Approximate orthogonality of complex fuzzy sets and approximately orthogonality preserving operators

Abstract

This paper presents the notion of approximate orthogonality of complex fuzzy sets, which is a generalization of the existing concept. Based on the new concept, we present some basic properties of the approximate orthogonality of complex fuzzy sets. Furthermore, we discuss approximately orthogonality preserving with respect to complement, union and intersection operations of complex fuzzy sets.

Keywords

Complex fuzzy sets operators approximate orthogonality approximately orthogonality preserving

1 Introduction

Complex fuzzy set theory introduced by Ramot et al. [17] is an increasingly popular generalization of fuzzy sets where traditional [0,1]-valued membership grades are extended to the unit disc of the complex plane. In recent years, complex fuzzy sets have been successfully used in various applications [1 , 18–21].

The membership grade of a complex fuzzy set can be expressed in the form of exponent, which contains an amplitude and a phase terms. In many theoretical and application studies, both terms are considered together. For example, distance measures of complex fuzzy set in [1 , 21] are defined by combining the difference between the amplitude terms and the difference between the phase terms.

As mentioned in Ramot et al. [17], the phase term of membership grade is the key feature which essentially distinguish complex fuzzy sets from traditional fuzzy sets. So some concepts and relationships in complex fuzzy theories are mostly depends on the phase term of membership grade. Dick [7] introduced the rotational invariance of operations of complex fuzzy sets. Bi et al. [2, 10] introduced the parallelity and approximate parallelity between complex fuzzy sets. Hu et al. [8] introduced the orthogonality relation of complex fuzzy sets, which is a dependency relation, i.e., is symmetric and reflexive, but is not transitive. Then they discussed orthogonality preserving with respect to various complex fuzzy operators.

However, the exact orthogonality of two complex fuzzy sets and operators which exactly preserve orthogonality restrict the potential applications of concepts of orthogonality in complex fuzzy sets. In this paper, we propose the concept of approximate orthogonality between complex fuzzy sets, which is a generalization of Hu et al.’s [8] concept. Based on the new concept, we discuss the approximately orthogonality preserving property of Ramot et al.’s [13] complex fuzzy operations.

The remainder of this paper is organized as follows. In Section 2, we review the orthogonality and orthogonality preserving operators introduced by Hu et al. [8]. The concepts of approximate parallelity of complex fuzzy sets are introduced in Section 3. Then, we discuss whether parallelity can be preserved for Ramot et al.’s [17] complex fuzzy operations in Section 4. Finally, conclusions are presented in Section 5.

2 Related work

2.1 Orthogonality and orthogonality preserving operators

Let X be a universe and D be the set of complex numbers whose modulus is less than or equal to 1. A mapping: X → D is called a complex fuzzy set of X, which can be given as $A = {(x, r_{A} (x) \cdot e^{j θ_{A} (x)} | x \in X}$ (1) where the phase term θ_A (x) is real-valued number and the amplitude term r_A (x) ∈ [0, 1]. Since e^jθ_A(x) is a periodic function and its periodicity law is 2π. In this paper, the phase term is limited to the interval [0, 2π), i.e., θ_A (x) ∈ [0, 2π). If it is out of range, then calculate the least positive residue modulo 2π to obtain a new phase term within valid range. We use the notation CF (X) to denote the set of all complex fuzzy sets on X.

In this paper, we only consider the phase term of membership vector. Unless otherwise stated, we always assume that r_A (x) ≠0 for all x ∈ X.

Hu et al. [8] introduced a dependency relation of complex fuzzy sets as follows,

Definition 1. Let A, B ∈ CF (X), and μ_A (x) and μ_B (x) be their membership, respectively. Then A and B are said to be orthogonal, denoted by A ⊥ B, if we have $< μ_{A} (x), μ_{B} (x) > = 0$ (2) for any x ∈ X, where <μ_A (x) , μ_B (x)> is the inner product of μ_A (x) , μ_B (x) ∈ D.

Moreover, Hu et al. [8] introduced the concept of orthogonality preserving mappings as follows,

Definition 2. A mapping f : CF (X) → CF (X) is orthogonality preserving if we have $f (A) ⊥ f (B)$ (3) for any A, B ∈ CF (X) with A ⊥ B.

Then they discussed the complex fuzzy operators which exactly preserve orthogonality.

3 Approximate orthogonality

This section introduces the concept of approximate orthogonality between complex fuzzy sets. For the orthogonality relation introduced above we may define its approximate extension.

