta}_{x}=\\{y\in
U:\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)\\}$ is called the soft
$\beta$-neighborhood of $x$ in $(U,F,A)_{\beta}$.

######  Theorem 3.19.

Let $\beta$ be an interval-valued number, $x\in\overline{SN}^{\beta}_{x}$ for
all $x\in U$.

Proof By Theorem 3.14, $\beta\leqslant\widetilde{SN}^{\beta}_{x}(x)$, then
$x\in\overline{SN}^{\beta}_{x}$.$\blacksquare$

######  Theorem 3.20.

Let $\beta$ be an interval-valued number, $y\in\overline{SN}^{\beta}_{x}$ if
and only if $\overline{SN}^{\beta}_{y}\sqsubset\overline{SN}^{\beta}_{x}$ for
all $x,y\in U$.

Proof If $y\in\overline{SN}^{\beta}_{x}$, i.e.,
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$, by Theorem 3.17 (1) we have
$\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}$, then
$\widetilde{SN}^{\beta}_{y}(z)\leqslant\widetilde{SN}^{\beta}_{x}(z)$ for each
$z\in U$.

If $\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)$, then
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)\leqslant\widetilde{SN}^{\beta}_{x}(z)$.
It means that if $z\in\overline{SN}^{\beta}_{y}$ then
$z\in\overline{SN}^{\beta}_{x}$, i.e.,
$\overline{SN}^{\beta}_{y}\sqsubset\overline{SN}^{\beta}_{x}$.

On the other hand, by the results of Theorem 3.19,
$y\in\overline{SN}^{\beta}_{y}$. If
$\overline{SN}^{\beta}_{y}\sqsubset\overline{SN}^{\beta}_{x}$, then
$y\in\overline{SN}^{\beta}_{x}$.$\blacksquare$

######  Theorem 3.21.

Let $\beta$ be an interval-valued number, if $y\in\overline{SN}^{\beta}_{x}$
and $z\in\overline{SN}^{\beta}_{y}$, then $z\in\overline{SN}^{\beta}_{x}$ for
all $x,y,z\in U$.

Proof $y\in\overline{SN}^{\beta}_{x}$ and $z\in\overline{SN}^{\beta}_{y}$,
i.e., $\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$ and
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)$. By Theorem 3.15, if
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$ and
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)$, then
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(z)$, i.e.,
$z\in\overline{SN}^{\beta}_{x}$.$\blacksquare$

######  Theorem 3.22.

Let $\beta$ be an interval-valued number, the following statements hold for
all $x,y\in U$.

(1) $\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}$ if and only
if $\overline{SN}^{\beta}_{y}\sqsubset\overline{SN}^{\beta}_{x}$.

(2) $\overline{SN}^{\beta}_{x}=\overline{SN}^{\beta}_{y}$ if and only if
$\widetilde{SN}^{\beta}_{x}=\widetilde{SN}^{\beta}_{y}$.

Proof (1) By Theorem 3.17 (1),
$\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}\Leftrightarrow\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)$.
By Definition 3.18,
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(y)\Leftrightarrow
y\in\overline{SN}^{\beta}_{x}$. By Theorem 3.20,
$y\in\overline{SN}^{\beta}_{x}\Leftrightarrow\overline{SN}^{\beta}_{y}\sqsubset\overline{SN}^{\beta}_{x}$.
Then,
$\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}\Leftrightarrow\overline{SN}^{\beta}_{y}\sqsubset\overline{SN}^{\beta}_{x}$.

(2) By the results of (1),
$\overline{SN}^{\beta}_{y}\sqsubset\overline{SN}^{\beta}_{x}$ and
$\overline{SN}^{\beta}_{x}\sqsubset\overline{SN}^{\beta}_{y}$ if and only if
$\widetilde{SN}^{\beta}_{y}\subset\widetilde{SN}^{\beta}_{x}$ and
$\widetilde{SN}^{\beta}_{x}\subset\widetilde{SN}^{\beta}_{y}$. It means that
$\overline{SN}^{\beta}_{y}=\overline{SN}^{\beta}_{x}$ if and only if
$\widetilde{SN}^{\beta}_{x}=\widetilde{SN}^{\beta}_{y}$.$\blacksquare$

######  Theorem 3.23.

Let $\beta$ be an interval-valued number, $\Lambda$ is a random set of
indicators, the following statements hold for all $x,y\in U$.

(1)
$\overline{SN}^{\beta}_{x}\sqcup\overline{SN}^{\beta}_{y}\sqsubset\overline{\widetilde{SN}^{\beta}_{x}\cup\widetilde{SN}^{\beta}_{y}}^{\beta}$.

(2)
$\overline{SN}^{\beta}_{x}\sqcap\overline{SN}^{\beta}_{y}=\overline{\widetilde{SN}^{\beta}_{x}\cap\widetilde{SN}^{\beta}_{y}}^{\beta}$.

(3)
$\sqcup_{v\in\Lambda}\overline{SN}_{x_{v}}^{a\beta}\sqsubset\overline{\bigcup_{v\in\Lambda}\widetilde{SN}_{x_{v}}^{a\beta}}^{a\beta}$.

(4)
$\sqcap_{v\in\Lambda}\overline{SN}_{x_{v}}^{a\beta}=\overline{\bigcap_{v\in\Lambda}\widetilde{SN}_{x_{v}}^{a\beta}}^{a\beta}$.

Proof (1) For each $z\in U$, if
$z\in\overline{SN}^{\beta}_{x}\sqcup\overline{SN}^{\beta}_{y}$, i.e.,
$\beta\leqslant\widetilde{SN}^{\beta}_{x}(z)$ or
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)$. By Lemma 3.13 (2),
$\beta\leqslant(\widetilde{SN}^{\beta}_{x}\cup\widetilde{SN}^{\beta}_{y})(z)$,
i.e.,
$z\in\overline{\widetilde{SN}^{\beta}_{x}\cup\widetilde{SN}^{\beta}_{y}}^{\beta}$.
Then,
$\overline{SN}^{\beta}_{x}\sqcup\overline{SN}^{\beta}_{y}\sqsubset\overline{\widetilde{SN}^{\beta}_{x}\cup\widetilde{SN}^{\beta}_{y}}^{\beta}$.

(2) By Lemma 3.13 (1), $\beta\leqslant\widetilde{SN}^{\beta}_{x}(z)$ and
$\beta\leqslant\widetilde{SN}^{\beta}_{y}(z)$ if and only if
$\beta\leqslant(\widetilde{SN}^{\beta}_{x}\cap\widetilde{SN}^{\beta}_{y})(z)$.
Then
$\overline{\widetilde{SN}^{\beta}_{x}\cap\widetilde{SN}^{\beta}_{y}}^{\beta}=\overline{SN}^{\beta}_{x}\sqcap\overline{SN}^{\beta}_{y}$.

(3) For each $z\in U$, if
$z\in\sqcup_{v\in\Lambda}\overline{SN}_{x_{v}}^{\beta}$, then there exists at
least one $v^{\prime}\in\Lambda$ such that
$z\in\overline{SN}_{x_{v^{\prime}}}^{\beta}$, i.e.,
$\beta\leqslant(\widetilde{SN}_{x_{v^{\prime}}}^{\beta})(z)$. By Lemma 3.13
(2), $\beta\leqslant(\bigcup_{v\in\Lambda}\widetilde{SN}_{x_{v}}^{\beta})(z)$,
i.e.,
$z\in\overline{\bigcup_{v\in\Lambda}\widetilde{SN}_{x_{v}}^{\beta}}^{\beta}$.
Then,
$\sqcup_{v\in\Lambda}\overline{SN}_{x_{v}}^{\beta}\sqsubset\overline{\bigcup_{v\in\Lambda}\widetilde{SN}_{x_{v}}^{\beta}}^{\beta}$.

(4) For each $z\in U$, $z\in\sqcap_{v\in\Lambda}\overline{SN}_{x_{v}}^{\beta}$
if and only if $z\in\overline{SN}_{x_{v}}^{\beta}$ for all $v\in\Lambda$,
i.e., $\beta\leqslant(\widetilde{SN}_{x_{v}}^{\beta})(z)$ holds for all
$v\in\Lambda$.

