Distances of complex fuzzy sets and continuity of complex fuzzy operations

Abstract

A complex fuzzy set is a set whose membership values are vectors in the unit circle in the complex plane. To enhance and extend the applicability of complex fuzzy sets, this paper investigates and develops different types of distance measures for complex fuzzy sets. Several distance measures of complex fuzzy sets are introduced. After that, the application of these distances to continuity problems of complex fuzzy operations is given.

Keywords

Distances continuity complex fuzzy sets complex fuzzy operations.

1 Introduction

Complex fuzzy sets [14] are an extended form of fuzzy set and have been successfully used in various fields, e.g., complex fuzzy logic systems [4 , 21], time series prediction [3 , 20] and image restoration [11].

In the aforementioned practical applications, distance measures between complex fuzzy sets are of high importance. In the modeling of complex neurofuzzy system [9], traditional measurement of complex numbers was used as a measure of model error. In the analysis of time-series forecasting using complex neurofuzzy system [3], several traditional measurements were used to measure the forecast error. A product-sum aggregation operator was developed for multiple periodic factor prediction problems [12], it is continuous with respect to the modulus of complex fuzzy sets. By definition of the complex fuzzy set [14], each complex grade of membership is defined by an amplitude term and a phase term. So a simpler method to calculate the difference between two complex fuzzy sets is combining the difference between the amplitude terms and the difference between the phase terms. In this way, Zhang et al. [22] proposed a distance measure for complex fuzzy sets and used to define δ-equality of complex fuzzy sets, Alkouri et al. [1] introduced several distance measures for complex fuzzy sets. However, the distance measure defined in [1, 22] ignored the fact that the phase term of a complex fuzzy set is a periodic function. They also ignored the the cyclic representation of complex membership vector in the polar plane. This may cause some results which are not consistent with our intuition. As shown in Fig. 1, A ≡ e^j0.2π, B ≡ e^j1.7π, C ≡ e^j1.9π, the distance between A and B is closer than the distance between A and C in our vision. But by distance measures in [1, 22], we have d (A, B) > d (A, C), which is not consistent with our intuition. Therefore, we propose new distance measures for complex fuzzy sets and compare them with some existing distance measures.

Fig.1

Membership vectors A, B and C.

In order to apply complex fuzzy sets to aforementioned practical applications more effectively, we shall pay more attention to distance measures of complex fuzzy sets. After giving the distance measures for complex fuzzy sets, we then apply them to continuity problems of complex fuzzy operations. We also examine continuity and uniform continuity properties of complex fuzzy inference methods in these metrics. The continuity results of this paper might serve as a certain criteria for choices of complex fuzzy inference methods in some of real applications.

The paper is organized as follows. In Section 2, we first review related work on complex fuzzy sets. In Section 3, we review some known distance measures and then introduced new several distance measures for complex fuzzy sets. In Section 4, equivalence of these metrics are established. In Section 5, we discuss continuity properties for complex fuzzy complement, complex fuzzy union and complex fuzzy intersection whereas in Section 6 we discuss continuity properties for complex fuzzy inference methods. Conclusions are presented in Section 7.

2 Distances for complex fuzzy sets

Let U = {x₁, x₂, …, x_n} be a universe of discourse, F^C (U) be the set of all complex fuzzy sets on U. The complex fuzzy set A may be represented as the set of ordered pairs $A = {(x, μ_{A} (x)) | x \in U}$ where membership function μ_A (x) is of the form r_A (x) · e^jw_A(x), $j = \sqrt{- 1}$ , the amplitude term r_A (x) and the phase term w_A (x) are both real-valued, and r_A (x) ∈ [0, 1]. Because e^jw_A(x) is a periodic function whose periodicity law is 2π, let w_A (x) ∈ [0, 2π). If a phase term is out of range, then compute the least positive residue modulo 2π in order to get a new phase term within valid range.

In this paper, we only consider the cases of finite universes. Naturally, some conclusions presented in this paper still are valid for cases of infinite universes.

Let A = {(x, μ_A (x)) |x ∈ U} and B = {(x, }{μ_B (x)) |x ∈ U} be two CFSs.