Definition 3. Let A, B ∈ CF (X), μ_A (x) = r_A (x) · e^jθ_A(x) and μ_B (x) = r_B (x) · e^jθ_B(x) be their membership functions respectively and ɛ ∈ [0, π/2). If we have

$min {| θ_{A} (x) - θ_{B} (x) |, 2 π - | θ_{A} (x) - θ_{B} (x) |} - \frac{π}{2} \leq ɛ$ (4) for each x ∈ X, then A and B are said to be ɛ-orthogonal, denoted A ⊥ _ɛB.

Remark 1. (i) In the above definition, min {|θ_A (x) - θ_B (x) |, 2π - |θ_A (x) - θ_B (x) |} is the included angle between membership functions μ_A (x) and μ_B (x).

(ii) ɛ represents the degree of orthogonal between A and B, for ɛ = 0, approximate orthogonality coincides with the notion of orthogonality.

See Fig. 1 for an example, two membership vectors A and C are orthogonality, and two membership vectors A and B are ɛ-orthogonality.

Obviously, A ⊥ B is a special case of A ⊥ _ɛB in which ɛ = 0.

Fig.1

A ⊥ C and A ⊥ _ɛB.

The following basic properties hold.

Theorem 1. Let A, B ∈ CF (X), and ɛ, ɛ₁, ɛ₂ ∈ [0, π/2).

A ⊥ _ɛB if and only if B ⊥ _ɛA

A ⊥ _ɛ
₁B ⇒ A ⊥ _ɛ
₂B if ɛ₁ ≤ ɛ₂.

Theorem 2. The approximate orthogonality of complex fuzzy sets is rotationally invariant, i.e., $A ⊥_{ɛ} B if and only if {Rot}_{θ} (A) ⊥_{ɛ} {Rot}_{θ} (B)$ (5) for any θ, ɛ and A, B ∈ CF (X). Where Rot_θ (x) = x · e^jθ for any x ∈ D.

Proof. This is clear: the included angle is invariant under a rotation. ■

Theorem 3. The approximate orthogonality of complex fuzzy sets is reflection invariant, i.e., $A ⊥_{ɛ} B if and only if Ref (A) ⊥_{ɛ} Ref (B)$ (6) for any ɛ and A, B ∈ CF (X). Where Ref (x) = x · e^j(-θ) for any x · e^jθ ∈ D.

Proof. This also is clear: the included angle is invariant under a reflection. ■

4 (ɛ, δ)-Orthogonality preserving operators

In this section, we first extend the definition of orthogonality preserving operators to (ɛ, δ)-orthogonality preserving operators in the following manner.

Definition 4. Let f : CF (X) ⁿ → CF (X) be an n-order function, and ɛ, δ ∈ [0, π/2). f is (ɛ, δ)-orthogonality preserving if $f (A_{1}, A_{2}, . . ., A_{n}) ⊥_{δ} f (B_{1}, B_{2}, . . ., B_{n})$ (7) for any A_i ⊥ _ɛB_i (i = 1, 2, . . . , n).

Then we study the approximate orthogonality preserving property of Ramot et al.’s [17] complex fuzzy operations.

4.1 Set rotation and reflection operations

Set rotation and reflection operations are two complex fuzzy set operations introduced by Ramot et al. [17].

Let A ∈ CF (X), and μ_A (x) = r_A (x) · e^jθ_A(x) be its membership function.

The Rotation of A by θ radians, denoted Rot_θ (A), is defined as ${Rot}_{θ} (μ_{A} (x)) = r_{A} (x) \cdot e^{j (θ_{A} (x) + θ)}$ (8)

The reflection of A, denoted Ref (A), is defined as $Ref (μ_{A} (x)) = r_{A} (x) \cdot e^{- j θ_{A} (x)}$ (9)

Theorem 4. For any ɛ ∈ [0, π/2), the set rotation and reflection operations are (ɛ, ɛ)-orthogonality preserving.

Proof. Trivial form Theorems 2 and 3. ■

4.2 Complex fuzzy complement

Let A ∈ CF (X), and μ_A (x) = r_A (x) · e^jθ_A(x) be its membership function. The following three complex fuzzy complement are defined as follows (see [17]). $μ_{\neg_{1} A} (x) = (1 - r_{A} (x)) \cdot e^{j (- θ_{A} (x))},$ (10) $μ_{\neg_{2} A} (x) = (1 - r_{A} (x)) \cdot e^{j (θ_{A} (x))},$ (11) $μ_{\neg_{3} A} (x) = (1 - r_{A} (x)) \cdot e^{j (θ_{A} (x) + π)} .$ (12)

Theorem 5. The complex fuzzy complement ¬_i (i = 1, 2, 3) are (ɛ, ɛ)-orthogonality preserving.