By Lemma 3.13 (1),
$\beta\leqslant(\bigcap_{v\in\Lambda}\widetilde{SN}_{x_{v}}^{\beta})(z)$ if
and only if $\beta\leqslant(\widetilde{SN}_{x_{v}}^{\beta})(z)$ for all
$v\in\Lambda$, i.e.,
$\sqcap_{v\in\Lambda}\overline{SN}_{x_{v}}^{\beta}=\overline{\bigcap_{v\in\Lambda}\widetilde{SN}_{x_{v}}^{\beta}}^{\beta}$.$\blacksquare$

######  Example 3.24.

Let $\beta=[0.5,0.6]$. If $\widetilde{SN}_{x}^{\beta}(z)=[0.3,0.6]$ and
$\widetilde{SN}_{y}^{\beta}(z)=[0.5,0.55]$, then
$(\widetilde{SN}_{x}^{\beta}\cup\widetilde{SN}_{y}^{\beta})(z)=[0.3,0.6]\vee[0.5,0.55]=[0.5,0.6]$.
$z\not\in\overline{SN}^{\beta}_{x}$ and $z\not\in\overline{SN}^{\beta}_{y}$,
however,
$z\in\overline{\widetilde{SN}^{\beta}_{x}\cup\widetilde{SN}^{\beta}_{y}}^{\beta}$.
Hence, the (1) and (3) of Theorem 3.23 cannot take the equal sign.

##  4 Four kinds of interval-valued fuzzy soft $\beta$-coverings based fuzzy
rough sets and their relationships

###  4.1 The 1st kind of interval-valued fuzzy soft $\beta$-coverings based
fuzzy rough sets

######  Definition 4.25.

Let $\beta$ be an interval-valued number, $X\in IF(U)$, $x,y\in U$. Let

$\left\\{\begin{array}[]{l}\widetilde{SA}_{-1}^{\beta}(X)(x)=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee X(y)\\},\\\
\widetilde{SA}_{+1}^{\beta}(X)(x)=\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge X(y)\\}.\end{array}\right.$

$\widetilde{SA}_{-1}^{\beta}(X)$ and $\widetilde{SA}_{+1}^{\beta}(X)$ are the
lower approximation and upper approximation of the interval-valued fuzzy set
$X$ in $(U,F,A)_{\beta}$, respectively.

If $\widetilde{SA}_{-1}^{\beta}(X)\neq\widetilde{SA}_{+1}^{\beta}(X)$, then
$X$ is called the 1st type of interval-valued fuzzy soft $\beta$-covering
based interval-valued fuzzy rough sets; otherwise $X$ is called interval-
valued fuzzy definable.

######  Theorem 4.26.

Let $\beta$ be an interval-valued number, $X,Y\in IF(U)$, $x,y\in U$, the
following statements hold.

(1) $\widetilde{SA}_{-1}^{\beta}(I^{U})=I^{U}$,
$\widetilde{SA}_{+1}^{\beta}(I^{\emptyset})=I^{\emptyset}$.

(2) $\widetilde{SA}_{-1}^{\beta}(X^{c})=(\widetilde{SA}_{+1}^{\beta}(X))^{c}$,
$\widetilde{SA}_{+1}^{\beta}(X^{c})=(\widetilde{SA}_{-1}^{\beta}(X))^{c}$.

(3) $\widetilde{SA}_{-1}^{\beta}(X\cap
Y)=\widetilde{SA}_{-1}^{\beta}(X)\cap\widetilde{SA}_{-1}^{\beta}(Y)$,
$\widetilde{SA}_{+1}^{\beta}(X\cup
Y)=\widetilde{SA}_{+1}^{\beta}(X)\cup\widetilde{SA}_{+1}^{\beta}(Y)$.

(4) If $X\subset Y$, then
$\widetilde{SA}_{-1}^{\beta}(X)\subset\widetilde{SA}_{-1}^{\beta}(Y)$,
$\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(Y)$.

(5)
$\widetilde{SA}_{-1}^{\beta}(X)\cup\widetilde{SA}_{-1}^{\beta}(Y)\subset\widetilde{SA}_{-1}^{\beta}(X\cup
Y)$, $\widetilde{SA}_{+1}^{\beta}(X\cap
Y)\subset\widetilde{SA}_{+1}^{\beta}(X)\cap\widetilde{SA}_{+1}^{\beta}(Y)$.

(6) If $(\widetilde{SN}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant\widetilde{SN}_{x}^{\beta}(x)$ for all $x\in U$, then
$\widetilde{SA}_{-1}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+1}^{\beta}(X)$.

(7) If $(\widetilde{SN}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant\widetilde{SN}_{x}^{\beta}(x)$ for all $x\in U$, then
$\widetilde{SA}_{-1}^{\beta}(\widetilde{SA}_{-1}^{\beta}(X))\subset\widetilde{SA}_{-1}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(\widetilde{SA}_{+1}^{\beta}(X))$.

(8) If $X\subset Y$,
$\widetilde{SA}_{-1}^{\beta}(X)\cup\widetilde{SA}_{-1}^{\beta}(Y)=\widetilde{SA}_{-1}^{\beta}(X\cup
Y)$, $\widetilde{SA}_{+1}^{\beta}(X\cap
Y)=\widetilde{SA}_{+1}^{\beta}(X)\cap\widetilde{SA}_{+1}^{\beta}(Y)$.

Proof (1) For all $x,y\in U$, since $I^{U}(y)=[1,1]$, then
$\widetilde{SA}_{-1}^{\beta}(I^{U})(x)=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
I^{U}(y)\\}=\bigwedge\limits_{y\in U}\\{[1,1]\\}=[1,1]=I^{U}(x)$, i.e.,
$\widetilde{SA}_{-1}^{\beta}(I^{U})=I^{U}$.

For all $x,y\in U$, since $I^{\emptyset}(y)=[0,0]$, then
$\widetilde{SA}_{+1}^{\beta}(I^{\emptyset})(x)=\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge
I^{\emptyset}(y)\\}=\bigvee\limits_{y\in
U}\\{[0,0]\\}=[0,0]=I^{\emptyset}(x)$, i.e.,
$\widetilde{SA}_{+1}^{\beta}(I^{\emptyset})=I^{\emptyset}$.

(2) We prove the case of $|U|=2$ at first. Suppose $U=\\{y_{1},y_{2}\\}$.

By the results of Theorem 2.9 (4) and (5),
$\widetilde{SA}_{-1}^{\beta}(X^{c})(y_{1})=\bigwedge\limits_{y_{i}\in
U}\\{(\widetilde{SN}_{y_{1}}^{\beta})^{c}(y_{i})\vee
X^{c}(y_{i})\\}=[(\widetilde{SN}_{y_{1}}^{\beta})^{c}(y_{1})\vee
X^{c}(y_{1})]\wedge[(\widetilde{SN}_{y_{1}}^{\beta})^{c}(y_{2})\vee
X^{c}(y_{2})]=[\widetilde{SN}_{y_{1}}^{\beta}(y_{1})\wedge
X(y_{1})]^{c}\wedge[\widetilde{SN}_{y_{1}}^{\beta}(y_{2})\wedge
X(y_{2})]^{c}=\\{[\widetilde{SN}_{y_{1}}^{\beta}(y_{1})\wedge
X(y_{1})]\vee[\widetilde{SN}_{y_{1}}^{\beta}(y_{2})\wedge
X(y_{2})]\\}^{c}=(\widetilde{SA}_{+1}^{\beta}(X))^{c}(y_{1})$.

$\widetilde{SA}_{-1}^{\beta}(X^{c})(y_{2})=(\widetilde{SA}_{+1}^{\beta}(X))^{c}(y_{2})$
can be proved as above.

To sum up,
$\widetilde{SA}_{-1}^{\beta}(X^{c})=(\widetilde{SA}_{+1}^{\beta}(X))^{c}$ hold
for the case of $|U|=2$. Other cases of $|U|<\infty$ can be proved by using
the mathematical induction.

