Definition 3.1 Let U and V be two finite non-empty universes of discourse and R ∈ P(U × V) be a binary relation from U to V. The ordered triple (U, V, R) is called a (two-universe) approximation space. Let B ∈ [V]w be a multi set drawn from V.
The lower and upper approximation of B,\( {\underset{\_}{\ R}}_s(B) \) and \( {\overline{R}}_s(B) \), with respect to the approximation space are multi set of U whose membership functions, for each a ∈ U, are defined, respectively, by:
$$ {C}_{{\underset{\_}{R}}_s(B)}(a)=\mathit{\min}\left\{{C}_B(b):b\in F(a)\right\} $$
$$ {C}_{{\overline{R}}_s(B)}(a)=\mathit{\max}\left\{{C}_B(b):b\in F(a)\right\} $$
where F(a) is the successor neighborhood of a.
The ordered set pair \( \left({\underset{\_}{R}}_s(B),{\overline{R}}_s(B)\right) \) is referred to as a generalized rough multiset with respect to successor neighborhood, and \( {\underset{\_}{R}}_s:P(V)\longrightarrow P(U) \) and \( {\overline{R}}_s:P(V)\longrightarrow P(U) \) are referred to as lower and upper generalized rough multi approximation operators, respectively.
Definition 3.2 Let (U, V, R) be a two-universe approximation space. Then, the lower and upper approximations of A ∈ [U]w are defined, respectively, as follows:
$$ {C}_{{\underset{\_}{R}}_P(A)}(b)=\mathit{\min}\left\{{C}_A(a):a\in G(b)\right\} $$
$$ {C}_{{\overline{R}}_P(A)}(b)=\mathit{\max}\left\{{C}_A(a):a\in G(b)\right\} $$
where G(b) is the predecessor neighborhood of b.
The pair \( \left({\underset{\_}{R}}_P(A),{\overline{R}}_P(A)\right) \) is referred to as a generalized rough multiset with respect to the predecessor neighborhood, and \( {\underset{\_}{R}}_P:P(U)\longrightarrow P(V) \) and \( {\overline{R}}_P:P(U)\longrightarrow P(V) \) are referred to as lower and upper rough multi approximation operators, respectively. If \( {\underset{\_}{R}}_P(A)={\overline{R}}_P(A) \), then A is called an exact multiset; otherwise, A is a rough multiset.
Proposition 3.1 In a two-universe model (U, V, R) with the binary relation R, the approximation operators \( {\underset{\_}{R}}_P \) and \( {\overline{R}}_P \) satisfy the following properties for all A, A1, A2 ∈ [U]w:
\( {\displaystyle \begin{array}{ll}\left({L}_1\right)\kern0.5em {\underset{\_}{R}}_P(A)={\left({\overline{R}}_P\left({A}^c\right)\right)}^c.& \left({L}_2\right)\kern0.5em {\underset{\_}{R}}_P(U)=V.\\ {}\left({L}_3\right)\kern0.5em {\underset{\_}{R}}_P\left({A}_1\cap {A}_2\right)={\underset{\_}{R}}_P\left({A}_1\right)\cap {\underset{\_}{R}}_P\left({A}_2\right).& \left({L}_4\right)\ {\underset{\_}{R}}_P\left({A}_1\cup {A}_2\right)\supseteq {\underset{\_}{R}}_P\left({A}_1\right)\cup {\underset{\_}{R}}_P\left({A}_2\right).\\ {}\left({L}_5\right)\ {A}_1\subseteq {A}_2\Longrightarrow {\underset{\_}{R}}_P\left({A}_1\right)\subseteq {\underset{\_}{R}}_P\left({A}_2\right).& \left({U}_1\right)\kern0.5em {\overline{R}}_P(A)={\left({\underset{\_}{R}}_P\left({A}^c\right)\right)}^c.\\ {}\left({U}_2\right)\kern0.5em {\overline{R}}_P\left(\phi \right)=\phi .& \left({U}_3\right)\kern0.5em {\overline{R}}_P\left({A}_1\cup {A}_2\right)={\overline{R}}_P\left({A}_1\right)\cup {\overline{R}}_P\left({A}_2\right).\\ {}\left({U}_4\right)\kern0.5em {\overline{R}}_P\left({A}_1\cap {A}_2\right)\subseteq {\overline{R}}_P\left({A}_1\right)\cap {\overline{R}}_P\left({A}_2\right).& \left({U}_5\right)\kern0.5em {A}_1\subseteq {A}_2\Longrightarrow {\overline{R}}_P\left({A}_1\right)\subseteq {\overline{R}}_P\left({A}_2\right).\end{array}} \).