Proof. Trivial form Theorems 2 and 3. ■

From this result, we have ¬_iA ⊥ _ɛ ¬ _iB (i ∈ {1, 2, 3}) from A ⊥ _ɛB.

Example 1. Suppose that two complex fuzzy sets A and B on X = {x₁, x₂} are given as follows: $\begin{matrix} A = & \frac{0.1 e^{j 0.51 π}}{x_{1}} + \frac{0.4 e^{j 0.71 π}}{x_{2}}, \\ B = & \frac{0.5 e^{j π}}{x_{1}} + \frac{0.7 e^{j 1.2 π}}{x_{2}} \end{matrix}$ Then we have A ⊥ _0.01πB and $\begin{matrix} \neg_{1} A = & \frac{0.9 e^{j 1.49 π}}{x_{1}} + \frac{0.6 e^{j 1.29 π}}{x_{2}}, \\ \neg_{1} B = & \frac{0.5 e^{j π}}{x_{1}} + \frac{0.3 e^{j 0.8 π}}{x_{2}}, \\ \neg_{2} A = & \frac{0.9 e^{j 0.51 π}}{x_{1}} + \frac{0.6 e^{j 0.71 π}}{x_{2}}, \\ \neg_{2} B = & \frac{0.5 e^{j π}}{x_{1}} + \frac{0.3 e^{j 1.2 π}}{x_{2}}, \\ \neg_{3} A = & \frac{0.9 e^{j 1.51 π}}{x_{1}} + \frac{0.6 e^{j 1.71 π}}{x_{2}}, \\ \neg_{3} B = & \frac{0.5 e^{j 2 π}}{x_{1}} + \frac{0.3 e^{j 0.2 π}}{x_{2}} . \end{matrix}$ Then we can verify that ¬_iA ⊥ _0.01π ¬ _iB for all i ∈ {1, 2, 3}.

4.3 Complex fuzzy union

Let A, B ∈ CF (X), μ_A (x) = r_A (x) · e^jθ_A(x) and μ_B (x) = r_B (x) · e^jθ_B(x) be their membership functions respectively. Ramot et al. [17] introduced the complex fuzzy union of A and B as follows: $μ_{A \cup B} (x) = (r_{A} (x) \oplus r_{B} (x)) \cdot e^{j θ_{A \otimes B} (x)}$ (13) where ⊕ represents a t-conorm.

The following functions are possibilities of θ_A⊗B.

$Sum : θ_{A + B} = θ_{A} + θ_{B}$ (14) $Max : θ_{A \lor B} = max (θ_{A}, θ_{B})$ (15) $Min : θ_{A \land B} = min (θ_{A}, θ_{B})$ (16) $Product : θ_{A • B} = θ_{A} \cdot θ_{B}$ (17) $Difference : θ_{A - B} = θ_{A} - θ_{B}$ (18)

Remark 2. Since the phase terms are limited to the interval [0, 2π), then the above functions +, •and - are addition modulo 2π, multiplication modulo 2π and subtraction modulo 2π, respectively.

Theorem 6. Suppose A ⊥ _ɛB, then we have $A \cup C ⊥_{ɛ} B \cup C$ where ⊗ ∈ {+ , -}.

Proof. If ⊗ =+. Then for any x ∈ X, θ_A∪C (x) = θ_A (x) + θ_C (x) and θ_B∪C (x) = θ_B (x) + θ_C (x) if ⊗ =+. Since the included angle is invariant under a rotation. The included angle between membership vectors μ_A∪C (x) and μ_B∪C (x) is the same to that between μ_A (x) and μ_B (x). Thus, we have A ∪ C ⊥ _ɛB ∪ C from A ⊥ _ɛB when ⊗ =+.

Similarly, we can prove the case of ⊗ =-. ■

Example 2. Suppose that three complex fuzzy sets A, B and C on X = {x₁, x₂, x₃} are given as follows: $\begin{matrix} A = & \frac{0.4 e^{j 0.21 π}}{x_{1}} + \frac{0.4 e^{j 0.1 π}}{x_{2}} + \frac{0.3 e^{j 0.3 π}}{x_{3}}, \\ B = & \frac{0.7 e^{j 0.7 π}}{x_{1}} + \frac{0.6 e^{j 0.61 π}}{x_{2}} + \frac{0.7 e^{j 0.8 π}}{x_{3}}, \\ C = & \frac{0.5 e^{j 0.3 π}}{x_{1}} + \frac{0.1 e^{j 0.5 π}}{x_{2}} + \frac{0.7 e^{j 0.6 π}}{x_{3}} . \end{matrix}$

Then we have A ⊥ _0.01πB.