Zhang et al. [22] proposed the following distance measure between CFSs A and B:

$\begin{matrix} d_{Z} (A, B) = & max (sup_{x \in U} | r_{A} (x) - r_{B} (x) |, \\ \frac{1}{2 π} sup_{x \in U} | w_{A} (x) - w_{B} (x) |) . \end{matrix}$ (1)

Alkouri et al. [1] defined the following distance measures between CFSs A and B:

The Hamming distance:

$\begin{matrix} d_{H} (A, B) = & \frac{1}{2} (\sum_{i = 1}^{n} | r_{A} (x_{i}) - r_{B} (x_{i}) | \\ + \frac{1}{2 π} \sum_{i = 1}^{n} | w_{A} (x_{i}) - w_{B} (x_{i}) |) . \end{matrix}$ (2)

The normalized Hamming distance:

$\begin{matrix} d_{nH} (A, B) = & \frac{1}{2 n} (\sum_{i = 1}^{n} | r_{A} (x_{i}) - r_{B} (x_{i}) | \\ + \frac{1}{2 π} \sum_{i = 1}^{n} | w_{A} (x_{i}) - w_{B} (x_{i}) |) . \end{matrix}$ (3)

The Euclidean distance:

$\begin{matrix} d_{E} (A, B) = & [\frac{1}{2} (\sum_{i = 1}^{n} ((r_{A} (x_{i}) - r_{B} (x_{i}))^{2} \\ + \frac{1}{4 π^{2}} (w_{A} (x_{i}) - w_{B} (x_{i}))^{2}))]^{1 / 2} . \end{matrix}$ (4)

The normalized Euclidean distance:

$\begin{matrix} d_{nE} (A, B) = & [\frac{1}{2 n} (\sum_{i = 1}^{n} ((r_{A} (x_{i}) - r_{B} (x_{i}))^{2} \\ + \frac{1}{4 π^{2}} (w_{A} (x_{i}) - w_{B} (x_{i}))^{2}))]^{1 / 2} . \end{matrix}$ (5)

Clearly, it is easy to notice that the distances formulas (1)-(5) have the following results: $\begin{matrix} 0 \leq d_{Z} (A, B) \leq 1, \\ 0 \leq d_{H} (A, B) \leq n, \\ 0 \leq d_{nH} (A, B) \leq 1, \\ 0 \leq d_{E} (A, B) \leq \sqrt{n}, \\ 0 \leq d_{nE} (A, B) \leq 1, \end{matrix}$

Distance measure is a term that describes the difference between complex fuzzy sets. Now, we propose some new distance measures. Let A, B be two complex fuzzy sets in U and p be a parameter satisfying 1≤ p ≤ ∞, denote

The Minkowski distances: $\begin{matrix} d_{p} (A, B) = \\ [\frac{1}{2} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{B} (x_{i}) |^{p} \\ + \frac{1}{π^{p}} (min (| w_{A} (x_{i}) - w_{B} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{B} (x_{i}) |)^{p}))]^{1 / p} . \end{matrix}$ (6) The normalized Minkowski distances: $\begin{matrix} l_{p} (A, B) = \\ [\frac{1}{2 n} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{B} (x_{i}) |^{p} \\ + \frac{1}{π^{p}} (min (| w_{A} (x_{i}) - w_{B} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{B} (x_{i}) |)^{p}))]^{1 / p} . \end{matrix}$ (7) When p = 1, d₁ is called the Hamming distance of complex fuzzy sets: $\begin{matrix} d_{1} (A, B) = \\ \frac{1}{2} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{B} (x_{i}) | \\ + \frac{1}{π} (min (| w_{A} (x_{i}) - w_{B} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{B} (x_{i}) |))) . \end{matrix}$ (8) and l₁ is called the normalized Hamming distance of complex fuzzy sets: $\begin{matrix} l_{1} (A, B) = \\ \frac{1}{2 n} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{B} (x_{i}) | \\ + \frac{1}{π} (min (| w_{A} (x_{i}) - w_{B} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{B} (x_{i}) |))) . \end{matrix}$ (9) When p = 2, d₂ is called the Euclidean distance of complex fuzzy sets: $\begin{matrix} d_{2} (A, B) = \\ [\frac{1}{2} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{B} (x_{i}) |^{2} \\ + \frac{1}{π^{2}} (min (| w_{A} (x_{i}) - w_{B} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{B} (x_{i}) |)^{2}))]^{1 / 2} . \end{matrix}$ (10) and l₂ is called the normalized Euclidean distance of complex fuzzy sets: $\begin{matrix} l_{2} (A, B) = \\ [\frac{1}{2 n} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{B} (x_{i}) |^{2} \\ + \frac{1}{π^{2}} (min (| w_{A} (x_{i}) - w_{B} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{B} (x_{i}) |)^{2}))]^{1 / 2} . \end{matrix}$ (11) When p =∞, d_∞ called the Chebyshev distance of complex fuzzy sets: $\begin{matrix} d_{\infty} (A, B) = \\ sup_{x \in U} [max (| r_{A} (x) - r_{B} (x) |, \\ \frac{1}{π} (min (| w_{A} (x) - w_{B} (x) |, \\ 2 π - | w_{A} (x) - w_{B} (x) |)))] . \end{matrix}$ (12) To facilitate future discussion, we introduce the following lemmas (see [8,14,9 , 8,14,9]).