Proof By the duality of approximation operators, we only need to prove the properties L1 − L5.
(L1) For all b ∈ V, according to Definition 3.2, we can obtain:
$$ {\displaystyle \begin{array}{c}{C}_{{\left[{\overline{R}}_P\left({A}^c\right)\right]}^c}(b)=w-\left\{\mathit{\max}\left\{{C}_{A^c}(a):a\in G(b)\right\}\right\}\\ {}=w-\left\{\mathit{\max}\left\{w-{C}_A(a):a\in G(b)\right\}\right\}\\ {}=w-\left\{w-\mathit{\min}\left\{{C}_A\left(\mathrm{a}\right):a\in G(b)\right\}\right\}\\ {}=w-w+\mathit{\min}\left\{{C}_A(a):a\in G(b)\right\}\\ {}=\mathit{\min}\left\{{C}_A(a):a\in G(b)\right\}\\ {}={C}_{{\underset{\_}{R}}_P(A)}(b).\end{array}} $$
Therefore, \( {\underset{\_}{R}}_P(A)={\left({\overline{R}}_P\left({A}^c\right)\right)}^c \).
(L2) Since CU(a) = 1 ∀ a ∈ U and G(b) ⊆ U, the min{CU(a) : a ∈ G(b)} = 1. Thus, \( {C}_{{\underset{\_}{R}}_P(U)}(b)=\min \left\{{C}_U(a):a\in G(b)\right\}=1 \) for all b ∈ V. Therefore, \( {\underset{\_}{R}}_P(U)=V \).
(L3) Since ∀b ∈ V,
$$ {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_P\left({A}_1\cap {A}_2\right)}(b)=\mathit{\min}\left\{{C}_{\left({A}_1\cap {A}_2\right)}(a):a\in G(b)\right\}\\ {}=\mathit{\min}\left\{\mathit{\min}\left\{{C}_{A_1}(a),{C}_{A_2}(a)\right\}:a\in G(b)\right\}\\ {}=\mathit{\min}\left\{\mathit{\min}\left\{{C}_{A_1}(a):a\in G(b)\right\},\mathit{\min}\left\{{C}_{A_2}(a):a\in G(b)\right\}\right\}\\ {}=\mathit{\min}\left\{{C}_{{\underset{\_}{R}}_P\left({A}_1\right)}(b),{C}_{{\underset{\_}{R}}_P\left({A}_2\right)}(b)\right\}\\ {}={C}_{{\underset{\_}{R}}_P\left({A}_1\right)\cap {\underset{\_}{R}}_P\left({A}_2\right)}(b).\end{array}} $$
Therefore, \( {\underset{\_}{R}}_P\left({A}_1\cap {A}_2\right)={\underset{\_}{R}}_P\left({A}_1\right)\cap {\underset{\_}{R}}_P\left({A}_2\right) \).
(L4) For all b ∈ V, we can have:
$$ {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_P\left({A}_1\cup {A}_2\right)}(b)=\mathit{\min}\left\{{C}_{\left({A}_1\cup {A}_2\right)}(a):a\in G(b)\right\}\\ {}=\mathit{\min}\left\{\mathit{\max}\left\{{C}_{A_1}(a),{C}_{A_2}(a)\right\}:a\in G(b)\right\}\\ {}\ge \mathit{\max}\left\{\mathit{\min}\left\{{C}_{A_1}(a):a\in G(b)\right\},\mathit{\min}\left\{{C}_{A_2}(a):a\in G(b)\right\}\right\}\\ {}=\mathit{\max}\left\{{C}_{{\underset{\_}{R}}_P\left({A}_1\right)}(b),{C}_{{\underset{\_}{R}}_P\left({A}_2\right)}(b)\right\}\\ {}={C}_{{\underset{\_}{R}}_P\left({A}_1\right)\cap {\underset{\_}{R}}_P\left({A}_2\right)}(b).\end{array}} $$
Hence, \( {\underset{\_}{R}}_P\left({A}_1\cup {A}_2\right)\supseteq {\underset{\_}{R}}_P\left({A}_1\right)\cup {\underset{\_}{R}}_P\left({A}_2\right) \).