If ⊕ is max union function and ⊗ =+, then we have $\begin{matrix} \begin{matrix} A \cup C = \frac{0.5 e^{j 0.51 π}}{x_{1}} + \frac{0.4 e^{j 0.6 π}}{x_{2}} + \frac{0.7 e^{j 0.9 π}}{x_{3}}, \\ B \cup C = \frac{0.7 e^{j π}}{x_{1}} + \frac{0.6 e^{j 1.11 π}}{x_{2}} + \frac{0.8 e^{j 1.4 π}}{x_{3}} . \end{matrix} \end{matrix}$ We can verify that A ∪ C ⊥ _0.01πB ∪ C.

Theorem 7. If ⊗ ∈ {∨ , ∧ , •}. Then, for any ɛ, there exists complex fuzzy sets A, B, C such that A ⊥ _ɛB and A ∪ C = B ∪ C.

Proof. Let A ≡ 0.5e^j0.5π+ɛ and B ≡ 0.5e^jπ. We can verify that A ⊥ _ɛB.

If ⊗ =∨. Let C ≡ 0.5e^j1.5π. We can verify that A ∪ C = B ∪ C.

If ⊗ =∧. Let C ≡ 0.5e^j0.2π. We can verify that A ∪ C = B ∪ C.

If ⊗ =•. Let C ≡ 0.5e^j0π. We can verify that A ∪ C = B ∪ C. ■

Theorem 8. Suppose A ⊥ _ɛB, A ⊥ _ɛC. Then we have $A ⊥_{ɛ} B \cup C,$ where ⊗ ∈ {∨ , ∧}.

Proof. If ⊗ =∨. Then, for any x ∈ X, θ_B∪C (x) = max {θ_B (x) , θ_C (x)}. Then the included angle between membership vectors μ_A (x) and μ_B∪C (x) is the included angle between μ_A (x) and μ_B (x) or the included angle between μ_A (x) and μ_C (x). This is clear A ⊥ _ɛB ∪ C from A ⊥ _ɛB, A ⊥ _ɛC if ⊗ =∨.

Similarly, we can prove the case of ⊗ =∧. ■

Corollary 1. Suppose A ⊥ _{ɛ
₁}B, A ⊥ _{ɛ
₂}C. Then we have $A ⊥_{ɛ} B \cup C,$ where ɛ = max(ɛ₁, ɛ₂) and ⊗ ∈ {∨ , ∧}.

Proof. From Theorem 1(ii), we have A ⊥ _ɛB from A ⊥ _{ɛ
₁}B, and A ⊥ _ɛC from A ⊥ _{ɛ
₁}C. Then we can prove this corollary from above theorem 8. ■

Example 3. Suppose that three complex fuzzy sets A, B and C in U = {x₁, x₂, x₃} are given as follows: $\begin{matrix} A = & \frac{0.3 e^{j 0.21 π}}{x_{1}} + \frac{0.5 e^{j 0.11 π}}{x_{2}} + \frac{0.2 e^{j 1.3 π}}{x_{3}}, \\ B = & \frac{0.2 e^{j 1.7 π}}{x_{1}} + \frac{0.1 e^{j 0.6 π}}{x_{2}} + \frac{0.4 e^{j 1.81 π}}{x_{3}}, \\ C = & \frac{0.5 e^{j 0.7 π}}{x_{1}} + \frac{0.6 e^{j 0.63 π}}{x_{2}} + \frac{0.8 e^{j 0.82 π}}{x_{3}} . \end{matrix}$ Then we have A ⊥ _0.01πB and A ⊥ _0.02πC.

If ⊕ is max union function and ⊗ =∨, then $\begin{matrix} \begin{matrix} B \cup C = \frac{0.5 e^{j 1.7 π}}{x_{1}} + \frac{0.6 e^{j 0.63 π}}{x_{2}} + \frac{0.8 e^{j 1.81 π}}{x_{3}} . \end{matrix} \end{matrix}$

We can verify that A ⊥ _0.02π (B ∪ C).

Corollary 2. Suppose A ⊥ _{ɛ
_i}B_i (i = 1, 2, . . . , n). Then we have $A ⊥_{ɛ} ⋃_{i = 1}^{n} B_{i},$ where $ɛ = max_{i} (ɛ_{i})$ and ⊗ ∈ {∨ , ∧}.

