![Fuzzifying Topological System
1
2.preliminaries
In this section, we introduce some abstraction and results in fuzzifying topology, which
will be used in the outcome. For the details, we present [7,23].First we describe a verity of fuzzy logical
symbols and properties. For any formulas𝜑, the symbol [𝜑] means the truth value of 𝜑, where the set of
truth values in the unit interval [0,1]. A formula is valid, we write ⊨ 𝜑 if and only if [𝜑]=1 for every
interpretation. Let x be a universe of discourse, and let P(X) and 𝓅(𝑋) denote the classes of all crisp
subsets and fuzzy subsets of X, respectively, Then, for 𝐴
̃,𝐵
̃ 𝜖 𝓅(𝑋), and any x𝜖X,
(1) [¬𝛼]=1-[𝛼];
[𝛼 ∧ 𝛽] = min ([𝛼], [𝛽]);
[𝛼 → 𝛽] = min (1,1 − [𝛼] + [𝛽]);
[∀𝑥 ∝ (𝑥)] = 𝑖𝑛𝑓𝑥𝜖𝑋 [∝ (𝑥)]
[𝑥 ∈ 𝐴
̃] = 𝐴
̃(𝑥)[∃𝑥 ∝ (𝑥)] ∶= [¬(∀𝑥¬∝ (𝑥))]
(2) [𝛼 ∨ 𝛽] ∶= [¬(¬𝛼 ∧ ¬𝛽)];
[𝛼 ↔ 𝛽] ∶= [(𝛼 → 𝛽) ∧ (𝛽 → 𝛼)];
[𝐴
̃ ⊆ 𝐵
̃] ∶= [∀𝑥(𝑥 ∈ 𝐴
̃ → 𝑥 ∈ 𝐵
̃)] 0
= 𝑖𝑛𝑓𝑥∈𝑋 min (1,1 − 𝐴
̃(𝑥) + 𝐵
̃(𝑥));
[𝐴
̃ ≡ 𝐵
̃] ∶= [(𝐴
̃ ⊆ 𝐵
̃)⋀(𝐵
̃ ⊆ 𝐴
̃)];
[𝛼 ⩒ 𝛽] ∶= [¬(𝛼 → ¬𝛽)] = min (1, (𝛼) + (𝛽));
[𝛼 ⩑ 𝛽] ∶= [¬𝛼 → 𝛽)] = 𝑚𝑎𝑥(0, [𝛼] + [𝛽] − 1).
Second, we give the following definitions and results in fuzzifying topology which are useful in the rest of the
present paper.
Definition 1. (cf. [𝟕])
Let X be a set. If 𝓅𝜖𝓅(𝑃(𝑋)) satisfies the following conditions:
(1) 𝓅(X) = 𝓅(ϕ) = 1;
(2) For any A, B,𝓅(𝐴 ∩ 𝐵) ≥ 𝓅(𝐴) ∧ 𝓅(𝐵);
(3) For any {𝐴𝜆: 𝜆 𝜖𝐴},𝓅(⋃ 𝐴𝜆
𝜆∈𝐴
) ≥ ⋀𝜆𝜖𝐴𝓅(𝐴𝜆).
Then 𝓅 is called a fuzzifying topology, (X, 𝓅)is called a fuzzifying topological space, A𝜖𝑃(𝑋) with 𝓅(𝐴) > 0 is
called a fuzzifying open set (write as 𝐴𝜖
𝓅), and 𝓅(𝐴) is called the degree of open of A.
Definition 2. (cf.[𝟕, 𝟐𝟑])
Let (X, 𝓅) be a fuzzifying topological space. Then
(1) 𝓅𝜖𝓅(𝑃(𝑋)), defined by 𝓅𝑐(𝐴) = 𝓅(𝑋~𝐴)(∀𝐴 ∈ 𝑃(𝑋)), is called a fuzzifying cotopology, B∈ 𝑃(𝑋) with
𝓅(𝑋~𝐴) > 0 is called a fuzzifying closed set (write as 𝐵𝜖
𝓅𝑐
), and 𝓅𝑐
(𝐴) is called the degree of close of
A, where X~A is the complement of A.
(2) 𝒩
𝑥 ∈ 𝓅(𝑃(𝑋)) defined by
𝒩
𝑥(A)= ⋁ 𝓅(𝐵)
𝑥𝜖𝐵⊆𝐴 (∀𝐴𝜖𝑃(𝑋))
is called the fuzzifying neighborhood system of a point x∈X, C∈ p(X) with𝒩𝑥(C)> 0 is called a fuzzifying
neighborhood of x (write as 𝐶∈
𝒩
𝑥), and 𝒩
𝑥(𝐶) is called the degree of C being a neighborhood of x.