$\widetilde{SA}_{+1}^{\beta}(X^{c})(y_{1})=\bigvee\limits_{y_{i}\in
U}\\{\widetilde{SN}_{y_{1}}^{\beta}(y_{i})\wedge
X^{c}(y_{i})\\}=[\widetilde{SN}_{y_{1}}^{\beta}(y_{1})\wedge
X^{c}(y_{1})]\vee[\widetilde{SN}_{y_{1}}^{\beta}(y_{2})\wedge
X^{c}(y_{2})]=\\{[(\widetilde{SN}_{y_{1}}^{\beta})^{c}(y_{1})\vee
X(y_{1})]\wedge[(\widetilde{SN}_{y_{1}}^{\beta})^{c}(y_{2})\vee
X(y_{2})]\\}^{c}=(\widetilde{SA}_{-1}^{\beta}(X))^{c}(y_{1})$.

$\widetilde{SA}_{+1}^{\beta}(X^{c})(y_{2})=(\widetilde{SA}_{-1}^{\beta}(X))^{c}(y_{2})$
can be proved as above.

To sum up,
$\widetilde{SA}_{+1}^{\beta}(X^{c})=(\widetilde{SA}_{-1}^{\beta}(X))^{c}$ hold
for the case of $|U|=2$. Other cases of $|U|<\infty$ can be proved by using
the mathematical induction.

(3) For each $x\in U$, by Theorem 2.9 (2), $\widetilde{SA}_{-1}^{\beta}(X\cap
Y)(x)=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee(X\cap
Y)(y)\\}=\bigwedge\limits_{y\in U}\\{[(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)]\wedge[(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
Y(y)]\\}=\\{\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)\\}\\}\wedge\\{\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
Y(y)\\}\\}=\widetilde{SA}_{-1}^{\beta}(X)(x)\wedge\widetilde{SA}_{-1}^{\beta}(Y)(x)$.
Then, $\widetilde{SA}_{-1}^{\beta}(X\cap
Y)=\widetilde{SA}_{-1}^{\beta}(X)\cap\widetilde{SA}_{-1}^{\beta}(Y)$.

For each $x\in U$, by Theorem 2.9 (2), $\widetilde{SA}_{+1}^{\beta}(X\cup
Y)(x)=\bigvee\limits_{y\in U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge(X\cup
Y)(y)\\}=\bigvee\limits_{y\in U}\\{[\widetilde{SN}_{x}^{\beta}(y)\wedge
X(y)]\vee[\widetilde{SN}_{x}^{\beta}(y)\wedge Y(y)]\\}=\\{\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge X(y)\\}\\}\vee\\{\bigcup\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge
Y(y)\\}\\}=\widetilde{SA}_{+1}^{\beta}(X)(x)\vee\widetilde{SA}_{+1}^{\beta}(Y)(x)$.
Then, $\widetilde{SA}_{+1}^{\beta}(X\cup
Y)=\widetilde{SA}_{+1}^{\beta}(X)\cup\widetilde{SA}_{+1}^{\beta}(Y)$.

(4) Since $X\subset Y$, thus $(\widetilde{SN}_{x}^{\beta})^{c}\cup
X\subset(\widetilde{SN}_{x}^{\beta})^{c}\cup Y$, then
$(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)\leqslant(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee Y(y)$ for all $x,y\in U$,
it implies that
$\widetilde{SA}_{-1}^{\beta}(X)(x)\leqslant\widetilde{SA}_{-1}^{\beta}(Y)(x)$
for all $x\in U$, i.e.,
$\widetilde{SA}_{-1}^{\beta}(X)\subset\widetilde{SA}_{-1}^{\beta}(Y)$.

Since $X\subset Y$, thus $\widetilde{SN}_{x}^{\beta}\cap
X\subset\widetilde{SN}_{x}^{\beta}\cap Y$, then
$\widetilde{SN}_{x}^{\beta}(y)\wedge
X(y)\leqslant\widetilde{SN}_{x}^{\beta}(y)\wedge Y(y)$ for all $x,y\in U$, it
implies that
$\widetilde{SA}_{+1}^{\beta}(X)(x)\leqslant\widetilde{SA}_{+1}^{\beta}(Y)(x)$
for all $x\in U$, i.e.,
$\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(Y)$.

(5) $X\subset X\cup Y$ and $Y\subset X\cup Y$. By the results of (4),
$\widetilde{SA}_{-1}^{\beta}(X)\subset\widetilde{SA}_{-1}^{\beta}(X\cup Y)$
and $\widetilde{SA}_{-1}^{\beta}(Y)\subset\widetilde{SA}_{-1}^{\beta}(X\cup
Y)$. Then
$\widetilde{SA}_{-1}^{\beta}(X)\cup\widetilde{SA}_{-1}^{\beta}(Y)\subset\widetilde{SA}_{-1}^{\beta}(X\cup
Y)$.

$X\cap Y\subset X$ and $X\cap Y\subset Y$. By the results of (4),
$\widetilde{SA}_{+1}^{\beta}(X\cap Y)\subset\widetilde{SA}_{+1}^{\beta}(X)$
and $\widetilde{SA}_{+1}^{\beta}(X\cap
Y)\subset\widetilde{SA}_{+1}^{\beta}(Y)$. Then
$\widetilde{SA}_{+1}^{\beta}(X\cap
Y)\subset\widetilde{SA}_{+1}^{\beta}(X)\cap\widetilde{SA}_{+1}^{\beta}(Y)$.

(6) Since $(\widetilde{SN}_{x}^{\beta})^{c}(x)\leqslant X(x)$ for all $x\in
U$, then $(\widetilde{SN}_{x}^{\beta})^{c}(x)\vee X(x)\leqslant X(x)\vee
X(x)=X(x)$, then $\widetilde{SA}_{-1}^{\beta}(X)(x)=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)\\}\leqslant(\widetilde{SN}_{x}^{\beta})^{c}(x)\vee X(x)\leqslant X(x)$,
i.e., $\widetilde{SA}_{-1}^{\beta}(X)\subset X$.

Since $X(x)\leqslant\widetilde{SN}_{x}^{\beta}(x)$ for all $x\in U$, then
$X(x)=X(x)\wedge X(x)\leqslant X(x)\wedge(\widetilde{SN}_{x}^{\beta})(x)$,
then $X(x)\leqslant
X(x)\wedge\widetilde{SN}_{x}^{\beta}(x)\leqslant\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge
X(y)\\}=\widetilde{SA}_{+1}^{\beta}(X)(x)$, i.e.,
$X\subset\widetilde{SA}_{+1}^{\beta}(X)$.

Hence, $\widetilde{SA}_{-1}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+1}^{\beta}(X)$.

(7) $(\widetilde{SN}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant\widetilde{SN}_{x}^{\beta}(x)$ for all $x\in U$, by the results
of (6), we have $\widetilde{SA}_{-1}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+1}^{\beta}(X)$. By the results of (4), we have
$\widetilde{SA}_{-1}^{\beta}(\widetilde{SA}_{-1}^{\beta}(X))\subset\widetilde{SA}_{-1}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(\widetilde{SA}_{+1}^{\beta}(X))$.

(8) If $X\subset Y$, then $X\cup Y\subset Y\cup Y=Y$. By the results of (4),
$\widetilde{SA}_{-1}^{\beta}(X\cup
Y)\subset\widetilde{SA}_{-1}^{\beta}(Y)\subset\widetilde{SA}_{-1}^{\beta}(X)\cup\widetilde{SA}_{-1}^{\beta}(Y)$.

By the results of (5),
$\widetilde{SA}_{-1}^{\beta}(X)\cup\widetilde{SA}_{-1}^{\beta}(Y)\subset\widetilde{SA}_{-1}^{\beta}(X\cup
Y)$.

Then,
$\widetilde{SA}_{-1}^{\beta}(X)\cup\widetilde{SA}_{-1}^{\beta}(Y)=\widetilde{SA}_{-1}^{\beta}(X\cup
Y)$.