Lemma 1. Let f and g be bounded, real valued functions on a set U. Then $\begin{matrix} | sup_{x \in U} f (x) - sup_{x \in U} g (x) | \leq sup_{x \in U} | f (x) - g (x) |, \\ | inf_{x \in U} f (x) - inf_{x \in U} g (x) | \leq sup_{x \in U} | f (x) - g (x) | . \end{matrix}$

Lemma 2. (Minkowski’s inequality) Let (a₁, a₂, …, a_n), (b₁, b₂, …, b_n) ∈ Rⁿ, and 1≤ p < ∞. Then $(\sum_{k = 1}^{n} | a_{k} + b_{k} |^{p})^{1 / p} \leq (\sum_{k = 1}^{n} | a_{k} |^{p})^{1 / p} + (\sum_{k = 1}^{n} | b_{k} |^{p})^{1 / p}$

Lemma 3. Let a and b be non-negative real numbers, and 1 ≤ p. Then $lim_{p \to \infty} (a^{p} + b^{p})^{1 / p} = max (a, b) .$

Lemma 4. Let a > 0, and 1 ≤ p. Then $lim_{p \to \infty} (a)^{1 / p} = 1 .$

Lemma 5. Let a, b ∈ [0, 2π), denote $a \overset{2 π}{-} b = min (| a - b |, 2 π - | a - b |),$

$a +_{2 π} b = {\begin{matrix} a + b, & a + b < 2 π \\ a + b - 2 π, & a + b \geq 2 π \end{matrix}$ and $a -_{2 π} b = {\begin{matrix} a - b + 2 π, & a - b < 0 \\ a - b, & a - b \geq 0 \end{matrix}$

Then, for any a, b ∈ [0, 2π) $\begin{matrix} (i) & a -_{2 π} b = a +_{2 π} (2 π - b) \\ (ii) & (a \overset{2 π}{-} b) + (b \overset{2 π}{-} c) \geq a \overset{2 π}{-} c \\ (iii) & (a +_{2 π} c) \overset{2 π}{-} (b +_{2 π} c) = a \overset{2 π}{-} b \\ (iv) & (a -_{2 π} c) \overset{2 π}{-} (b -_{2 π} c) = a \overset{2 π}{-} b \\ (v) & (a +_{2 π} c) \overset{2 π}{-} (b +_{2 π} d) \leq (a \overset{2 π}{-} b) + (c \overset{2 π}{-} d) \\ (vi) & (a -_{2 π} c) \overset{2 π}{-} (b -_{2 π} d) \leq (a \overset{2 π}{-} b) + (c \overset{2 π}{-} d) \end{matrix}$

Proof. Property (i) can be easily proved.

(ii) We have cases to prove it, for instance a ≥ b ≥ c.

Case 1, a - b ≤ π, b - c ≤ π then we have

$\begin{matrix} min (| a - b |, 2 π - | a - b |) \\ + min (| b - c |, 2 π - | b - c |) \\ = | a - b | + | b - c | \geq | a - c | . \end{matrix}$

Case 2, a - b ≤ π, b - c > π then we have a - c ≥ b - c > π and min(|a - c|, 2π - |a - c|) =2π - a + c, so

$\begin{matrix} min (| a - b |, 2 π - | a - b |) \\ + min (| b - c |, 2 π - | b - c |) \\ = a - b + 2 π - b + c \\ \geq 2 π - b + c \\ \geq 2 π - a + c \\ = min (| a - c |, 2 π - | a - c |) . \end{matrix}$ Analogously, for the proof of other cases.

The proofs of (iii)-(iv) are similar to that of (ii).