(3) The fuzzifying cluster operator Cl(A)𝜖 𝓅(𝑋) of a set A⊆X is defined by Cl(A)(x) = 1-𝒩
𝑥(X~A) (∀𝑥 ∈ 𝑋),
and Cl(A)(x) is called the degree of x being a cluster point of A.
(4) The fuzzifying interior operator Int(A)∈ 𝓅(𝑋) of a set A⊆ 𝑋 is defined by Int(A)(x) = 𝒩
𝑥(𝑋~𝐴)(∀𝑥 ∈ 𝑋),
and Int(A)(x) is called the degree of x being an interior point of A.](https://image.slidesharecdn.com/fuzzyalgebrafirstwork3rdsmstr-210916110309/85/Fuzzy-algebra-1-320.jpg)
![Fuzzifying Topological System
2
Definition 3.(𝒄𝒇. [𝟖])
Let (X, 𝓅) be a fuzzifying topological space. Then
(1) A ∈ P(X) satisfying the following condition is called fuzzifying e-open set (write as𝐴𝜖
𝓅𝑒):
𝓅𝑒(𝐴) = ⋀𝜆𝜖𝐴(𝐼𝑛𝑡(𝐶𝑙𝛿(𝐴)))(x)⋁ 𝐶𝑙(𝐼𝑛𝑡𝛿(𝐴))(x))> 0,
Where 𝓅𝑒(A) is called the degree of e-open of A.
(2) A𝜖 𝑃(𝑋) satisfying
𝓅𝑒
𝑐
(A)= 𝓅𝑒(X~A)> 0
is called a fuzzifying e-closed set (write as𝐴𝜖
𝓅𝑒
𝑐
), and 𝓅𝑒
𝑐
(A) is called the degree of e-closed of A.
(3) 𝒩
𝑥
𝑒
∈ 𝓅(𝑃(𝑋)), defined by
𝒩𝑥
𝑒
(𝐴)=⋁𝑥∈𝐵⊆𝐴 𝓅𝑒(B) (∀𝐴 ∈ 𝑃(𝑋))
Is called a fuzzifying e-neighborhood system of a point x𝜖X, C ∈ P(X) with𝒩
𝑥
𝑒
(C)> 0 is called a
fuzzifying e-neighborhood of x (write as𝐶∈
𝒩
𝑥
𝑒
), and 𝒩
𝑥
𝑒
(C) is called the degree of C being an e-
neighborhood of x.
(4) C𝑙𝑒(A) 𝜖, 𝓅(𝑋), defined by
C 𝑙𝑒(A)(x)= 1 − 𝒩
𝑥
𝑒
(X~A) (∀𝑥 ∈ 𝑋),
is called a fuzzifying e-cluster point operator of a subset A⊆X, x∈X with C𝑙𝑒(A)(x)> 0 is called a
fuzzifying e-cluster point of A (write as 𝑥∈
𝐶𝑙𝑒(𝐴)), and C𝑙𝑒(A)(x) is called the degree of x being a
fuzzifying e-cluster point of A.
(5) 𝐼𝑛𝑡𝑒(A)∈ 𝓅(𝑋), defined by
𝐼𝑛𝑡𝑒(A)(x)= 𝒩
𝑥
𝑒
(A) (∀𝑥 ∈ 𝑋),
Is called a fuzzifying e-interior point operator of a subset A⊆X, x∈X with 𝑖𝑛𝑡𝑒(A)(x)> 0 is called a
fuzzifying e-cluster point of A (write as𝑥𝜖
𝐼𝑛𝑡𝑒(𝐴)), and 𝐼𝑛𝑡𝑒(A)(x) is called the degree of x being
fuzzifying e-interior point of A.
Definition 4. (cf.[𝟖])
Let (X, 𝓅) and (Υ, 𝜇) be two fuzzifying topological spaces, 𝑓𝜖Υ𝑥
Satisfying the following condition is called fuzzifying e-continuous function (write as 𝑓𝜖
𝓅):
𝓅∈(𝑓)=⋀𝜇∈𝑝(Υ)min (1,1-𝜇(𝑈) + 𝓅𝑒(𝑓−1
(𝑈)))> 0,
Where 𝓅(𝑓) is called the degree of 𝑓 being an e-continuous function.