If $X\subset Y$, then $X=X\cap X\subset X\cap Y$. By the results of (4),
$\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(X\cap Y)$.
Then, we have
$\widetilde{SA}_{+1}^{\beta}(X)\cap\widetilde{SA}_{+1}^{\beta}(Y)\subset\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(X\cap
Y)$.

By the results of (5), $\widetilde{SA}_{+1}^{\beta}(X\cap
Y)\subset\widetilde{SA}_{+1}^{\beta}(X)\cap\widetilde{SA}_{+1}^{\beta}(Y)$.

Hence, $\widetilde{SA}_{+1}^{\beta}(X\cap
Y)=\widetilde{SA}_{+1}^{\beta}(X)\cap\widetilde{SA}_{+1}^{\beta}(Y)$.$\blacksquare$

######  Definition 4.27.

Let $\beta$ be an interval-valued number and $X\sqsubset U$ be an object set,
$x,y\in U$. $\overline{SA}_{-1}^{\beta}(X)=\\{x\in
U:\overline{SN}^{\beta}_{x}\sqsubset X\\}$ and
$\overline{SA}_{+1}^{\beta}(X)=\\{x\in U:\overline{SN}^{\beta}_{x}\sqcap
X\neq\emptyset\\}$ are the lower approximation and upper approximation of the
object set $X$ in $(U,F,A)_{\beta}$, respectively.

If $\overline{SA}_{-1}^{\beta}(X)\neq\overline{SA}_{+1}^{\beta}(X)$, then $X$
is called the 1st type of soft $\beta$-covering based rough sets. Otherwise,
$X$ is called definable.

######  Theorem 4.28.

Let $\beta$ be an interval-valued number and $X,Y\sqsubset U$, the the
following statements hold.

(1) $\overline{SA}_{-1}^{\beta}(\emptyset)=\emptyset$,
$\overline{SA}_{-1}^{\beta}(U)=U$.

(2) $\overline{SA}_{+1}^{\beta}(\emptyset)=\emptyset$,
$\overline{SA}_{+1}^{\beta}(U)=U$.

(3) If $X\sqsubset Y$, then
$\overline{SA}_{-1}^{\beta}(X)\sqsubset\overline{SA}_{-1}^{\beta}(Y)$ and
$\overline{SA}_{+1}^{\beta}(X)\sqsubset\overline{SA}_{+1}^{\beta}(Y)$

(4)
$\overline{SA}_{-1}^{\beta}(X)\sqcup\overline{SA}_{-1}^{\beta}(Y)\sqsubset\overline{SA}_{-1}^{\beta}(X\sqcup
Y)$, $\overline{SA}_{+1}^{\beta}(X\sqcap
Y)\sqsubset\overline{SA}_{+1}^{\beta}(X)\sqcap\overline{SA}_{+1}^{\beta}(Y)$.

(5)
$\overline{SA}_{-1}^{\beta}(X)\sqcap\overline{SA}_{-1}^{\beta}(Y)=\overline{SA}_{-1}^{\beta}(X\sqcap
Y)$, $\overline{SA}_{+1}^{\beta}(X\sqcup
Y)=\overline{SA}_{+1}^{\beta}(X)\sqcup\overline{SA}_{+1}^{\beta}(Y)$.

(6) $\overline{SA}_{-1}^{\beta}(X^{c})=(\overline{SA}_{+1}^{\beta}(X))^{c}$,
$\overline{SA}_{+1}^{\beta}(X^{c})=(\overline{SA}_{-1}^{\beta}(X))^{c}$.

(7) $\overline{SA}_{-1}^{\beta}(X)\sqsubset
X\sqsubset\overline{SA}_{+1}^{\beta}(X)$.

Proof (1) and (2) are obvious.

(3) If $x\in\overline{SA}_{-1}^{\beta}(X)$, then
$\overline{SN}^{\beta}_{x}\sqsubset X\sqsubset Y$, i.e.,
$x\in\overline{SA}_{-1}^{\beta}(Y)$. Then
$\overline{SA}_{-1}^{\beta}(X)\sqsubset\overline{SA}_{-1}^{\beta}(Y)$.

If $x\in\overline{SA}_{+1}^{\beta}(X)$, then
$\emptyset\neq\overline{SN}^{\beta}_{x}\sqcap
X\sqsubset\overline{SN}^{\beta}_{x}\sqcap Y$, i.e.,
$x\in\overline{SA}_{+1}^{\beta}(Y)$. Then
$\overline{SA}_{+1}^{\beta}(X)\sqsubset\overline{SA}_{+1}^{\beta}(Y)$.

(4) Since $X\sqsubset X\sqcup Y$ and $Y\sqsubset X\sqcup Y$, by the results of
(3), $\overline{SA}_{-1}^{\beta}(X)\sqsubset\overline{SA}_{-1}^{\beta}(X\sqcup
Y)$ and
$\overline{SA}_{-1}^{\beta}(Y)\sqsubset\overline{SA}_{-1}^{\beta}(X\sqcup Y)$.
Then
$\overline{SA}_{-1}^{\beta}(X)\sqcup\overline{SA}_{-1}^{\beta}(Y)\sqsubset\overline{SA}_{-1}^{\beta}(X\sqcup
Y)$.

Since $X\sqcap Y\sqsubset X$ and $X\sqcap Y\sqsubset Y$, by the results of
(3), $\overline{SA}_{+1}^{\beta}(X\sqcap
Y)\sqsubset\overline{SA}_{+1}^{\beta}(X)$ and
$\overline{SA}_{+1}^{\beta}(X\sqcap Y)\sqsubset\overline{SA}_{+1}^{\beta}(Y)$.
Then $\overline{SA}_{+1}^{\beta}(X\sqcap
Y)\sqsubset\overline{SA}_{+1}^{\beta}(X)\sqcap\overline{SA}_{+1}^{\beta}(Y)$.

(5) If $x\in\overline{SA}_{-1}^{\beta}(X)\sqcap\overline{SA}_{-1}^{\beta}(Y)$,
i.e., $x\in\overline{SA}_{-1}^{\beta}(X)$ and
$x\in\overline{SA}_{-1}^{\beta}(Y)$, then $\overline{SN}^{\beta}_{x}\sqsubset
X$ and $\overline{SN}^{\beta}_{x}\sqsubset Y$. It means that
$\overline{SN}^{\beta}_{x}\sqsubset X\sqcap Y$, i.e.,
$x\in\overline{SA}_{-1}^{\beta}(X\sqcap Y)$. Then
$\overline{SA}_{-1}^{\beta}(X)\sqcap\overline{SA}_{-1}^{\beta}(Y)\sqsubset\overline{SA}_{-1}^{\beta}(X\sqcap
Y)$.

On the other hand, $X\sqcap Y\sqsubset X$ and $X\sqcap Y\sqsubset Y$, by the
results of (3), $\overline{SA}_{-1}^{\beta}(X\sqcap
Y)\sqsubset\overline{SA}_{-1}^{\beta}(X)\sqcap\overline{SA}_{-1}^{\beta}(Y)$.

Hence, $\overline{SA}_{-1}^{\beta}(X\sqcap
Y)=\overline{SA}_{-1}^{\beta}(X)\sqcap\overline{SA}_{-1}^{\beta}(Y)$.

If $x\in\overline{SA}_{+1}^{\beta}(X\sqcup Y)$, i.e.,
$\overline{SN}^{\beta}_{x}\sqcap(X\sqcup Y)\neq\emptyset$, then there is $y\in
U$ such that $y\in\overline{SN}^{\beta}_{x}$ and $y\in X\sqcup Y$, it means
that at least one of $y\in X$ and $y\in Y$ holds, then we can obtain that at
least one of $\overline{SN}^{\beta}_{x}\sqcap X\neq\emptyset$ and
$\overline{SN}^{\beta}_{x}\sqcap Y\neq\emptyset$ holds, i.e.,
$x\in\overline{SA}_{+1}^{\beta}(X)\sqcup\overline{SA}_{+1}^{\beta}(Y)$. Then
$\overline{SA}_{+1}^{\beta}(X\sqcup
Y)\sqsubset\overline{SA}_{+1}^{\beta}(X)\sqcup\overline{SA}_{+1}^{\beta}(Y)$.