Theorem 9. If ⊗ ∈ {+ , -}. Then, for any ɛ, there exists complex fuzzy sets A, B, C such that A ⊥ _ɛB, A ⊥ _ɛC and A = B ∪ C.

Proof. If ⊗ =+. Let A ≡ 0.5e^jπ, B ≡ 0.5e^j1.5π+ɛ and C ≡ 0.5e^j1.5π-ɛ. We can verify that A ⊥ _ɛB, A ⊥ _ɛC and A = B ∪ C.

If ⊗ =-. Let A ≡ 0.5e^jπ, B ≡ 0.5e^j1.5π+ɛ and C ≡ 0.5e^j0.5π+ɛ. We can verify that A ⊥ _ɛB, A ⊥ _ɛC and A = B ∪ C. ■

4.4 Complex fuzzy intersection

Let A, B ∈ CF (X), μ_A (x) = r_A (x) · e^jθ_A(x) and μ_B (x) = r_B (x) · e^jθ_B(x) be their membership functions respectively. Ramot et al. [17] introduced the complex fuzzy intersection of A and B as follows: $μ_{A \cup B} (x) = (r_{A} (x) ★ r_{B} (x)) \cdot e^{j θ_{A \otimes B} (x)}$ (19) where ★ represents a t-norm, θ_A⊗B is one of functions in (14)-(18).

Theorem 10. Suppose A ⊥ _ɛB, then we have $A \cap C ⊥_{ɛ} B \cap C$ where ⊗ ∈ {+ , -}.

Proof. Similar to Theorem 6. ■

Theorem 11. If ⊗ ∈ {∨ , ∧ , •}. Then, for any ɛ, there exists complex fuzzy sets A, B, C such that A ⊥ _ɛB and A ∩ C = B ∩ C.

Proof. Similar to Theorem 7. ■

Theorem 12. Suppose A ⊥ _ɛB, A ⊥ _ɛC. Then we have $A ⊥_{ɛ} B \cap C,$ where ⊗ ∈ {∨ , ∧}.

Proof. Similar to Theorem 8. ■

Corollary 3. Suppose A ⊥ _{ɛ
_i}B_i (i = 1, 2, . . . , n). Then we have $A ⊥_{ɛ} ⋂_{i = 1}^{n} B_{i},$ where $ɛ = max_{i} (ɛ_{i})$ and ⊗ ∈ {∨ , ∧}.

Proof. Similar to Corollary 2. ■

Theorem 13. If ⊗ ∈ {+ , -}. Then, for any ɛ, there exists complex fuzzy sets A, B, C such that A ⊥ _ɛB, A ⊥ _ɛC and A = B ∩ C.

Proof. Similar to Theorem 9. ■

5 Conclusion

In this paper, we considered a weaker orthogonality relationship, denoted by “approximate” orthogonality, and discussed approximately orthogonality preserving operators. Obviously, the new concepts proposed in the present paper are extensions of the old concepts obtained by Hu et al. [8]. The new concepts are more comprehensive than the old ones because the latter are special cases of the former respectively.

We presented some results on approximately orthogonality preserving with respect to various operators of complex fuzzy sets, which can be summarized as follows (also see Table 1)

Table 1

Approximately preserve orthogonality of complex fuzzy operators

	+	-	∨	∧	•
A ⊥ _ɛB ⇒ A ∘ C ⊥ _ɛB ∘ C	√	√	×	×	×
A ⊥ _ɛB, A ⊥ _ɛC ⇒ A ⊥ _ɛB ∘ C	×	×	√	√	×

√ and × represent the corresponding property holds and does not hold respectively.

(i) A ⊥ _ɛB ⇒ A ∘ C ⊥ _ɛB ∘ C, where ⊗ ∈ {+ , -} (see Theorems 6 and 10);

(ii) A ⊥ _{ɛ
₁}B, A ⊥ _{ɛ
₂}C ⇒ A ⊥ _{max(ɛ₁,ɛ₂)}B ∘ C, where ⊗ ∈ {∨ , ∧} (see Theorems 7 and 11).

In this paper, we only considered approximately orthogonality preserving with respect to several Ramot et al.’s complex fuzzy operations. Naturally, a more detailed discussion of other complex fuzzy operations, such as the operations proposed by Zhang et al. [21], will be necessary and interesting. Another problem we need to investigate in the future is the approximately orthogonality preserving with respect to complex fuzzy reasoning methods.

Footnotes

Acknowledgments

This project was supported by the Opening Foundation of Guangxi Colleges and Universities Key Laboratory of Complex System Optimization and Big Data Processing (Grant No. 2017CSOBDP0103).

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