Definition 5. (cf. [𝟗])
Let (X,𝓅) and (Υ, 𝜇) be two fuzzifying topological spaces, 𝑓𝜖Υ𝑥
satisfying the
following condition is called fuzzifying e-irresolute function (write as𝑓𝜖
𝓅𝑒):
𝓅𝑒(𝑓) = ⋀𝑢∈𝑝(Υ)min(1,1-𝜇𝑒(U)+ 𝓅𝑒(𝑓−1
(𝑈)) > 0,
where 𝓅𝑒(𝑓) is called the degree of f being an e-irresolute function.
3. Fuzzifying e-𝜃-neighborhood system of a point, fuzzifying e-𝜃closure of a set, fuzzifying e-𝜃-
interior of a set, fuzzifying e-𝜃-open sets, and fuzzifying e-𝜃-closed sets.
Definition 6. 𝒩𝒙
𝒆𝜽
∈ 𝓅(𝑃(𝑋)), defined by
𝒩
𝒙
𝒆𝜽
(A) = ⋁𝒚∈𝒑(𝑿) max (0, 𝒩
𝑥
𝑒
(B)+∧𝑦∈𝐴 𝒩𝑦
𝑒
(X~B)-1),](https://image.slidesharecdn.com/fuzzyalgebrafirstwork3rdsmstr-210916110309/85/Fuzzy-algebra-2-320.jpg)
![Fuzzifying Topological System
3
is called a fuzzifying e-neighborhood system of a point x∈ X, C∈ P(X) with 𝒩
𝒙
𝒆𝜽
(C)> 0 is called a
fuzzifying e-neighborhood of x (write as 𝐶∈
𝒩
𝒙
𝒆𝜽
), and 𝒩
𝒙
𝒆𝜽
(C) is called the degree of C being an e-
neighborhood of x.
Theorem 7. The mapping 𝒩𝑒𝜃
: 𝑋 → 𝓅𝑁
(𝑃(𝑋))( the set of all normal fuzzy subsets of P(X)), defined by
𝒩𝑒𝜃
(x)= 𝒩
𝒙
𝒆𝜽
(∀𝑥 ∈ 𝑋), has the following properties:
(1) ⊨ 𝐴∈
𝒩
𝒙
𝒆𝜽
→x∈A (∀𝑥 ∈ 𝑋 , ∀∈ 𝑃(𝑋));
(2) ⊨ A⊆B→ (𝐴∈
𝒩
𝒙
𝒆𝜽
→ 𝐵∈
𝒩
𝒙
𝒆𝜽
)(∀𝑥 ∈ 𝑋, ∀𝐴, 𝐵 ∈ 𝑃(𝑋));
(3) ⊨ (𝐴∈
𝒩𝒙
𝒆𝜽
)⋀( 𝐵∈
𝒩
𝒙
𝒆𝜽
)↔ 𝐴⋂𝐵∈
𝒩
𝒙
𝒆𝜽
(∀𝑥 ∈ 𝑋, ∀𝐴, 𝐵 ∈ 𝑃(𝑋)).
Proof. (1) If[𝐴∈
𝒩
𝒙
𝒆𝜽] = 0, then the result holds. If [𝐴∈
𝒩
𝒙
𝒆𝜽] > 0, then
∨𝐵∈𝑃(𝑋)max (0, 𝒩𝑥
𝑒(𝐵) +∧𝑦∈𝐴 𝒩𝑦
𝑒(𝑋~𝐵) − 1)
=∨𝐵∈𝑃(𝑋)max (0, 𝒩
𝑥
𝑒(𝐵) + [ C 𝑙𝑒(B) ⊆ A] − 1)> 0,
and thus exists a C∈ 𝑃(𝑋) such that 𝒩
𝒙
𝒆
(C)+[C 𝑙𝑒(C) ⊆ A] − 1 > 0. From Theorems 4.2(1) and 5.5(3) in [8],
we have [C 𝑙𝑒(C) ⊆ A] > 1 − 𝒩
𝒙
𝒆
(C)≥ 1 − |𝑥 ∈ 𝐶| ≥ 1 − C 𝑙𝑒(C)(x). Therefore,
[C 𝑙𝑒(C) ⊆ A] = ⋀𝑦∈𝐴(1 − C 𝑙𝑒(C)(y) > 1 − C 𝑙𝑒(C)(x),
and so x∈A. Otherwise, if x∈A, then
⋀𝑦∈𝐴(1 − C 𝑙𝑒(C)(y) > 1 − C 𝑙𝑒(C)(x).](https://image.slidesharecdn.com/fuzzyalgebrafirstwork3rdsmstr-210916110309/85/Fuzzy-algebra-3-320.jpg)
Fuzzy algebra
- 1. Fuzzifying Topological System 1 2.preliminaries In this section, we introduce some abstraction and results in fuzzifying topology, which will be used in the outcome. For the details, we present [7,23].First we describe a verity of fuzzy logical symbols and properties. For any formulas𝜑, the symbol [𝜑] means the truth value of 