On the other hand, $X\sqsubset X\sqcup Y$ and $Y\sqsubset X\sqcup Y$, by the
results of (3),
$\overline{SA}_{+1}^{\beta}(X)\sqcup\overline{SA}_{+1}^{\beta}(Y)\sqsubset\overline{SA}_{+1}^{\beta}(X\sqcup
Y)$.

Hence,
$\overline{SA}_{+1}^{\beta}(X)\sqcup\overline{SA}_{+1}^{\beta}(Y)=\overline{SA}_{+1}^{\beta}(X\sqcup
Y)$.

(6) If $x\in\overline{SA}_{-1}^{\beta}(X^{c})$, i.e.,
$\overline{SN}^{\beta}_{x}\sqsubset X^{c}$, then
$\overline{SN}^{\beta}_{x}\sqcap X=\emptyset$, i.e.,
$x\not\in\overline{SA}_{+1}^{\beta}(X)$, thus
$x\in(\overline{SA}_{+1}^{\beta}(X))^{c}$. Then
$\overline{SA}_{-1}^{\beta}(X^{c})\sqsubset(\overline{SA}_{+1}^{\beta}(X))^{c}$.

On the other hand, $x\in(\overline{SA}_{+1}^{\beta}(X))^{c}$, i.e.,
$x\not\in\overline{SA}_{+1}^{\beta}(X)$, thus $\overline{SN}^{\beta}_{x}\sqcap
X=\emptyset$. Then $\overline{SN}^{\beta}_{x}\sqsubset X^{c}$, i.e.,
$x\in\overline{SA}_{-1}^{\beta}(X^{c})$. Then
$(\overline{SA}_{+1}^{\beta}(X))^{c}\sqsubset\overline{SA}_{-1}^{\beta}(X^{c})$.

Hence,
$\overline{SA}_{-1}^{\beta}(X^{c})=(\overline{SA}_{+1}^{\beta}(X))^{c}$.

If $x\in\overline{SA}_{+1}^{\beta}(X^{c})$, i.e.,
$\overline{SN}^{\beta}_{x}\sqcap X^{c}\neq\emptyset$, then
$\overline{SN}^{\beta}_{x}\not\sqsubset X$, i.e.,
$x\not\in\overline{SA}_{-1}^{\beta}(X)$ and
$x\in(\overline{SA}_{-1}^{\beta}(X))^{c}$. Then
$\overline{SA}_{+1}^{\beta}(X^{c})\sqsubset(\overline{SA}_{-1}^{\beta}(X))^{c}$.

On the other hand, if $x\in(\overline{SA}_{-1}^{\beta}(X))^{c}$, i.e.,
$x\not\in\overline{SA}_{-1}^{\beta}(X)$ and
$\overline{SN}^{\beta}_{x}\not\sqsubset X$. Thus
$\overline{SN}^{\beta}_{x}\sqcap X^{c}\neq\emptyset$, i.e.,
$x\in\overline{SA}_{+1}^{\beta}(X^{c})$. Then
$(\overline{SA}_{-1}^{\beta}(X))^{c}\sqsubset\overline{SA}_{+1}^{\beta}(X^{c})$.

Hence,
$\overline{SA}_{+1}^{\beta}(X^{c})=(\overline{SA}_{-1}^{\beta}(X))^{c}$.

(7) By Theorem 3.19, $x\in\overline{SN}^{\beta}_{x}$ for all $x\in U$.

For a random $y\not\in X$ ($y\in X^{c}$), since
$y\in\overline{SN}^{\beta}_{y}$, then $\overline{SN}^{\beta}_{y}\not\sqsubset
X$, i.e., $y\not\in\overline{SA}_{-1}^{\beta}(X)$
($y\in(\overline{SA}_{-1}^{\beta}(X))^{c}$), then
$X^{c}\sqsubset(\overline{SA}_{-1}^{\beta}(X))^{c}$, i.e.,
$\overline{SA}_{-1}^{\beta}(X)\sqsubset X$.

For all $x\in X$, since $x\in\overline{SN}^{\beta}_{x}$, then
$\emptyset\neq\\{x\\}\sqsubset\overline{SN}^{\beta}_{x}\sqcap X$, i.e.,
$x\in\overline{SA}_{+1}^{\beta}(X)$. Then
$X\sqsubset\overline{SA}_{+1}^{\beta}(X)$.

To sum up, $\overline{SA}_{-1}^{\beta}(X)\sqsubset
X\sqsubset\overline{SA}_{+1}^{\beta}(X)$.$\blacksquare$

######  Theorem 4.29.

Let $\beta$ be an interval-valued number, $X\sqsubset U$, $A\sqsubset E$,
$B\sqsubset E$. In $T_{1}=(U,F,A)_{\beta}$ and $T_{2}=(U,F,B)_{\beta}$, if
$\overline{SN}^{1\beta}_{x}=\overline{SN}^{2\beta}_{x}$ for all $x\in U$, then
$\overline{SA}_{-1}^{1\beta}(X)=\overline{SA}_{-1}^{2\beta}(X)$ and
$\overline{SA}_{+1}^{1\beta}(X)=\overline{SA}_{+1}^{2\beta}(X)$.

Proof (1) Since $\overline{SN}^{1\beta}_{x}=\overline{SN}^{2\beta}_{x}$ for
all $x\in U$, then $\overline{SN}^{1\beta}_{x}\sqsubset X$ if and only if
$\overline{SN}^{2\beta}_{x}\sqsubset X$, then
$\overline{SA}_{-1}^{1\beta}(X)=\overline{SA}_{-1}^{2\beta}(X)$.

$\overline{SN}^{1\beta}_{x}\sqcap X\neq\emptyset$ if and only if
$\overline{SN}^{2\beta}_{x}\sqcap X\neq\emptyset$, then
$\overline{SA}_{+1}^{1\beta}(X)=\overline{SA}_{+1}^{2\beta}(X)$.$\blacksquare$

###  4.2 The 2nd kind of interval-valued fuzzy soft $\beta$-coverings based
fuzzy rough sets

######  Definition 4.30.

Let $\beta$ be an interval-valued number, and $U$ and $E$ be the sets of
objects and parameters, respectively. $\widetilde{SM}^{\beta}_{x}$ is called
the interval-valued fuzzy soft complementary $\beta$-neighborhood of $x$ in
$(U,F,A)_{\beta}$, where
$\widetilde{SM}^{\beta}_{x}(y)=\widetilde{SN}^{\beta}_{y}(x)$.

######  Definition 4.31.

Let $\beta$ be an interval-valued number, $X\in IF(U)$, $x,y\in U$.
$\widetilde{SA}_{-2}^{\beta}(X)$ and $\widetilde{SA}_{+2}^{\beta}(X)$ are
obtained by

$\left\\{\begin{array}[]{l}\widetilde{SA}_{-2}^{\beta}(X)(x)=\bigwedge\limits_{y\in
U}\\{(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee X(y)\\},\\\
\widetilde{SA}_{+2}^{\beta}(X)(x)=\bigvee\limits_{y\in
U}\\{\widetilde{SM}_{x}^{\beta}(y)\wedge X(y)\\}.\end{array}\right.$

If $\widetilde{SA}_{-2}^{\beta}(X)\neq\widetilde{SA}_{+2}^{\beta}(X)$, then
$X$ is called the 2nd type of interval-valued fuzzy soft $\beta$-covering
based interval-valued fuzzy rough sets; otherwise $X$ is called interval-
valued fuzzy definable.

######  Theorem 4.32.

Let $\beta$ be an interval-valued number, $X,Y\in IF(U)$, $x,y\in U$, the
following statements hold.

(1) $\widetilde{SA}_{-2}^{\beta}(I^{U})=I^{U}$,
$\widetilde{SA}_{+2}^{\beta}(I^{\emptyset})=I^{\emptyset}$.