(L5) Since A1 ⊆ A2, then \( \forall a\in U,{C}_{A_1}(a)\le {C}_{A_2}(a) \). Thus, \( {C}_{{\underset{\_}{R}}_P\left({A}_1\right)}(b)=\mathit{\min}\left\{{C}_{A_1}(a):a\in G(b)\right\}\le \mathit{\min}\left\{{C}_{A_2}(a):a\in G(b)\right\}={C}_{{\underset{\_}{R}}_P\left({A}_2\right)}(b) \).
Therefore, \( {\underset{\_}{R}}_P\left({A}_1\right)\subseteq {\underset{\_}{R}}_P\left({A}_2\right) \).
The next proposition gives us characterizations of the rough multi lower and rough multi upper approximation operators based on different types of relations.
Proposition 3.2. Let R ∈ P(U × V) be an arbitrary binary relation. Then, ∀A ∈ [U]w:
(i) R is inverse serial \( \Longleftrightarrow \left({L}_6\right){\underset{\_}{R}}_P\left(\phi \right)=\phi \)\( \Longleftrightarrow \left({U}_6\right){\overline{R}}_P(U)=V \)\( \Longleftrightarrow (LU){\underset{\_}{R}}_P(A)\subseteq {\overline{R}}_P(A) \).
If U = V, then:
(ii) R is reflexive \( \Longleftrightarrow \left({L}_7\right){\underset{\_}{R}}_P(A)\subseteq A\Longleftrightarrow \left({U}_7\right)\ A\subseteq {\overline{R}}_P(A) \)
(iii) R is symmetric \( \Longleftrightarrow \left({L}_8\right)\ A\subseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\Longleftrightarrow \left({U}_8\right){\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq A \)
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(iv)
R is transitive \( \Longleftrightarrow \left({L}_9\right){\underset{\_}{R}}_P(A)\subseteq {\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right)\Longleftrightarrow \left({U}_9\right){\overline{R}}_P\left({\overline{R}}_P(A)\right)\subseteq {\overline{R}}_P(A) \)
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(v)
R is left Euclidean \( \Longleftrightarrow \left({L}_{10}\right){\overline{R}}_P(A)\subseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\Longleftrightarrow \left({U}_{10}\right){\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq {\underset{\_}{R}}_P(A) \)
Proof (i) Supposing that R is an inverse serial relation, then for any b ∈ V, we have G(b) ≠ ϕ. Thus, \( {C}_{{\underset{\_}{R}}_P\left(\phi \right)}(b)=\mathit{\min}\left\{{C}_{\phi }(a):a\in G(b)\right\}=0\forall b\in V \). Therefore, \( {\underset{\_}{R}}_P\left(\phi \right)=\phi \).
Conversely, assuming that \( {\underset{\_}{R}}_P\left(\phi \right)=\phi \) ,i.e., \( {C}_{{\underset{\_}{R}}_P\left(\phi \right)}(b)=\mathit{\min}\left\{{C}_{\phi }(a):a\in G(b)\right\}=0\kern0.5em \forall b\in V \). If there exists b∘ ∈ V such that G(b∘) = ϕ then \( {C}_{{\underset{\_}{R}}_P\left(\phi \right)}\left({b}_{\circ}\right)=\mathit{\min}\left\{{C}_{\phi }(a):a\in G\left({b}_{\circ}\right)\right\}=\min \left\{\kern1em \right\}= undefined \) which contradicts the assumption. Thus, G(b) ≠ ϕ ∀ b ∈ V,i.e., R is an inverse serial. We can prove that R is an inverse serial if and only if \( \left({U}_6\right)\ {\overline{R}}_P(U)=V \) by the duality of approximation operators. For the third part, R is inverse serial \( if\ and\ only\ if\ (LU)\ {\underset{\_}{R}}_P(A)\subseteq {\overline{R}}_P(A) \), and the proof is obvious.
(ii) By the duality, it is only to prove that R is reflexive if and only if \( \left({L}_7\right)\ {\underset{\_}{R}}_P(A)\subseteq A \). Since R is reflexive, then ∀b ∈ V, b ∈ G(b), i.e., min{CA(a) : a ∈ G(b)} ≤ CA(b) which implies that \( {\underset{\_}{R}}_P(A)\subseteq A \).
Conversely, assuming \( {\underset{\_}{R}}_P(A)\subseteq A \) for all multi subset A of U. Because a crisp set is a special case of a multiset, then \( {\underset{\_}{R}}_P(A)\subseteq A \) for all A ⊆ U and by proposition 2.1, R is a reflexive relation.