(v) From (ii) and (iii), $\begin{matrix} (a +_{2 π} c) \overset{2 π}{-} (b +_{2 π} d) \\ \leq (a +_{2 π} c) \overset{2 π}{-} (c +_{2 π} b) + (c +_{2 π} b) \overset{2 π}{-} (b +_{2 π} d) \\ = (a \overset{2 π}{-} b) + (c \overset{2 π}{-} d) \end{matrix}$

The proof of (vi) is similar to that of(v).

Theorem 1. All distances in (6)-(12) are distance measures of complex fuzzy sets.

Proof. It is easy to see that all the distances above satisfy the following conditions (1) and (2), i.e., for any A, B ∈ F^C (U),

(1) d (A, B) ≥0, d (A, B) =0 if and only if A = B;

(2) d (A, B) = d (B, A).

So, we just go to prove the Triangular Inequality (T.I.) for Minkowski distances d_p, 1≤ p < ∞. Analogously, for the proof of other distances.

Let A, B, C ∈ F^C (U), then by Lemma 2 and Lemma 5, we have

$\begin{matrix} d (A, C) + d (C, B) \\ = [\frac{1}{2} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{C} (x_{i}) |^{p} \\ + \frac{1}{π^{p}} (min (| w_{A} (x_{i}) - w_{C} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{C} (x_{i}) |)^{p}))]^{1 / p} \\ + [\frac{1}{2} \sum_{i = 1}^{n} (| r_{C} (x_{i}) - r_{B} (x_{i}) |^{p} \\ + \frac{1}{π^{p}} (min (| w_{C} (x_{i}) - w_{B} (x_{i}) |, \\ 2 π - | w_{C} (x_{i}) - w_{B} (x_{i}) |)^{p}))]^{1 / p} \\ \geq [\frac{1}{2} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{C} (x_{i}) + r_{C} (x_{i}) - r_{B} (x_{i}) |^{p} \\ + \frac{1}{π^{p}} (min (| w_{A} (x_{i}) - w_{C} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{C} (x_{i}) |) \\ + min (| w_{B} (x_{i}) - w_{C} (x_{i}) |, \\ 2 π - | w_{B} (x_{i}) - w_{C} (x_{i}) |)^{p}))]^{1 / p} \\ \geq [\frac{1}{2} \sum_{i = 1}^{n} (| r_{A} (x_{i}) - r_{B} (x_{i}) |^{p} \\ + min (| w_{A} (x_{i}) - w_{B} (x_{i}) |, \\ 2 π - | w_{A} (x_{i}) - w_{B} (x_{i}) |)^{p})]^{1 / p} \\ = d (A, B) . \end{matrix}$

Clearly, it is easy to notice that the distances formulas (6)–(12) have the following results: $\begin{matrix} 0 \leq d_{p} (A, B) \leq n^{1 / p}, \\ 0 \leq l_{p} (A, B) \leq 1, \end{matrix}$

When p = 1, we have $\begin{matrix} 0 \leq d_{1} (A, B) \leq n, \\ 0 \leq l_{1} (A, B) \leq 1, \end{matrix}$

When p = 2, we have $\begin{matrix} 0 \leq d_{2} (A, B) \leq \sqrt{n}, \\ 0 \leq l_{2} (A, B) \leq 1, \end{matrix}$

When p =∞, we have $\begin{matrix} 0 \leq d_{\infty} (A, B) \leq 1 . \end{matrix}$

Example 1. Let A, B and C in U = {x} be complex fuzzy sets given by $A = e^{j 0.2 π}, B = e^{j 1.7 π}, C = e^{j 1.9 π}$

Using formulas (1)-(5), we obtain the following: $\begin{matrix} d_{Z} (A, B) = 0.75, d_{Z} (A, C) = 0.85, \\ d_{nH} (A, B) = d_{H} (A, B) = 0.375, \\ d_{nH} (A, C) = d_{H} (A, C) = 0.425, \\ d_{nE} (A, B) = d_{E} (A, B) = 0.5625, \\ d_{nE} (A, C) = d_{E} (A, C) = 0.7225 . \end{matrix}$ Applying formulas (6)-(12), we obtain $\begin{matrix} d_{1} (A, B) = l_{1} (A, B) = 0.25, \\ d_{1} (A, C) = l_{1} (A, C) = 0.15, \\ d_{2} (A, B) = l_{2} (A, B) = 0.125, \\ d_{2} (A, C) = l_{2} (A, C) = 0.045, \\ d_{\infty} (A, B) = 0.5, d_{\infty} (A, B) = 0.3 . \end{matrix}$