(2) $\widetilde{SA}_{-2}^{\beta}(X^{c})=(\widetilde{SA}_{+2}^{\beta}(X))^{c}$,
$\widetilde{SA}_{+2}^{\beta}(X^{c})=(\widetilde{SA}_{-2}^{\beta}(X))^{c}$.

(3) $\widetilde{SA}_{-2}^{\beta}(X\cap
Y)=\widetilde{SA}_{-2}^{\beta}(X)\cap\widetilde{SA}_{-2}^{\beta}(Y)$,
$\widetilde{SA}_{+2}^{\beta}(X\cup
Y)=\widetilde{SA}_{+2}^{\beta}(X)\cup\widetilde{SA}_{+2}^{\beta}(Y)$.

(4) If $X\subset Y$, then
$\widetilde{SA}_{-2}^{\beta}(X)\subset\widetilde{SA}_{-2}^{\beta}(Y)$,
$\widetilde{SA}_{+2}^{\beta}(X)\subset\widetilde{SA}_{+2}^{\beta}(Y)$.

(5)
$\widetilde{SA}_{-2}^{\beta}(X)\cup\widetilde{SA}_{-2}^{\beta}(Y)\subset\widetilde{SA}_{-2}^{\beta}(X\cup
Y)$, $\widetilde{SA}_{+2}^{\beta}(X\cap
Y)\subset\widetilde{SA}_{+2}^{\beta}(X)\cap\widetilde{SA}_{+2}^{\beta}(Y)$.

(6) If $(\widetilde{SM}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant\widetilde{SM}_{x}^{\beta}(x)$ for all $x\in U$, then
$\widetilde{SA}_{-2}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+2}^{\beta}(X)$.

(7) If $(\widetilde{SM}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant\widetilde{SM}_{x}^{\beta}(x)$ for all $x\in U$, then
$\widetilde{SA}_{-2}^{\beta}(\widetilde{SA}_{-2}^{\beta}(X))\subset\widetilde{SA}_{-2}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+2}^{\beta}(X)\subset\widetilde{SA}_{+2}^{\beta}(\widetilde{SA}_{+2}^{\beta}(X))$.

(8) If $X\subset Y$,
$\widetilde{SA}_{-2}^{\beta}(X)\cup\widetilde{SA}_{-2}^{\beta}(Y)=\widetilde{SA}_{-2}^{\beta}(X\cup
Y)$, $\widetilde{SA}_{+2}^{\beta}(X\cap
Y)=\widetilde{SA}_{+2}^{\beta}(X)\cap\widetilde{SA}_{+2}^{\beta}(Y)$.

Proof It can be proved similarly to Theorem 4.26 by using
$\widetilde{SM}^{\beta}_{x}$ and $(\widetilde{SM}_{x}^{\beta})^{c}$ to instead
of $\widetilde{SN}^{\beta}_{x}$ and $(\widetilde{SN}^{\beta}_{x})^{c}$ in
Theorem 4.26, respectively.$\blacksquare$

######  Definition 4.33.

Let $\beta$ be an interval-valued number and $X\sqsubset U$ be an object set,
$x,y\in U$. $\overline{SA}_{-2}^{\beta}(X)=\\{x\in
U:\overline{SM}^{\beta}_{x}\sqsubset X\\}$ and
$\overline{SA}_{+2}^{\beta}(X)=\\{x\in U:\overline{SM}^{\beta}_{x}\sqcap
X\neq\emptyset\\}$ are the lower approximation and upper approximation of the
object set $X$ in $(U,F,A)_{\beta}$, respectively.

If $\overline{SA}_{-2}^{\beta}(X)\neq\overline{SA}_{+2}^{\beta}(X)$, then $X$
is called the 2nd type of soft $\beta$-covering based rough sets. Otherwise,
$X$ is called definable.

###  4.3 The 3rd kind of interval-valued fuzzy soft $\beta$-coverings based
fuzzy rough sets

######  Definition 4.34.

Let $\beta$ be an interval-valued number, $X\in IF(U)$, $x,y\in U$.
$\widetilde{SA}_{-3}^{\beta}(X)$ and $\widetilde{SA}_{+3}^{\beta}(X)$ are
obtained by

$\left\\{\begin{array}[]{l}\widetilde{SA}_{-3}^{\beta}(X)(x)=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)\\},\\\ \widetilde{SA}_{+3}^{\beta}(X)(x)=\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)\\}.\end{array}\right.$

######  Theorem 4.35.

Let $\beta$ be an interval-valued number, $X,Y\in IF(U)$, $x,y\in U$, the
following statements hold.

(1) $\widetilde{SA}_{-3}^{\beta}(I^{U})=I^{U}$,
$\widetilde{SA}_{+3}^{\beta}(I^{\emptyset})=I^{\emptyset}$.

(2) $\widetilde{SA}_{-3}^{\beta}(X^{c})=(\widetilde{SA}_{+3}^{\beta}(X))^{c}$,
$\widetilde{SA}_{+3}^{\beta}(X^{c})=(\widetilde{SA}_{-3}^{\beta}(X))^{c}$.

(3) $\widetilde{SA}_{-3}^{\beta}(X\cap
Y)=\widetilde{SA}_{-3}^{\beta}(X)\cap\widetilde{SA}_{-3}^{\beta}(Y)$,
$\widetilde{SA}_{+3}^{\beta}(X\cup
Y)=\widetilde{SA}_{+3}^{\beta}(X)\cup\widetilde{SA}_{+3}^{\beta}(Y)$.

(4) If $X\subset Y$, then
$\widetilde{SA}_{-3}^{\beta}(X)\subset\widetilde{SA}_{-3}^{\beta}(Y)$,
$\widetilde{SA}_{+3}^{\beta}(X)\subset\widetilde{SA}_{+3}^{\beta}(Y)$.

(5)
$\widetilde{SA}_{-3}^{\beta}(X)\cup\widetilde{SA}_{-3}^{\beta}(Y)\subset\widetilde{SA}_{-3}^{\beta}(X\cup
Y)$, $\widetilde{SA}_{+3}^{\beta}(X\cap
Y)\subset\widetilde{SA}_{+3}^{\beta}(X)\cap\widetilde{SA}_{+3}^{\beta}(Y)$.

(6) If
$(\widetilde{SN}_{x}^{\beta}\cap\widetilde{SM}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant(\widetilde{SN}_{x}^{\beta}\cap\widetilde{SM}_{x}^{\beta})(x)$
for all $x\in U$, then $\widetilde{SA}_{-3}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+3}^{\beta}(X)$.

(7) If
$(\widetilde{SN}_{x}^{\beta}\cap\widetilde{SM}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant(\widetilde{SN}_{x}^{\beta}\cap\widetilde{SM}_{x}^{\beta})(x)$
for all $x\in U$, then
$\widetilde{SA}_{-3}^{\beta}(\widetilde{SA}_{-3}^{\beta}(X))\subset\widetilde{SA}_{-3}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+3}^{\beta}(X)\subset\widetilde{SA}_{+3}^{\beta}(\widetilde{SA}_{+3}^{\beta}(X))$.

(8) If $X\subset Y$,
$\widetilde{SA}_{-3}^{\beta}(X)\cup\widetilde{SA}_{-3}^{\beta}(Y)=\widetilde{SA}_{-3}^{\beta}(X\cup
Y)$, $\widetilde{SA}_{+3}^{\beta}(X\cap
Y)=\widetilde{SA}_{+3}^{\beta}(X)\cap\widetilde{SA}_{+3}^{\beta}(Y)$.

Proof It can be proved similarly to Theorem 4.26 by using
$\widetilde{SN}_{x}^{\beta}\cap\widetilde{SM}_{x}^{\beta}$ and
$(\widetilde{SN}_{x}^{\beta}\cap\widetilde{SM}_{x}^{\beta})^{c}$ to instead of
$\widetilde{SN}^{\beta}_{x}$ and $(\widetilde{SN}^{\beta}_{x})^{c}$ in Theorem
4.26, respectively.$\blacksquare$

######  Definition 4.36.

Let $\beta$ be an interval-valued number and $X\sqsubset U$ be an object set,
$x,y\in U$. $\overline{SA}_{-3}^{\beta}(X)=\\{x\in
U:\overline{SN}^{\beta}_{x}\sqsubset X\ or\ \overline{SM}^{\beta}_{x}\sqsubset
X\\}$ and $\overline{SA}_{+3}^{\beta}(X)=\\{x\in
U:\overline{SN}^{\beta}_{x}\sqcap X\neq\emptyset\ and\
\overline{SM}^{\beta}_{x}\sqcap X\neq\emptyset\\}$ are the lower approximation
and upper approximation of the object set $X$ in $(U,F,A)_{\beta}$,
respectively.