(iii) Assuming that R is symmetric, then for all a ∈ G(b), we have b ∈ G(a). So, max{min{CA(c) : c ∈ G(a)} : a ∈ G(b)} ≤ CA(b).
Therefore, \( {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\subseteq A \).
Conversely, assuming \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq A \) for all multi subset A of U. Because a crisp set is a special case of a multiset, then \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq A \) for all A ⊆ U and by proposition 2.1, R is a symmetric relation. For the other statement, the proof is similar.
(iv) Supposing that R is a transitive relation, then for all a ∈ G(b), we have G(a) ⊆ G(b). Thus, \( {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right)}(b)=\mathit{\min}\left\{\mathit{\min}\left\{{C}_A(a):c\in G(a)\right\}:a\in G(b)\right\}\\ {}\ge \mathit{\min}\left\{\mathit{\min}\left\{{C}_A(c):c\in G(b)\right\}:a\in G(b)\right\}\\ {}=\mathit{\min}\left\{{C}_A(c):c\in G(b)\right\}\\ {}={\underset{\_}{R}}_{\mathcal{P}}(A)(b).\end{array}} \)
Therefore, \( {\underset{\_}{R}}_P(A)\subseteq {\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right) \).
The proof of the other side is similar to (iii).
(v) Assuming that R is a left Euclidean relation, then for all a ∈ G(b), we have G(b) ⊆ G(a). So, \( {\displaystyle \begin{array}{c}{C}_{{\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)}(b)=\mathit{\max}\left\{\mathit{\min}\left\{{C}_A(c):c\in G(a)\right\}:a\in G(b)\right\}\\ {}\le \mathit{\max}\left\{\mathit{\min}\left\{{C}_A(c):c\in G(b)\right\}:a\in G(b)\right\}\\ {}=\mathit{\min}\left\{{C}_A(c):c\in G(b)\right\}={C}_{{\underset{\_}{R}}_P(A)}(b).\end{array}} \)
Therefore, \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq {\underset{\_}{R}}_P(A) \).
The proof of the other side is like (iii).
Remark 3.1 If R ∈ P(U × V) is a serial relation in a two-universe approximation space (U, V, R), then the properties L6, U6, and LU are not true in general, as shown in the following example:
Example 3.1 Let U = {a1, a2, a3, a4}, V = {b1, b2, b3, b4, b5}, and R be a binary relation from U to V defined as:
$$ \mathrm{R}=\left\{\left({a}_1,{b}_2\right),\left({a}_1,{b}_4\right),\left({a}_2,{b}_3\right),\left({a}_2,{b}_4\right),\left({a}_3,{b}_3\right),\left({a}_4,{b}_1\right),\left({a}_4,{b}_2\right)\right\}. $$
If A ∈ [U]w is a multiset drawn from U. Let A = {2/a1, 3/a2, 4/a4}.
Then, we have:
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\( {C}_{{\underset{\_}{R}}_{\mathcal{P}}(A)}(b) \)
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\( {C}_{{\overline{R}}_{\mathcal{P}}(A)}(b) \)
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\( {C}_{{\underset{\_}{R}}_{\mathcal{P}}\left(\phi \right)}(b) \)
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\( {C}_{{\overline{R}}_{\mathcal{P}}(U)}(b) \)
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Hence, \( {\underset{\_}{R}}_P\left(\phi \right)\ne \phi \), \( {\overline{R}}_P(U)\ne V \), and \( {\underset{\_}{R}}_P(A)\ne {\overline{R}}_P(A) \), i.e., L6, U6, and LU do not hold.
Remark 3.2 Let R be any reflexive relation, then ∀A ∈ [U]w the properties L8 − L10 and U8 − U10 are not true in general. The following example shows this remark.
Example 3. 2 Let U = {a1, a2, a3, a4, a5} and R be a reflexive relation on U defined as R = {(a1, a1), (a1, a2), (a2, a1), (a2, a2), (a2, a4), (a3, a3), (a3, a5),
$$ \left({a}_4,{a}_2\right),\left({a}_4,{a}_4\right),\left({a}_5,{a}_2\right),\left({a}_5,{a}_5\right)\Big\}. $$
If A and B are multisets drawn from U defined as A = {2/a2, 3/a3, 4/a5} and B = {2/a1, 3/a2, 1/a4, 4/a5}, then we have:
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\( {C}_{{\underset{\_}{R}}_P(B)}(a) \)
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\( {C}_{{\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right)}(a) \)
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\( {C}_{{\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)}(a) \)
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\( {C}_{{\overline{R}}_P(A)}(a) \)
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\( {C}_{{\overline{R}}_P\left({\overline{R}}_P(B)\right)}(a) \)
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\( {C}_{{\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)}(a) \)
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Hence,\( A\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \), \( {\underset{\_}{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right) \), \( {\overline{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \),\( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq A \), \( {\overline{R}}_P\left({\overline{R}}_P(A)\right)\nsubseteq {\overline{R}}_P(A) \), \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq {\underset{\_}{R}}_P(A) \),i.e., L8 − L10, U8 − U10 do not hold.