Because periodicity of the phase term of complex fuzzy sets, in the distance measures (6)-(12), we use min(|w_A (x) - w_B (x) |, 2π - |w_A (x) - w_B (x) |) to measure the difference of the phase terms, instead of |w_A (x) - w_B (x) |. As shown in Fig. 1, the distance between A and B is closer than the distance between A and C in our vision. But the results of this example demonstrate that the distance measure described by Zhang et al. [22] and the distance measures described by Alkouri et al. [1] cannot get d (A, B) ≤ d (A, C), which is not consistent with our intuition. By distance measures in (6)-(12), we can calculate d (A, B) > d (A, C), which is more reasonable in the intuition.

3 Continuity of complex fuzzy operations

3.1 Complex Fuzzy Complement

The complement of a complex fuzzy set is an extension of the definition of traditional fuzzy complement. Ramot et al. [14] introduced following three complex fuzzy complement.

Let A be a complex fuzzy sets on U, and μ_A (x) = r_A (x) · e^jw_A(x) its membership functions. The complex fuzzy complement of A, denoted ¬A is specified by a function $μ_{\neg_{1} A} (x) = c (r_{A} (x)) \cdot e^{j (- w_{A} (x))}$ (13) $μ_{\neg_{2} A} (x) = c (r_{A} (x)) \cdot e^{j (w_{A} (x))}$ (14) $μ_{\neg_{3} A} (x) = c (r_{A} (x)) \cdot e^{j (w_{A} (x) + π)}$ (15) where c (r_A (x)) =1 - r_A (x).

Definition 1. Let d be a distance between complex fuzzy sets, ¬ : F^C (U) → F^C (U) be a complex fuzzy complement.

(i) Complex fuzzy complement ¬ is said to be uniformly continuous in metric d, if for any ɛ > 0, there exists δ > 0 such that $d (\neg A_{1}^{*}, \neg A_{2}^{*}) < ɛ$ whenever $d (A_{1}^{*}, A_{2}^{*}) < δ$ for any $A_{1}^{*}, A_{2}^{*} \in F^{C} (U)$ ;

(ii) Complex fuzzy complement ¬ is said to be continuous at A ∈ F^C (U) in metric d, if for any ɛ > 0, there exists δ > 0 such that d (¬ A^*, ¬ A) < ɛ whenever d (A^*, A) < δ for any A^* ∈ F^C (U).

From Definition 1 we know that if a complex fuzzy complement ¬ is uniformly continuous in a metric d then it is continuous in this metric, but not vice versa.

Theorem 2. Complex fuzzy complement ¬_i, i = 1, 2, is uniformly continuous in metric

d ∈ {d_p, l_p, ρ_p, d_Z, d_H, d_nH, d_E, d_nE}. Complex fuzzy complement ¬₃ is uniformly continuous in metric d ∈ {d_p, l_p, ρ_p}, but not uniformly continuous in metric d ∈ {d_Z, d_H, d_nH, d_E, d_nE}.

Proof. It is easy to verify, for

d ∈ {d_p, l_p, d_Z, d_H, d_nH, d_E, d_nE}, we have

$\begin{matrix} d (\neg_{1} A_{1}^{*}, \neg_{1} A_{2}^{*}) = d (A_{1}^{*}, A_{2}^{*}), \\ d (\neg_{2} A_{1}^{*}, \neg_{2} A_{2}^{*}) = d (A_{1}^{*}, A_{2}^{*}), \end{matrix}$ for any $A_{1}^{*}, A_{2}^{*} \in F^{C} (U)$ . And for d ∈ {d_p, ρ_p}, $\begin{matrix} d (\neg_{3} A_{1}^{*}, \neg_{3} A_{2}^{*}) = d (A_{1}^{*}, A_{2}^{*}), \end{matrix}$ for any $A_{1}^{*}, A_{2}^{*} \in F^{C} (U)$ . Thus complex fuzzy complement ¬_i, i = 1, 2, is uniformly continuous in metric d ∈ {d_p, l_p, d_Z, d_H, d_nH, d_E, d_nE}, Complex fuzzy complement ¬₃ is uniformly continuous in metrics d_p, l_p. By Definition 1, complex fuzzy complement ¬_i, i = 1, 2, 3, is uniformly continuous in metric ρ_p.