𝜑, where the set of truth values in the unit interval [0,1]. A formula is valid, we write ⊨ 𝜑 if and only if [𝜑]=1 for every interpretation. Let x be a universe of discourse, and let P(X) and 𝓅(𝑋) denote the classes of all crisp subsets and fuzzy subsets of X, respectively, Then, for 𝐴 ̃,𝐵 ̃ 𝜖 𝓅(𝑋), and any x𝜖X, (1) [¬𝛼]=1-[𝛼]; [𝛼 ∧ 𝛽] = min ([𝛼], [𝛽]); [𝛼 → 𝛽] = min (1,1 − [𝛼] + [𝛽]); [∀𝑥 ∝ (𝑥)] = 𝑖𝑛𝑓𝑥𝜖𝑋 [∝ (𝑥)] [𝑥 ∈ 𝐴 ̃] = 𝐴 ̃(𝑥)[∃𝑥 ∝ (𝑥)] ∶= [¬(∀𝑥¬∝ (𝑥))] (2) [𝛼 ∨ 𝛽] ∶= [¬(¬𝛼 ∧ ¬𝛽)]; [𝛼 ↔ 𝛽] ∶= [(𝛼 → 𝛽) ∧ (𝛽 → 𝛼)]; [𝐴 ̃ ⊆ 𝐵 ̃] ∶= [∀𝑥(𝑥 ∈ 𝐴 ̃ → 𝑥 ∈ 𝐵 ̃)] 0 = 𝑖𝑛𝑓𝑥∈𝑋 min (1,1 − 𝐴 ̃(𝑥) + 𝐵 ̃(𝑥)); [𝐴 ̃ ≡ 𝐵 ̃] ∶= [(𝐴 ̃ ⊆ 𝐵 ̃)⋀(𝐵 ̃ ⊆ 𝐴 ̃)]; [𝛼 ⩒ 𝛽] ∶= [¬(𝛼 → ¬𝛽)] = min (1, (𝛼) + (𝛽)); [𝛼 ⩑ 𝛽] ∶= [¬𝛼 → 𝛽)] = 𝑚𝑎𝑥(0, [𝛼] + [𝛽] − 1). Second, we give the following definitions and results in fuzzifying topology which are useful in the rest of the present paper. Definition 1. (cf. [𝟕]) Let X be a set. If 𝓅𝜖𝓅(𝑃(𝑋)) satisfies the following conditions: (1) 𝓅(X) = 𝓅(ϕ) = 1; (2) For any A, B,𝓅(𝐴 ∩ 𝐵) ≥ 𝓅(𝐴) ∧ 𝓅(𝐵); (3) For any {𝐴𝜆: 𝜆 𝜖𝐴},𝓅(⋃ 𝐴𝜆 𝜆∈𝐴 ) ≥ ⋀𝜆𝜖𝐴𝓅(𝐴𝜆). Then 𝓅 is called a fuzzifying topology, (X, 𝓅)is called a fuzzifying topological space, A𝜖𝑃(𝑋) with 𝓅(𝐴) > 0 is called a fuzzifying open set (write as 𝐴𝜖 𝓅), and 𝓅(𝐴) is called the degree of open of A. Definition 2. (cf.[𝟕, 𝟐𝟑]) Let (X, 𝓅) be a fuzzifying topological space. Then (1) 𝓅𝜖𝓅(𝑃(𝑋)), defined by 𝓅𝑐(𝐴) = 𝓅(𝑋~𝐴)(∀𝐴 ∈ 𝑃(𝑋)), is called a fuzzifying cotopology, B∈ 𝑃(𝑋) with 𝓅(𝑋~𝐴) > 0 is called a fuzzifying closed set (write as 𝐵𝜖 𝓅𝑐 ), and 𝓅𝑐 (𝐴) is called the degree of close of A, where X~A is the complement of A. (2) 𝒩 𝑥 ∈ 𝓅(𝑃(𝑋)) defined by 𝒩 𝑥(A)= ⋁ 𝓅(𝐵) 𝑥𝜖𝐵⊆𝐴 (∀𝐴𝜖𝑃(𝑋)) is called the fuzzifying neighborhood system of a point x∈X, C∈ p(X) with𝒩𝑥(C)> 0 is called a fuzzifying neighborhood of x (write as 𝐶∈ 𝒩 𝑥), and 𝒩 𝑥(𝐶) is called the degree of C being a neighborhood of x. (3) The fuzzifying cluster operator Cl(A)𝜖 𝓅(𝑋) of a set A⊆X is defined by Cl(A)(x) = 1-𝒩 𝑥(X~A) (∀𝑥 ∈ 𝑋), and Cl(A)(x) is called the degree of x being a cluster point of A. (4) The fuzzifying interior operator Int(A)∈ 𝓅(𝑋) of a set A⊆ 𝑋 is defined by Int(A)(x) = 𝒩 𝑥(𝑋~𝐴)(∀𝑥 ∈ 𝑋), and Int(A)(x) is called the degree of x being an interior point of A.