Remark 3.3 Let R be any symmetric relation, then ∀A ∈ [U]w the properties L6, L7, L9, L10, U6, U7, U9, U10 and LU are not true in general. The following example shows this remark.
Example 3.3 Let U = {a1, a2, a3, a4, a5} and R be a symmetric relation on U defined as R = {(a1, a1), (a1, a2), (a2, a1), (a2, a4), (a4, a2), (a4, a4), (a5, a5)}.
If A is a multiset drawn from U defined as A = {4/a1, 2/a2, 3/a4, 1/a5}, then we have:
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\( {\underset{\_}{R}}_{\mathcal{P}}\left({\underset{\_}{R}}_{\mathcal{P}}(A)\right)(a) \)
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\( {\overline{R}}_{\mathcal{P}}\left({\underset{\_}{R}}_{\mathcal{P}}(A)\right)(a) \)
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\( {\overline{R}}_{\mathcal{P}}(A)(a) \)
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\( {\overline{R}}_{\mathcal{P}}\left({\overline{R}}_{\mathcal{P}}(A)\right)(a) \)
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Hence, \( {\underset{\_}{R}}_P\left(\phi \right)\ne \phi \), \( {\underset{\_}{R}}_P(A)\nsubseteq A \), \( {\underset{\_}{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\underset{\_}{R}}_P(A)\right) \), \( {\overline{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \), \( {\overline{R}}_P(U)\ne U \),\( A\nsubseteq {\overline{R}}_P(A) \), \( {\overline{R}}_P\left({\overline{R}}_P(A)\right)\nsubseteq {\overline{R}}_P(A) \),\( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq {\underset{\_}{R}}_P(A) \) and \( {\underset{\_}{R}}_P(A)\nsubseteq {\overline{R}}_P(A) \), i.e., L6, L7, L9, L10 and U6, U7, U9, U10 and LU do not hold.
Remark 3.4 Let R be any transitive relation, then ∀A ∈ [U]w the properties L6, L7, L8, L10, U6, U7, U8, U10 and LU do not hold in general. The following example shows this remark.
Example 3.4 Let U = {a1, a2, a3, a4, a5} and R be a transitive relation on U defined as R = {(a1, a2), (a1, a3), (a2, a3), (a4, a4), (a5, a2), (a5, a3)}.
If A is a multiset drawn from U defined as A = {3/a1, 4/a3, 2/a5} and B = {3/a1, 1/a2, 2/a4, 4/a5}, then we have:
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\( {\underset{\_}{R}}_{\mathcal{P}}(A)(a) \)
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\( {\overline{R}}_{\mathcal{P}}(A)(a) \)
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\( {\underset{\_}{R}}_{\mathcal{P}}(B)(a) \)
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\( {\underset{\_}{R}}_{\mathcal{P}}\left({\overline{R}}_{\mathcal{P}}(A)\right)(a) \)
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\( {\underset{\_}{R}}_{\mathcal{P}}\left(\phi \right)(a) \)
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\( {\overline{R}}_{\mathcal{P}}(U)(a) \)
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Hence, \( {\underset{\_}{R}}_P\left(\phi \right)\ne \phi \), \( {\underset{\_}{R}}_P(A)\nsubseteq A \), \( A\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \), \( {\overline{R}}_P(A)\nsubseteq {\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right) \), \( {\overline{R}}_P(U)\ne V \), \( A\nsubseteq {\overline{R}}_P(A) \), \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq A \), \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\nsubseteq {\underset{\_}{R}}_P(A) \) and \( {\underset{\_}{R}}_P(A)\nsubseteq {\overline{R}}_P(A) \), i.e., L6, L7, L8, L10, U6, U7, U8, U10 and LU do not hold.