Let ɛ = 0.5, for any π/2 > δ > 0, denote $A_{1}^{*} (x) = 0.5 e^{j (π + \frac{δ}{4})}$ , $A_{2}^{*} (x) = 0.5 e^{j (π - \frac{δ}{4})}$ for any x ∈ U. Then $d_{Z} (A_{1}^{*}, A_{2}^{*}) < δ$ , but

$d_{Z} (\neg_{3} A_{1}^{*}, \neg_{3} A_{2}^{*}) = \frac{1}{2 π} (\frac{δ}{4} - (2 π - \frac{δ}{4})) = 1 - \frac{δ}{4 π} > 0.5 .$

So ¬₃ is not uniformly continuous in metric d_Z. Hence, ¬₃ is not uniformly continuous in metric d ∈ {d_Z, d_H, d_nH, d_E, d_nE}.

From above theorem, for some ¬, it might be d (A, B) and d (¬ A, ¬ B) are both small in one distance, and d (A, B) is small but d (¬ A, ¬ B) is big in another distance, see the following Example 2,

Example 2. Assume two complex fuzzy sets A and B are given by $\begin{matrix} A = & 0.1 e^{j 1.02 π} / x_{1} + 0.2 e^{j 0.95 π} / x_{2} \\ B = & 0.1 e^{j 0.96 π} / x_{1} + 0.2 e^{j 1.03 π} / x_{2} \end{matrix}$ Then we have $\begin{matrix} \neg_{1} A = & 0.9 e^{j 0.98 π} / x_{1} + 0.8 e^{j 1.05 π} / x_{2} \\ \neg_{1} B = & 0.9 e^{j 1.04 π} / x_{1} + 0.8 e^{j 0.97 π} / x_{2} \\ \neg_{2} A = & 0.9 e^{j 1.02 π} / x_{1} + 0.8 e^{j 0.95 π} / x_{2} \\ \neg_{2} B = & 0.9 e^{j 0.96 π} / x_{1} + 0.8 e^{j 1.03 π} / x_{2} \\ \neg_{3} A = & 0.9 e^{j 0.02 π} / x_{1} + 0.8 e^{j 1.95 π} / x_{2} \\ \neg_{3} B = & 0.9 e^{j 1.96 π} / x_{1} + 0.8 e^{j 0.03 π} / x_{2} \end{matrix}$ It is easy to verify $\begin{matrix} d_{1} (A, B) = d_{1} (\neg_{i} A, \neg_{i} B) = 0.07, i = 1, 2, 3 \\ l_{1} (A, B) = l_{1} (\neg_{i} A, \neg_{i} B) = 0.035, i = 1, 2, 3 \\ d_{2} (A, B) = d_{2} (\neg_{i} A, \neg_{i} B) = 0.0353, i = 1, 2, 3 \\ l_{2} (A, B) = l_{2} (\neg_{i} A, \neg_{i} B) = 0.05, i = 1, 2, 3 \\ d_{\infty} (A, B) = d_{\infty} (\neg_{i} A, \neg_{i} B) = 0.08, i = 1, 2, 3 \end{matrix}$ and $\begin{matrix} d_{H} (A, B) = 0.035, \\ d_{nH} (A, B) = 0.0175, \\ d_{E} (A, B) = 0.05, \\ d_{nE} (A, B) = 0.0177, \\ d_{Z} (A, B) = 0.04 \end{matrix}$ but $\begin{matrix} d_{H} (\neg_{3} A, \neg_{3} B) = 0.965, \\ d_{nH} (\neg_{3} A, \neg_{3} B) = 0.4825, \\ d_{E} (\neg_{3} A, \neg_{3} B) = 1.3647, \\ d_{nE} (\neg_{3} A, \neg_{3} B) = 0.9651, \\ d_{Z} (\neg_{3} A, \neg_{3} B) = 0.97 \end{matrix}$

Set rotation and reflection operations, described in [14], are another two complex fuzzy operations from F^C (U) to F^C (U).