- 2. Fuzzifying Topological System 2 Definition 3.(𝒄𝒇. [𝟖]) Let (X, 𝓅) be a fuzzifying topological space. Then (1) A ∈ P(X) satisfying the following condition is called fuzzifying e-open set (write as𝐴𝜖 𝓅𝑒): 𝓅𝑒(𝐴) = ⋀𝜆𝜖𝐴(𝐼𝑛𝑡(𝐶𝑙𝛿(𝐴)))(x)⋁ 𝐶𝑙(𝐼𝑛𝑡𝛿(𝐴))(x))> 0, Where 𝓅𝑒(A) is called the degree of e-open of A. (2) A𝜖 𝑃(𝑋) satisfying 𝓅𝑒 𝑐 (A)= 𝓅𝑒(X~A)> 0 is called a fuzzifying e-closed set (write as𝐴𝜖 𝓅𝑒 𝑐 ), and 𝓅𝑒 𝑐 (A) is called the degree of e-closed of A. (3) 𝒩 𝑥 𝑒 ∈ 𝓅(𝑃(𝑋)), defined by 𝒩𝑥 𝑒 (𝐴)=⋁𝑥∈𝐵⊆𝐴 𝓅𝑒(B) (∀𝐴 ∈ 𝑃(𝑋)) Is called a fuzzifying e-neighborhood system of a point x𝜖X, C ∈ P(X) with𝒩 𝑥 𝑒 (C)> 0 is called a fuzzifying e-neighborhood of x (write as𝐶∈ 𝒩 𝑥 𝑒 ), and 𝒩 𝑥 𝑒 (C) is called the degree of C being an e- neighborhood of x. (4) C𝑙𝑒(A) 𝜖, 𝓅(𝑋), defined by C 𝑙𝑒(A)(x)= 1 − 𝒩 𝑥 𝑒 (X~A) (∀𝑥 ∈ 𝑋), is called a fuzzifying e-cluster point operator of a subset A⊆X, x∈X with C𝑙𝑒(A)(x)> 0 is called a fuzzifying e-cluster point of A (write as 𝑥∈ 𝐶𝑙𝑒(𝐴)), and C𝑙𝑒(A)(x) is called the degree of x being a fuzzifying e-cluster point of A. (5) 𝐼𝑛𝑡𝑒(A)∈ 𝓅(𝑋), defined by 𝐼𝑛𝑡𝑒(A)(x)= 𝒩 𝑥 𝑒 (A) (∀𝑥 ∈ 𝑋), Is called a fuzzifying e-interior point operator of a subset A⊆X, x∈X with 𝑖𝑛𝑡𝑒(A)(x)> 0 is called a fuzzifying e-cluster point of A (write as𝑥𝜖 𝐼𝑛𝑡𝑒(𝐴)), and 𝐼𝑛𝑡𝑒(A)(x) is called the degree of x being fuzzifying e-interior point of A. Definition 4. (cf.[𝟖]) Let (X, 𝓅) and (Υ, 𝜇) be two fuzzifying topological spaces, 𝑓𝜖Υ𝑥 Satisfying the following condition is called fuzzifying e-continuous function (write as 𝑓𝜖 𝓅): 𝓅∈(𝑓)=⋀𝜇∈𝑝(Υ)min (1,1-𝜇(𝑈) + 𝓅𝑒(𝑓−1 (𝑈)))> 0, Where 𝓅(𝑓) is called the degree of 𝑓 being an e-continuous function. Definition 