Definition 3.4 A multi constant \( \hat{\alpha} \) is a multiset in U defined as:
$$ {C}_{\hat{\alpha}}(a)=\alpha \forall a\in U,\alpha \in N. $$
Proposition 3.3 Let (U, V, R) be a two- universe approximation space, the rough multi lower and upper approximation operators have the following properties for all Aj ∈ [U]w, j ∈ J which is an finite index set and for all α ∈ {1, 2, 3, …},
$$ \left(\mathrm{i}\right){\underset{\_}{R}}_P\left({\cap}_{j\in J}{A}_j\right)={\cap}_{j\in J}{\underset{\_}{R}}_P\left({A}_j\right). $$
$$ \left(\mathrm{ii}\right){\underset{\_}{R}}_P\left({\cup}_{j\in J}{A}_{\mathrm{j}}\right)\supseteq {\cup}_{j\in J}{\underset{\_}{R}}_P\left({A}_j\right). $$
$$ \left(\mathrm{iii}\right){\underset{\_}{R}}_P\left(A\cup \hat{\alpha}\right)={\underset{\_}{R}}_P(A)\cup \hat{\alpha}. $$
$$ \left(\mathrm{iv}\right){\overline{R}}_P\left({\cup}_{j\in J}{A}_j\right)={\cup}_{j\in J}{\overline{R}}_P\left({A}_j\right). $$
$$ \left(\mathrm{v}\right){\overline{\mathrm{R}}}_P\left({\cap}_{\mathrm{j}\in \mathrm{J}}{\mathrm{A}}_{\mathrm{j}}\right)\subseteq {\cap}_{j\in J}{\overline{R}}_P\left({A}_j\right). $$
Proof By the duality of approximation operators, we only need to prove the properties (i) − (iii).
(i) For each b ∈ V, we have:
$$ {C}_{{\underset{\_}{R}}_P\left({\cap}_{j\in J}{A}_j\right)}(b)=\mathit{\min}\left\{{C}_{\left({\cap}_{j\in J}{A}_j\right)}(a):(a)\in G(b)\right\} $$
$$ =\mathit{\min}\left\{\mathit{\min}\left\{{C}_{A_j}(a):j\in J\right\}:a\in G(b)\right\} $$
$$ =\mathit{\min}\left\{\mathit{\min}\left\{{C}_{A_j}(a):a\in G(b)\right\}:j\in J\right\}=\mathit{\min}\left\{{C}_{{\underset{\_}{R}}_P\left({A}_j\right)}(b):j\in J\right\} $$
$$ ={C}_{\cap_{j\in J}{\underset{\_}{R}}_P\left({A}_j\right)}(b). $$
(ii) Since ∀(b) ∈ V,
$$ {C}_{{\underset{\_}{R}}_P\left({\cup}_{j\in J}{A}_j\right)}(b)=\mathit{\min}\left\{{C}_{\left({\cup}_{j\in J}{A}_j\right)}(a):(a)\in G(b)\right\} $$
$$ =\mathit{\min}\left\{\mathit{\max}\left\{{C}_{A_j}(a):j\in J\right\}:(a)\in G(b)\right\} $$
$$ \ge \mathit{\min}\left\{{C}_{B_j}(c):(c)\in G(b)\right\},\forall j\in J={C}_{{\underset{\_}{R}}_P\left({A}_j\right)}(b),\forall j\in J. $$
Therefore, \( {C}_{{\underset{\_}{R}}_P\left({\cup}_{j\in J}{A}_j\right)}(b)\ge \mathit{\max}\left\{{\underset{\_}{R}}_P\left({A}_j\right)(b),\forall j\in J\right\}={C}_{\cup_{j\in J}{\underset{\_}{R}}_P\left({A}_j\right)}(b). \)
(iii) For each (b) ∈ V, we have:
$$ {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_P\left(A\cup \hat{\alpha}\right)}(b)=\mathit{\min}\left\{{C}_{\left(A\cup \hat{\alpha}\right)}(a):(a)\in G(b)\right\}\\ {}=\mathit{\min}\left\{\mathit{\max}\left\{{C}_A(a),{C}_{\hat{\alpha}}(a)\right\}:(a)\in G(b)\right\}\\ {}=\mathit{\max}\left\{\mathit{\min}\left\{{\mathrm{C}}_A(a):a\in G(b)\right\},{C}_{\hat{\alpha}}(a)\right\}\\ {}={C}_{\left({\underset{\_}{R}}_P(A)\cup \hat{\alpha}\right)}(b).\end{array}} $$
Proposition 3.4 Let (U, V, R) be a two-universe approximation space. Then, the following are equivalent ∀α ∈ N
(i) R is an inverse serial relation,
$$ \left(\mathrm{ii}\right){\underset{\_}{R}}_P\left(\hat{\alpha}\right)=\hat{\alpha}, $$
$$ \left(\mathrm{iii}\right){\overline{R}}_P\left(\hat{\alpha}\right)=\hat{\alpha}. $$
Proof (i) ⟹ (ii) Let R be an inverse serial relation, then we have \( {\underset{\_}{R}}_P\left(\hat{\alpha}\right)={\underset{\_}{R}}_P\left(\hat{\alpha}\cup \phi \right)=\hat{\alpha}\cup {\underset{\_}{R}}_P\left(\phi \right)=\hat{\alpha}\cup \phi =\hat{\alpha .} \)
(ii) ⟹ (iii) Coming from the duality of approximation operators.