Let A be a complex fuzzy sets on U, and μ_A (x) = r_A (x) · e^jw_A(x) its membership functions. The Rotation of A by θ radians, denoted Rot_θ (A) is defined as ${Rot}_{θ} (μ_{A} (x)) = r_{A} (x) \cdot e^{j (w_{A} (x) + θ)}$ (16)

The reflection of A, denoted Ref (A) is defined as $Ref (μ_{A} (x)) = r_{A} (x) \cdot e^{- {jw}_{A} (x)}$ (17)

Theorem 3. Reflection operation Ref is uniformly continuous in metric

d ∈ {d_p, l_p, ρ_p, d_Z, d_H, d_nH, d_E, d_nE}. Set rotation Rot_θ, 0 < θ < 2π, is uniformly continuous in metric d ∈ {d_p, l_p, ρ_p}, but not uniformly continuous in metric d ∈ {d_Z, d_H, d_nH, d_E, d_nE}.

Proof. Trivial.

3.2 Complex Fuzzy Union

The complex fuzzy union of complex fuzzy sets are reviewed as follows (see [14]).

Let A and B be two complex fuzzy sets on U, and μ_A (x) = r_A (x) · e^jw_A(x) and μ_B (x) = r_B (x) · e^jw_B(x) their membership functions, respectively. The complex fuzzy union of A and B, denoted A ∪ B, is specified by a function $μ_{A \cup B} (x) = (r_{A} (x) \oplus r_{B} (x)) \cdot e^{{jw}_{A \otimes B} (x)}$ (18) where ⊕ represents a t-conorm function.

The functions given below are possibilities for calculating w_A⊗B.

$Sum : w_{A \otimes B} = w_{A} + w_{B}$ (19) $Max : w_{A \otimes B} = max (w_{A}, w_{B})$ (20) $Min : w_{A \otimes B} = min (w_{A}, w_{B})$ (21) $Winner Take All : w_{A \otimes B} = {\begin{matrix} w_{A}, & r_{A} > r_{B} \\ w_{B}, & r_{A} < r_{B} \end{matrix}$ (22) $Average : w_{A \otimes B} = (w_{A} + w_{B}) / 2$ (23) $Product : w_{A \otimes B} = w_{A} \cdot w_{B}$ (24) $Difference : w_{A \otimes B} = w_{A} - w_{B}$ (25)

Remark 1. In this paper, the phase term is confined to the interval [0, 2π), so the above functions Sum, Product and Difference are addition modulo 2π, multiplication modulo 2π and subtraction modulo 2π, respectively.

Definition 2. Let d be a distance between complex fuzzy sets, ∪ : F^C (U) × F^C (U) → F^C (U) be a complex fuzzy union. ∪ is said to be continuous in metric d, if for any ɛ > 0, there exists δ > 0 such that d (A₁ ∪ B₁, A₂ ∪ B₂) < ɛ whenever d (A₁, A₂) < δ and d (B₁, B₂) < δ for any A₁, A₂, B₁, B₂, ∈ F^C (U).

Theorem 4. Let μ_A (x) = r_A (x) · e^jw_A(x) and μ_B (x) = r_B (x) · e^jw_B(x) The complex fuzzy union of A and B, denoted A ∪ B, is given by $μ_{A \cup B} (x) = (r_{A} (x) \oplus r_{B} (x)) \cdot e^{{jw}_{A \otimes B} (x)}$ (26) where ⊕ represents a t-conorm function. If ⊕ is a continuous t-conorm function, then the continuous results of complex fuzzy union in metric d ∈ {d_p, l_p, ρ_p, d_Z, d_H, d_nH, d_E, d_nE} depend on the choices of w_A⊗B and they are given in Table 1.

Table 1

Continuity results of complex fuzzy union and intersection

	d_p, l_p, ρ_p	d_Z, d_H, d_nH, d_E, d_nE
Sum,Difference,Product	√	×
Average	×	√
Max,Min	√	√
Winner Take All	×	×

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

Proof. We only consider the case of ⊗ = Sum and d = d_∞.

Let δ′ = min(δ, πɛ/2), d_∞ (A₁, A₂) < δ′ and d_∞ (B₁, B₂) < δ′, i.e., for any x ∈ U, $\begin{matrix} | r_{A_{1}} (x) - r_{A_{2}} (x) | < δ^{'}, \\ | r_{B_{1}} (x) - r_{B_{2}} (x) | < δ^{'}, \end{matrix}$ and $\begin{matrix} \frac{1}{π} min (| w_{A_{1}} (x) - w_{A_{2}} (x) |, \\ 2 π - | w_{A_{1}} (x) - w_{A_{2}} (x) |) < δ^{'}, \end{matrix}$ $\begin{matrix} \frac{1}{π} min (| w_{B_{1}} (x) - w_{B_{2}} (x) |, \\ 2 π - | w_{B_{1}} (x) - w_{B_{2}} (x) |) < δ^{'}, \end{matrix}$ then, by the continuity of t-conorm ⊕.