5. (cf. [𝟗]) Let (X,𝓅) and (Υ, 𝜇) be two fuzzifying topological spaces, 𝑓𝜖Υ𝑥 satisfying the following condition is called fuzzifying e-irresolute function (write as𝑓𝜖 𝓅𝑒): 𝓅𝑒(𝑓) = ⋀𝑢∈𝑝(Υ)min(1,1-𝜇𝑒(U)+ 𝓅𝑒(𝑓−1 (𝑈)) > 0, where 𝓅𝑒(𝑓) is called the degree of f being an e-irresolute function. 3. Fuzzifying e-𝜃-neighborhood system of a point, fuzzifying e-𝜃closure of a set, fuzzifying e-𝜃- interior of a set, fuzzifying e-𝜃-open sets, and fuzzifying e-𝜃-closed sets. Definition 6. 𝒩𝒙 𝒆𝜽 ∈ 𝓅(𝑃(𝑋)), defined by 𝒩 𝒙 𝒆𝜽 (A) = ⋁𝒚∈𝒑(𝑿) max (0, 𝒩 𝑥 𝑒 (B)+∧𝑦∈𝐴 𝒩𝑦 𝑒 (X~B)-1),
- 3. Fuzzifying Topological System 3 is called a fuzzifying e-neighborhood system of a point x∈ X, C∈ P(X) with 𝒩 𝒙 𝒆𝜽 (C)> 0 is called a fuzzifying e-neighborhood of x (write as 𝐶∈ 𝒩 𝒙 𝒆𝜽 ), and 𝒩 𝒙 𝒆𝜽 (C) is called the degree of C being an e- neighborhood of x. Theorem 7. The mapping 𝒩𝑒𝜃 : 𝑋 → 𝓅𝑁 (𝑃(𝑋))( the set of all normal fuzzy subsets of P(X)), defined by 𝒩𝑒𝜃 (x)= 𝒩 𝒙 𝒆𝜽 (∀𝑥 ∈ 𝑋), has the following properties: (1) ⊨ 𝐴∈ 𝒩 𝒙 𝒆𝜽 →x∈A (∀𝑥 ∈ 𝑋 , ∀∈ 𝑃(𝑋)); (2) ⊨ A⊆B→ (𝐴∈ 𝒩 𝒙 𝒆𝜽 → 𝐵∈ 𝒩 𝒙 𝒆𝜽 )(∀𝑥 ∈ 𝑋, ∀𝐴, 𝐵 ∈ 𝑃(𝑋)); (3) ⊨ (𝐴∈ 𝒩𝒙 𝒆𝜽 )⋀( 𝐵∈ 𝒩 𝒙 𝒆𝜽 )↔ 𝐴⋂𝐵∈ 𝒩 𝒙 𝒆𝜽 (∀𝑥 ∈ 𝑋, ∀𝐴, 𝐵 ∈ 𝑃(𝑋)). Proof. (1) If[𝐴∈ 𝒩 𝒙 𝒆𝜽] = 0, then the result holds. If [𝐴∈ 𝒩 𝒙 𝒆𝜽] > 0, then ∨𝐵∈𝑃(𝑋)max (0, 𝒩𝑥 𝑒(𝐵) +∧𝑦∈𝐴 𝒩𝑦 𝑒(𝑋~𝐵) − 1) =∨𝐵∈𝑃(𝑋)max (0, 𝒩 𝑥 𝑒(𝐵) + [ C 𝑙𝑒(B) ⊆ A] − 1)> 0, and thus exists a C∈ 𝑃(𝑋) such that 𝒩 𝒙 𝒆 (C)+[C 𝑙𝑒(C) ⊆ A] − 1 > 0. From Theorems 4.2(1) and 5.5(3) in [8], we have [C 𝑙𝑒(C) ⊆ A] > 1 − 𝒩 𝒙 𝒆 (C)≥ 1 − |𝑥 ∈ 𝐶| ≥ 1 − C 𝑙𝑒(C)(x). Therefore, [C 𝑙𝑒(C) ⊆ A] = ⋀𝑦∈𝐴(1 − C 𝑙𝑒(C)(y) > 1 − C 𝑙𝑒(C)(x), and so x∈A. Otherwise, if x∈A, then ⋀𝑦∈𝐴(1 − C 𝑙𝑒(C)(y) > 1 − C 𝑙𝑒(C)(x).