Complex trapezoidal intuitionistic fuzzy numbers

: The paper introduces the Complex Trapezoidal Intuitionistic Fuzzy Numbers (CTrIFNs) in Cartesian form. Basic arithmetic operations are proposed based on ( α, β ) cuts and veriﬁed with suitable illustrations.


Introduction
The real world situations are often associated with built-in uncertainities. The exact values are not known but the range of possible values are known. Uncertainities expressed in the form of intervals are apt in formulating the reality as a mathematical model which makes TrIFN a better option. CTrIFNs may be helpful in dealing uncertainities which are multi-dimensional in real life situations. The need for CTrIFNs arises when visualizing the areas where uncertainities are continuous and multi-dimensional. For example, turbulence is a violent or unsteady movement of air, water or some other fluid. Turbulence is chaotic and varies with time and direction. Uncertainities arise from the pressure difference at various points and also from the direction which depends on the fluctuations of the turbulence. These critical points are more clear when expressed using interval numbers. The critical points are dependant on multiple issues like magnitude, direction, pressure and the human perception. Complex numbers provide a better understanding of uncertainities with multi-dimensional quantities. Hence, the need for extension of Trapezoidal Intuitionistic Fuzzy Numbers into complex number system arises. Intuitionistic fuzzy sets were introduced by Krassimir T. Atanassov [1]. J. J. Buckley defined the Fuzzy Complex Numbers [3]. Burillo et al. proposed the defintion of intuitionistic fuzzy numbers [6]. The arithmetic operations on Trapezoidal Intuitionistic Fuzzy Numbers using (α, β) cuts were defined by Thangaraj Beaula and M. Priyadharsini [4]. R. Parvathi and C. Malathi developed arithmetic operations on Symmetric Trapezoidal Intuitionistic Fuzzy Numbers [2]. Moore [5] developed the interval arithmetic as a formal system. With all these motivations, Complex Trapezoidal Intuitionistic Fuzzy Numbers (CTrIFNs) are introduced in this paper.

Notations and preliminaries
We remind the reader that Z = A + iB is a Complex Trapezoidal Intuitionistic Fuzzy Number, where A = [a, b, c, d; a , b, c, d ] and B = [p, q, r, s; p , q, r, s ] are TrIFNs such that a ≤ a ≤ b ≤ c ≤ d ≤ d . In this section, basic definitions relating to intuitionistic fuzzy sets, trapezoidal intuitionistic fuzzy numbers which are prerequisites for this study are dealt with.
determine the degree of membership and the degree of non-membership of the element x ∈ X, respectively and for every Definition 2.2. [6] An intuitionistic fuzzy number (IFN) A is defined as follows: • It is an intuitionistic fuzzy subset of the real line; • It is normal, that is, there exists a x ∈ R such that µ A (x) = 1, (So, ν A (x) = π A(x) = 0); • It is a convex set for the membership function µ A (x). That is, • It is a concave set for the non-membership function ν A (x). That is,
Note: In Definition 2.3, the following changes in notations are made and used in this paper. When a 1 = b, a 2 = c, a 1 − α = a, a 2 + β = d, a 1 − α = a and a 2 + β = d in Definition 2.3, then the membership and non-membership functions take the form is a crisp subset of R and is defined as Thus, A α is a closed interval and is denoted by is a crisp subset of R and is defined as Thus, A β is a closed interval and is denoted by Assumptions: Let R be the set of real numbers and x ∈ R. Let a , a, b, c, d, d , p , p, q, r, s,

Definition of CTrIFN
Let A and B be two TrIFNs. A Complex Trapezoidal Intuitionistic Fuzzy Number (CTrIFN) Z is a trapezoidal intuitionistic fuzzy number in the complex plane C and is of the form For every x ∈ R, Remark: A CTrIFN becomes a CTrFN by letting a = a and d = d and ν A = 1 − µ A .

Addition
is also a CTrIFN. Z takes the form with the membership and non-membership values of Re(Z) and Im(Z), for every x ∈ R as Proof. The membership and non-membership functions of Equating the components to x and expressing in terms of α gives the membership function of Re(Z) as, Equating both the components to x and expressing in terms of β gives the non-membership function of Re(Z) as, Now, the sum of the α-cut of B and the α-cut of D is Equating each term to x gives the membership function of Im(Z) as, Here, the sum of the β-cut of B and the β-cut of D is Equating each term to x gives the non-membership function of Im(Z) as,

Subtraction
Consider the two TrIFNs Z 1 and Z 2 such that Z 1 = A + iB and Z 2 = C + iD. The difference of two CTrIFNs denoted by with the membership and non-membership values of Re(Z) and Im(Z) as, Proof. The membership and non-membership functions of CTrIFN Z = (A − C) + i (B − D) can be found by the (α, β) cuts. Now, the α-cut of A minus the α-cut of C is Equating each term to x gives the membership function of Re(Z) as, Now, the β-cut of A minus the β-cut of C is and equating each term to x gives the non-membership function of Re(Z) as, Here, α-cut of B -α-cut of D = [α (q − p + s 1 − r 1 ) + p − s 1 , (s − p 1 ) − α (s − r + q 1 − p 1 )] and equating each term to x gives the membership function of Im(Z) as, +i [ap 1 + pa 1 , bq 1 + qb 1 , cr 1 + rc 1 , ds 1 + sd 1 ; a p 1 + p a 1 , bq 1 + qb 1 , cr 1 + rc 1 , d s 1 + s d 1 ]

Product of CTrIFNS
with the membership and non-membership values of Re(Z) and Im(Z) as, The membership and non-membership functions are found using (α, β) cuts. Now, finding the α-cut of AC minus the α-cut of BD and equating each term to x gives The membership function of Re(Z) is, Similarly, the non-membership function of Re(Z), Similarly, the membership function of Im(Z), Similarly, the non-membership function of Im(Z)

Scalar multiplication
Let Z = A + iB be a CTrIFN and k is a scalar then Therefore, the additive inverse does not exist for a CTrIFn.
Note: The additive inverse for a CTrIFN exists only if the interval is degenerate. That is, a = b = c = d and p = q = r = s. In this case, it is a complex number.

Additive property of a conjugate of CTrIFN
The additive property of a conjugate of a CTrIFN does not hold, that is

Multiplicative property of a conjugate of CTrIFN
The multiplicative property of a conjugate of CTrIFN also does not hold, that is, because of the lack of additive inverse).

Conclusion
In this paper, an attempt has been made to introduce Complex Trapezoidal Intuitionistic Fuzzy Numbers (CTrIFNs). Further, few properties are discussed and verified with suitable numerical examples. It is proposed to work on establishing few more properties of CTrIFNs and their applications in quantum mechanics, electrical engineering and fluid dynamics.