|r_{A
₁} (x) ⊕ r_{B
₁} (x) - r_{A
₂} (x) ⊕ r_{B
₂} (x) | < ɛ.

and by Lemma 5(vi), $\begin{matrix} \frac{1}{π} min (| w_{A_{1}} (x) \oplus w_{B_{1}} (x) - w_{A_{2}} (x) \oplus w_{B_{2}} (x) |, \\ 2 π - | w_{A_{1}} (x) \oplus w_{B_{1}} (x) - w_{A_{2}} (x) \oplus w_{B_{2}} (x) |) \\ \leq \frac{1}{π} min (| w_{A_{1}} (x) - w_{A_{2}} (x) |, \\ 2 π - | w_{A_{1}} (x) - w_{A_{2}} (x) |) \\ + \frac{1}{π} min (| w_{B_{1}} (x) - w_{B_{2}} (x) |, \\ 2 π - | w_{B_{1}} (x) - w_{B_{2}} (x) |) \\ \leq \frac{1}{π} (π ɛ / 2 + π ɛ / 2), \end{matrix}$ where ⊗ = Sum.

Thus d_∞ (A₁ ∪ B₁, A₂ ∪ B₂) < ɛ. Then ∪ is said to be continuous in metric d_∞, when ⊕ is a continuous t-conorm function and ⊗ = Sum.

3.3 Complex Fuzzy Intersection

The complex fuzzy intersection of complex fuzzy sets are reviewed as follows (see [14]).

Let A and B be two complex fuzzy sets on U, and μ_A (x) = r_A (x) · e^jw_A(x) and μ_B (x) = r_B (x) · e^jw_B(x) their membership functions, respectively. The complex fuzzy intersection of A and B, denoted A ∩ B, is specified by a function $μ_{A \cup B} (x) = (r_{A} (x) ★ r_{B} (x)) \cdot e^{{jw}_{A \otimes B} (x)}$ (27) where ★ signifies any t-norm function.

Possible choices for calculating w_A⊗B are given in (20)-(26). Then, we have following results.

Definition 3. Let d be a distance between complex fuzzy sets, ∩ : F^C (U) × F^C (U) → F^C (U) be a complex fuzzy intersection. ∩ is said to be continuous in metric d, if for any ɛ > 0, there exists δ > 0 such that d (A₁ ∩ B₁, A₂ ∩ B₂) < ɛ whenever d (A₁, A₂) < δ and d (B₁, B₂) < δ for any A₁, A₂, B₁, B₂, ∈ F^C (U).

Theorem 5. Let μ_A (x) = r_A (x) · e^jw_A(x) and μ_B (x) = r_B (x) · e^jw_B(x) The complex fuzzy intersection of A and B, denoted A ∩ B, is given by $μ_{A \cap B} (x) = (r_{A} (x) ★ r_{B} (x)) \cdot e^{{jw}_{A \otimes B} (x)}$ (28) where ★ represents a t-norm function. If ★ is a continuous t-norm function, then the continuity results of complex fuzzy intersection in metric

d ∈ {d_p, l_p, ρ_p, d_Z, d_H, d_nH, d_E, d_nE} depend on the choices of w_A⊗B and they are given in Table 1.

4 Conclusion

Many researchers have introduced various types of distance measures for complex fuzzy sets. However, many of these distance measures may ignore circularity and periodicity of the phase term of a complex fuzzy set. Therefore, this paper introduces several new distance measures. Based on these distances, we presented some continuity results of various complex fuzzy operations.

In this paper, we only considered distances of complex fuzzy sets and continuity of complex fuzzy operations. Recently, interval-valued complex fuzzy sets and interval-valued complex fuzzy operations are developed by Greenfield et al. [6]. Based on the concept of rotational invariance [4], Greenfield et al. [7] also discussed the choices of interval-valued complex fuzzy operations in interval-valued complex fuzzy inference systems. Naturally, we can consider distances between interval-valued complex fuzzy sets and continuity of interval-valued complex fuzzy operations, and then the continuity results can serve as valuable references for choices of operations in interval-valued complex fuzzy inference systems.

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) and the Guangxi University Science and Technology Research Project (Grant No. 2013YB193).

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