(iii) ⟹ (i) Assuming \( {\overline{R}}_P\left(\hat{\alpha}\right)=\hat{\alpha} \), since U is a special case of \( \hat{\alpha} \) which is α = w. Then by assumption, we have \( {\overline{R}}_P(U)=V, \) i.e., R is an inverse serial relation.
In the next three propositions, the connections of the approximation operators in definitions 2.7, and 3.1 are made, and the conditions under which these approximation operators made the equivalent are obtained.
Proposition 3.5 Let (U, V, R) be a two-universe approximation space, then the following holds for all A ∈ [U]wand B ∈ [V]w:
$$ \left(\mathrm{i}\right)\kern0.5em {\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)\subseteq A,A\subseteq {\underset{\_}{R}}_s\left({\overline{R}}_P(A)\right),\left(\mathrm{i}\mathrm{v}\right)\kern0.5em {\overline{R}}_s(B)={\overline{R}}_s\left({\underset{\_}{R}}_P\left({\overline{R}}_s(B)\right)\right), $$
$$ \left(\mathrm{ii}\right)\kern0.5em {\overline{R}}_P\left({\underset{\_}{R}}_s(B)\right)\subseteq B,B\subseteq {\underset{\_}{R}}_P\left({\overline{R}}_s(B)\right),\left(\mathrm{v}\right)\kern0.5em {\underset{\_}{R}}_P(A)={\underset{\_}{R}}_P\left({\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)\right), $$
$$ \left(\mathrm{iii}\right)\kern0.5em {\underset{\_}{R}}_s(B)={\underset{\_}{R}}_s\left({\overline{R}}_P\left({\underset{\_}{R}}_s(B)\right)\right),\left(\mathrm{vi}\right)\kern0.5em {\overline{R}}_P(A)={\overline{R}}_P\left({\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\right) $$
Proof (i) Since for every a ∈ U, we have either F(a) = ϕ or F(a) ≠ ϕ. If F(a) = ϕ, then \( {C}_{{\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)}(a)=\mathit{\max}\left\{\mathit{\min}\left\{{C}_A(a):c\in G(b)\right\}:b\in F(a)\right\}=0 \) and hence \( {\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)\subseteq A. \) If A(a) ≠ ϕ, then we have a ∈ G(b) ∀ b ∈ A(a). Thus, max{min{CA(c) : c ∈ G(b)}b ∈ A(a)} ≤ CA(a), hence \( {\overline{R}}_s\left({\underset{\_}{R}}_P(A)\right)\subseteq A. \) We can easily prove the other part by the duality of approximation operators.
(ii) is similar to (i).
(iii) − (vi) can be proved by the properties (i) and (ii).
Lemma 3.1 Let (U, V, R) be a two-universe approximation space, b ∈ V; if R is a strong inverse serial relation, then for all a1, a2 ∈ G(b),
$$ {C}_{{\underset{\_}{R}}_s(B)}\left({a}_1\right)={C}_{{\underset{\_}{R}}_s(B)}\left({a}_2\right);{C}_{{\overline{R}}_s(B)}\left({a}_1\right)={C}_{{\overline{R}}_s(B)}\left({a}_2\right). $$
Proof The proofs come directly from Lemma 2.1.
Proposition 3.6 Let (U, V, R) be a two-universe approximation space with a strong inverse serial relation, then the following holds for all A ∈ [U]w and B ∈ [V]w:
$$ \left(\mathrm{i}\right)\ {\overline{R}}_P\left({\underset{\_}{R}}_P(B)\right)={\underset{\_}{R}}_P\left({\overline{R}}_P(B)\right) $$
$$ \left(\mathrm{ii}\right)\ {\underset{\_}{R}}_P\left({\overline{R}}_P(B)\right)={\overline{R}}_P\left({\overline{R}}_P(B)\right). $$
Proof The proofs follow immediately from Lemma 3.1.