If $\overline{SA}_{-3}^{\beta}(X)\neq\overline{SA}_{+3}^{\beta}(X)$, then $X$
is called the 3rd type of soft $\beta$-covering based rough sets. Otherwise,
$X$ is called definable.

###  4.4 The 4th kind of interval-valued fuzzy soft $\beta$-coverings based
fuzzy rough sets

######  Definition 4.37.

Let $\beta$ be an interval-valued number, $X\in IF(U)$, $x,y\in U$.
$\widetilde{SA}_{-4}^{\beta}(X)$ and $\widetilde{SA}_{+4}^{\beta}(X)$ are
obtained by

$\left\\{\begin{array}[]{l}\widetilde{SA}_{-4}^{\beta}(X)(x)=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\wedge(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)\\},\\\ \widetilde{SA}_{+4}^{\beta}(X)(x)=\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\vee\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)\\}.\end{array}\right.$

######  Theorem 4.38.

Let $\beta$ be an interval-valued number, $X,Y\in IF(U)$, $x,y\in U$, the
following statements hold.

(1) $\widetilde{SA}_{-4}^{\beta}(I^{U})=I^{U}$,
$\widetilde{SA}_{+4}^{\beta}(I^{\emptyset})=I^{\emptyset}$.

(2) $\widetilde{SA}_{-4}^{\beta}(X^{c})=(\widetilde{SA}_{+4}^{\beta}(X))^{c}$,
$\widetilde{SA}_{+4}^{\beta}(X^{c})=(\widetilde{SA}_{-4}^{\beta}(X))^{c}$.

(3) $\widetilde{SA}_{-4}^{\beta}(X\cap
Y)=\widetilde{SA}_{-4}^{\beta}(X)\cap\widetilde{SA}_{-4}^{\beta}(Y)$,
$\widetilde{SA}_{+4}^{\beta}(X\cup
Y)=\widetilde{SA}_{+4}^{\beta}(X)\cup\widetilde{SA}_{+4}^{\beta}(Y)$.

(4) If $X\subset Y$, then
$\widetilde{SA}_{-4}^{\beta}(X)\subset\widetilde{SA}_{-4}^{\beta}(Y)$,
$\widetilde{SA}_{+4}^{\beta}(X)\subset\widetilde{SA}_{+4}^{\beta}(Y)$.

(5)
$\widetilde{SA}_{-4}^{\beta}(X)\cup\widetilde{SA}_{-4}^{\beta}(Y)\subset\widetilde{SA}_{-4}^{\beta}(X\cup
Y)$, $\widetilde{SA}_{+4}^{\beta}(X\cap
Y)\subset\widetilde{SA}_{+4}^{\beta}(X)\cap\widetilde{SA}_{+4}^{\beta}(Y)$.

(6) If
$(\widetilde{SN}_{x}^{\beta}\cup\widetilde{SM}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant(\widetilde{SN}_{x}^{\beta}\cup\widetilde{SM}_{x}^{\beta})(x)$
for all $x\in U$, then $\widetilde{SA}_{-4}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+4}^{\beta}(X)$.

(7) If
$(\widetilde{SN}_{x}^{\beta}\cup\widetilde{SM}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant(\widetilde{SN}_{x}^{\beta}\cup\widetilde{SM}_{x}^{\beta})(x)$
for all $x\in U$, then
$\widetilde{SA}_{-4}^{\beta}(\widetilde{SA}_{-4}^{\beta}(X))\subset\widetilde{SA}_{-4}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+4}^{\beta}(X)\subset\widetilde{SA}_{+4}^{\beta}(\widetilde{SA}_{+4}^{\beta}(X))$.

(8) If $X\subset Y$,
$\widetilde{SA}_{-4}^{\beta}(X)\cup\widetilde{SA}_{-4}^{\beta}(Y)=\widetilde{SA}_{-4}^{\beta}(X\cup
Y)$, $\widetilde{SA}_{+4}^{\beta}(X\cap
Y)=\widetilde{SA}_{+4}^{\beta}(X)\cap\widetilde{SA}_{+4}^{\beta}(Y)$.

Proof It can be proved similarly to Theorem 4.26 by using
$\widetilde{SN}_{x}^{\beta}\cup\widetilde{SM}_{x}^{\beta}$ and
$(\widetilde{SN}_{x}^{\beta}\cup\widetilde{SM}_{x}^{\beta})^{c}$ to instead of
$\widetilde{SN}^{\beta}_{x}$ and $(\widetilde{SN}^{\beta}_{x})^{c}$ in Theorem
4.26, respectively.$\blacksquare$

######  Definition 4.39.

Let $\beta$ be an interval-valued number and $X\sqsubset U$ be an object set,
$x,y\in U$. $\overline{SA}_{-4}^{\beta}(X)=\\{x\in
U:\overline{SN}^{\beta}_{x}\sqsubset X\ and\
\overline{SM}^{\beta}_{x}\sqsubset X\\}$ and
$\overline{SA}_{+4}^{\beta}(X)=\\{x\in U:\overline{SN}^{\beta}_{x}\sqcap
X\neq\emptyset\ or\ \overline{SM}^{\beta}_{x}\sqcap X\neq\emptyset\\}$ are the
lower approximation and upper approximation of the object set $X$ in
$(U,F,A)_{\beta}$, respectively.

If $\overline{SA}_{-4}^{\beta}(X)\neq\overline{SA}_{+4}^{\beta}(X)$, then $X$
is called the 4th type of soft $\beta$-covering based rough sets. Otherwise,
$X$ is called definable.

###  4.5 The relationships of four kinds of interval-valued fuzzy soft
$\beta$-coverings based fuzzy rough sets

######  Theorem 4.40.

Let $\beta$ be an interval-valued number, $X\in IF(U)$, the following
statements hold in $(U,F,A)_{\beta}$.

(1)
$\widetilde{SA}_{-3}^{\beta}(X)=\widetilde{SA}_{-1}^{\beta}(X)\cup\widetilde{SA}_{-2}^{\beta}(X)$.

(2)
$\widetilde{SA}_{+3}^{\beta}(X)=\widetilde{SA}_{+1}^{\beta}(X)\cap\widetilde{SA}_{+2}^{\beta}(X)$.

(3)
$\widetilde{SA}_{-4}^{\beta}(X)=\widetilde{SA}_{-1}^{\beta}(X)\cap\widetilde{SA}_{-2}^{\beta}(X)$.

(4)
$\widetilde{SA}_{+4}^{\beta}(X)=\widetilde{SA}_{+1}^{\beta}(X)\cup\widetilde{SA}_{+2}^{\beta}(X)$.

(5)
$\widetilde{SA}_{-4}^{\beta}(X)\subset\widetilde{SA}_{-1}^{\beta}(X)\subset\widetilde{SA}_{-3}^{\beta}(X)$.

(6)
$\widetilde{SA}_{-4}^{\beta}(X)\subset\widetilde{SA}_{-2}^{\beta}(X)\subset\widetilde{SA}_{-3}^{\beta}(X)$.

(7)
$\widetilde{SA}_{+3}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+4}^{\beta}(X)$.

(8)
$\widetilde{SA}_{+3}^{\beta}(X)\subset\widetilde{SA}_{+2}^{\beta}(X)\subset\widetilde{SA}_{+4}^{\beta}(X)$.

Proof (1) For all $x,y\in U$,
$\widetilde{SA}_{-1}^{\beta}(X)(x)\vee\widetilde{SA}_{-2}^{\beta}(X)(x)=(\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)\\})\vee(\bigwedge\limits_{y\in
U}\\{(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee X(y)\\})=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)\vee(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee X(y)\\}=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)\\}=\widetilde{SA}_{-3}^{\beta}(X)(x)$.