Proposition 3.7 Two pairs of lower approximation and upper approximation operators in definitions 2.7 and 3.2 are equivalent if and only if R is a symmetric relation.
Proof Let R be a symmetric relation on U, A ∈ [U]w. Then for all a ∈ U, we have F(a) = G(a), i.e., \( {\displaystyle \begin{array}{c}{C}_{{\underset{\_}{R}}_s(A)}(a)=\mathit{\min}\left\{{C}_A(b):b\in F(a)\right\}\\ {}=\mathit{\min}\left\{{C}_A(b):b\in G(a)\right\}={C}_{{\underset{\_}{R}}_P(A)}(a).\end{array}} \)
Conversely, assuming \( {\underset{\_}{R}}_s(A)={\underset{\_}{R}}_P(A) \), since by the proposition 3.4, we have \( {\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\subseteq A \), by proposition 3.1,and R is a symmetric relation.
Proposition 3.8 Let G = (U, R) be a generalized approximation space and A be a multisubset of U. Then, the following holds:
(i) If R is symmetric then:
$$ {\underset{\_}{R}}_P(A)={\underset{\_}{R}}_P\left({\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\right);{\overline{R}}_P(A)={\overline{R}}_P\left({\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\right). $$
(ii) If R is inverse serial and transitive then:
$$ {\underset{\_}{R}}_P(A)\subseteq {\underset{\_}{R}}_P\left({\overline{R}}_P\left({\underset{\_}{R}}_P(A)\right)\right);{\overline{R}}_P(A)\supseteq {\overline{R}}_P\left({\underset{\_}{R}}_P\left({\overline{R}}_P(A)\right)\right). $$
Proof Obvious
Example 3.5 Let U = {a1, a2, a3, a4} a set of four patients and V = {Fever(b1), Headache(b2), Stomachache (b3), Cough(b4), Myalgia (b5)} be five symptoms,if R = {(a1, b2), (a1, b4), (a2, b3), (a2, b4), (a3, b3), (a3, b5), (a4, b1), (a4, b2), (a4, b5)}
is a relation relating patients to symptoms. Let A = {3/a1, 0/a2, 3/a3, 5/a4} represents a multiset of patients and times of visiting the doctor. Thus, using definition 2.10, we have:
$$ G\left({b}_1\right)=\left\{{a}_4\right\},G\left({b}_2\right)=\left\{{a}_1,{a}_4\right\},G\left({b}_3\right)=\left\{{a}_2,{a}_3\right\},G\left({b}_4\right)=\left\{{a}_1,{a}_2\right\},G\left({b}_5\right)=\left\{{a}_3,{a}_4\right\} $$
and so, we get:
$$ {\underset{\_}{R}}_P(A)=\left\{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{${b}_1$}\right.,\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{${b}_2$}\right.,\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{${b}_3$}\right.,\raisebox{1ex}{$0$}\!\left/ \!\raisebox{-1ex}{${b}_4$}\right.,\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{${b}_5$}\right.\right\}\ and\ {\overline{R}}_p(A)=\left\{\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{${b}_1$}\right.,\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{${b}_2$}\right.,\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{${b}_3$}\right.,\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{${b}_4$}\right.,\raisebox{1ex}{$5$}\!\left/ \!\raisebox{-1ex}{${b}_5$}\right.\right\}. $$
If A = {a1, a3, a4} . By using the class U/R−1 = {{a4}, {a1, a4}, {a2, a3}, {a1, a2}, {a3, a4}} , the lower and upper approximations using rough sets on one universe U are \( \underset{\_}{R}(A)=\left\{{a}_1,{a}_3,\kern0.5em {a}_4\right\}=A \) and \( \overline{R}(A)=\left\{{a}_1,{a}_2,{a}_3,{a}_4\right\}=U \). Clearly, this method does not have any deviations between the effectiveness of symptoms. But by using the multi approximations over the two universes U and V, we have degree of effectiveness of b1 which is \( \frac{5}{5} \), b2 which is \( \frac{3}{5} \), b3 which is \( \frac{0}{2} \), b4 which is \( \frac{0}{3} \), and b5 which is \( \frac{2}{5} \).