(2) For all $x,y\in U$,
$\widetilde{SA}_{+1}^{\beta}(X)(x)\wedge\widetilde{SA}_{+2}^{\beta}(X)(x)=(\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge X(y)\\})\wedge(\bigvee\limits_{y\in
U}\\{\widetilde{SM}_{x}^{\beta}(y)\wedge X(y)\\})=\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)\wedge X(y)\\}=\widetilde{SA}_{+3}^{\beta}(X)(x)$.

(3) For all $x,y\in U$,
$\widetilde{SA}_{-1}^{\beta}(X)(x)\wedge\widetilde{SA}_{-2}^{\beta}(X)(x)=(\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)\\})\wedge(\bigwedge\limits_{y\in
U}\\{(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee X(y)\\})=\bigwedge\limits_{y\in
U}\\{[(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)]\wedge[(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)]\\}=\bigwedge\limits_{y\in
U}\\{[(\widetilde{SM}_{x}^{\beta})^{c}(y)\wedge(\widetilde{SN}_{x}^{\beta})^{c}(y)]\vee
X(y)\\}=\widetilde{SA}_{-4}^{\beta}(X)(x)$.

(4) For all $x,y\in U$,
$\widetilde{SA}_{+1}^{\beta}(X)(x)\vee\widetilde{SA}_{+2}^{\beta}(X)(x)=(\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge X(y)\\})\vee(\bigvee\limits_{y\in
U}\\{\widetilde{SM}_{x}^{\beta}(y)\wedge X(y)\\})=\bigvee\limits_{y\in
U}\\{[\widetilde{SN}_{x}^{\beta}(y)\vee\widetilde{SM}_{x}^{\beta}(y)]\wedge
X(y)\\}=\widetilde{SA}_{+4}^{\beta}(X)(x)$.

(5) For all $y\in U$,
$(\widetilde{SN}_{x}^{\beta})^{c}(y)\wedge(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)\leqslant(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee
X(y)\leqslant(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)$.

(6) For all $y\in U$,
$(\widetilde{SN}_{x}^{\beta})^{c}(y)\wedge(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)\leqslant(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)\leqslant(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)$.

(7) For all $y\in U$,
$\widetilde{SN}_{x}^{\beta}(y)\wedge\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)\leqslant\widetilde{SN}_{x}^{\beta}(y)\wedge
X(y)\leqslant\widetilde{SN}_{x}^{\beta}(y)\vee\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)$.

(8) For all $y\in U$,
$\widetilde{SN}_{x}^{\beta}(y)\wedge\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)\leqslant\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)\leqslant\widetilde{SN}_{x}^{\beta}(y)\vee\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)$.$\blacksquare$

######  Theorem 4.41.

Let $\beta$ be an interval-valued number, $X\in IF(U)$, the following
statements hold in $(U,F,A)_{\beta}$.

(1)
$\overline{SA}_{-3}^{\beta}(X)=\overline{SA}_{-1}^{\beta}(X)\sqcup\overline{SA}_{-2}^{\beta}(X)$.

(2)
$\overline{SA}_{+3}^{\beta}(X)=\overline{SA}_{+1}^{\beta}(X)\sqcap\overline{SA}_{+2}^{\beta}(X)$.

(3)
$\overline{SA}_{-4}^{\beta}(X)=\overline{SA}_{-1}^{\beta}(X)\sqcap\overline{SA}_{-2}^{\beta}(X)$.

(4)
$\overline{SA}_{+4}^{\beta}(X)=\overline{SA}_{+1}^{\beta}(X)\sqcup\overline{SA}_{+2}^{\beta}(X)$.

(5)
$\overline{SA}_{-4}^{\beta}(X)\sqsubset\overline{SA}_{-1}^{\beta}(X)\sqsubset\overline{SA}_{-3}^{\beta}(X)$.

(6)
$\overline{SA}_{-4}^{\beta}(X)\sqsubset\overline{SA}_{-2}^{\beta}(X)\sqsubset\overline{SA}_{-3}^{\beta}(X)$.

(7)
$\overline{SA}_{+3}^{\beta}(X)\sqsubset\overline{SA}_{+1}^{\beta}(X)\sqsubset\overline{SA}_{+4}^{\beta}(X)$.

(8)
$\overline{SA}_{+3}^{\beta}(X)\sqsubset\overline{SA}_{+2}^{\beta}(X)\sqsubset\overline{SA}_{+4}^{\beta}(X)$.

Proof It is clear from Definition 4.27, 4.33, 4.36 and 4.39.$\blacksquare$

######  Theorem 4.42.

Let $\beta$ be an interval-valued number, $X\in IF(U)$, if
$(\widetilde{SN}_{x}^{\beta})^{c}(x)\leqslant
X(x)\leqslant\widetilde{SN}_{x}^{\beta}(x)$ for all $x\in U$, then
$\widetilde{SA}_{-4}^{\beta}(X)\subset\widetilde{SA}_{-2}^{\beta}(X)\subset\widetilde{SA}_{-3}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+3}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+4}^{\beta}(X)$.

Proof For all $x,y\in U$,
$\widetilde{SM}_{x}^{\beta}(y)=\widetilde{SN}_{y}^{\beta}(x)$, then
$\widetilde{SM}_{x}^{\beta}(x)=\widetilde{SN}_{x}^{\beta}(x)$ and
$(\widetilde{SM}_{x}^{\beta})^{c}(x)=(\widetilde{SN}_{x}^{\beta})^{c}(x)$.

For all $x\in U$, $(\widetilde{SN}_{x}^{\beta})^{c}(x)\leqslant X(x)$, then
$(\widetilde{SN}_{x}^{\beta})^{c}(x)\vee(\widetilde{SM}_{x}^{\beta})^{c}(x)\vee
X(x)=(\widetilde{SN}_{x}^{\beta})^{c}(x)\vee X(x)\leqslant X(x)$. Then,
$\widetilde{SA}_{-3}^{\beta}(X)(x)=\bigwedge\limits_{y\in
U}\\{(\widetilde{SN}_{x}^{\beta})^{c}(y)\vee(\widetilde{SM}_{x}^{\beta})^{c}(y)\vee
X(y)\\}\leqslant(\widetilde{SN}_{x}^{\beta})^{c}(x)\vee(\widetilde{SM}_{x}^{\beta})^{c}(x)\vee
X(x)\leqslant X(x)$ for all $x\in U$, i.e.,
$\widetilde{SA}_{-3}^{\beta}(X)\subset X$.

For all $x\in U$, $X(x)\leqslant\widetilde{SN}_{x}^{\beta}(x)$, then
$X(x)\leqslant\widetilde{SN}_{x}^{\beta}(x)\wedge
X(x)=\widetilde{SN}_{x}^{\beta}(x)\wedge\widetilde{SM}_{x}^{\beta}(x)\wedge
X(x)$. Then, $\widetilde{SA}_{+3}^{\beta}(X)(x)=\bigvee\limits_{y\in
U}\\{\widetilde{SN}_{x}^{\beta}(y)\wedge\widetilde{SM}_{x}^{\beta}(y)\wedge
X(y)\\}\geqslant\widetilde{SN}_{x}^{\beta}(x)\wedge\widetilde{SM}_{x}^{\beta}(x)\wedge
X(x)\geqslant X(x)$ for all $x\in U$, i.e.,
$\widetilde{SA}_{+3}^{\beta}(X)\supset X$.

To combine the results of Theorem 4.40 (6) and (7),
$\widetilde{SA}_{-4}^{\beta}(X)\subset\widetilde{SA}_{-2}^{\beta}(X)\subset\widetilde{SA}_{-3}^{\beta}(X)\subset
X\subset\widetilde{SA}_{+3}^{\beta}(X)\subset\widetilde{SA}_{+1}^{\beta}(X)\subset\widetilde{SA}_{+4}^{\beta}(X)$
holds.$\blacksquare$

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