# Similarity Measures between Temporal Complex Intuitionistic Fuzzy Sets and Application in Pattern Recognition and Medical Diagnosis

Authors: Mohammed M. Khalaf; Sayer Obaid Alharbi; Wathek Chammam
Journal: Discrete Dynamics in Nature and Society
 (2019)

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## Abstract

This work addresses the issue of similarity measures between two temporal complex Atanassov’s intuitionistic fuzzy sets, many measures of similarity between complex Atanassov’s intuitionistic fuzzy sets. What was proposed before did not consider the abstention group influence, which may lead to counterintuitive results in some cases. A new structure of temporal complex Atanassov’s intuitionistic fuzzy sets is obtained. This set is formally generalized from a conventional Atanassov’s intuitionistic complex fuzzy sets. Here we analyze the limitations of the existing similarity measures. Then, a new similarity measure of temporal complex Atanassov’s intuitionistic fuzzy sets is proposed and several numeric examples are given to demonstrate the validity of the proposed measure. Finally, the proposed similarity measure is applied to pattern recognition and medical diagnosis.


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## Body

## 1. Introduction

Fuzzy set theory was conferred by Zadeh [1] to solve difficulties in dealing with uncertainties. Since then, the theories of fuzzy sets and fuzzy logic have been examined by many researchers to solve many real life problems involving ambiguous and uncertain environment. By adding a new component the idea of the concept of Atanassov’s intuitionistic fuzzy set (AIFS) was introduced [2]. Applications of these sets have been broadly studied in other aspects such as image processing [3], multicriteria decision making [4], pattern recognition [5], etc. Buckley [6] and Nguyen et al. [7] combined complex numbers with fuzzy sets. On the other hand, the innovative complex fuzzy set is introduced. The complex fuzzy set is characterized by a membership function, μA(x), whose range is not limited to [0,1] but extended to the unit circle in the complex plane. Hence, μA(x) is a complex-Valued function that assigns a grade of membership of the form rAxeiωAx, i=-1 to any element in the universe of discourse. The value of μS(x) is defined by the two variables, rA(x) and ωAx, both real-valued, with μAx∈[0,1]. Complex fuzzy set theory modifies the original concept of fuzzy membership by asserting that, at least in some instances, it is necessary to add a second dimension to the expression of membership. However, this added dimension does not alter the basic concept of fuzziness. Membership in a complex fuzzy set remains “as fuzzy” as membership in a traditional fuzzy set. The fuzziness of membership, i.e., the representation of membership as a value in the range [0,1], is retained in complex fuzzy sets through theamplitudeof the grade of membership, rAx. The novelty of complex fuzzy sets is manifested in the additional dimension of membership: thephaseof the grade of membership, ωAx. The properties ofmembership phaseare discussed at length in this section. Ramot et al. [8, 9] extended the range of membership to “unit circle in the complex plane”, unlike others who limited the range to [0,1]. Omar [10] studied similarity measures between temporal intuitionistic fuzzy sets. As the complex fuzzy membership grade is two-dimensional (amplitude and phase), a complex fuzzy set can be visually represented by a three-dimensional graph where the universe of discourse is the third axis. Figure 1 shows the complex fuzzy set.Figure 1
Complex fuzzy set defined in [11].We divide the paper into four main sections. In the first section preliminaries and basic definitions, we provide some details about the complex fuzzy sets. In the second section, detail is given about the complex version of temporal complex intuitionistic fuzzy set, which is an extension of complex intuitionistic fuzzy set by adding the times and studied the correlation coefficient between two temporal complex intuitionistic fuzzy set. In the third section, details is given about similarity measures between other extensions of temporal complex intuitionistic fuzzy set and extend the method proposed by Chaira [12] for intuitionistic fuzzy set based on the Sugeno [13] and Omar [10] intuitionistic fuzzy generator. In the fourth section, we give application in pattern recognition, medical diagnosis, and topology.


## 2. Preliminaries and Basic Definitions

Definition 1 (see [8]).
A complex fuzzy set (CFS)A defined on a universe X is an object of the form A defined on a universe of discourse X which is an object of the form (1)A=x,μAxeiωAx:x∈X,where i=-1, μAx∈[0,1], and 0≤ωAx≤2π.Definition 2 (see [2]).
A complex intuitionistic fuzzy set (CIFS)A defined on a universe of discourse X is an object of the form (2)A=x,μAxeiαAx,νAxeiβAx:x∈X,where i=-1, μAx,νAx∈[0,1], αAx,βAx∈0,2π, and 0≤μAx+vAx≤1.Definition 3 (see [14]).
LetA and B be two CIFSs in X, where (3)A=x,μAxeiαAx,νAxeiβAx:x∈XB=x,μBxeiαBx,νBxeiβBx:x∈X.Then, A∪B is given as(4)A∪B=x,μA∪BxeiαA∪Bx,νA∩BxeiβA∩Bx:x∈X
where(5)μA∪BxeiαA∪Bx=x,μAx∨νBxeiαAx∨αBx,νA∩BxeiαA∩Bx=x,νAx∧νBxeiβAx∧βBx.Definition 4 (see [15]).
LetA and B be two CIF-sets in X, where (6)A=x,μAxeiαAx,νAxeiβAx:x∈XB=x,μBxeiαBx,νBxeiβBx:x∈X.Then, for all x∈X,
(1)A⊂B if and only if μAx<μBx; νAx>νBx. For amplitude terms and αAx<αBx, βAx>βBx for phase terms.
(2)A=B if and only if μAx=μBx; νAx=νBx. For amplitude terms and αAx=αBx, βAx=βBx for phase terms.Definition 5 (see [16]).
LetA and B be two CIFSs in X, where (7)A=x,μAxeiαAx,νAxeiβAx:x∈XB=x,μBxeiαBx,νBxeiβBx:x∈X.Then, for all x∈X,
(1)A∩B=x,μA∩BxeiαA∩B(x),νA∪BxeiβA∪B(x):x∈X
where(8)μA∩BxeiαA∩Bx=x,μAx∧νBxeiαAx∧αBx,νA∪BxeiαA∪Bx=x,νAx∨νBxeiβAx∨βBx.
(2) The complex fuzzy complement ofA, denoted by A-, is specified by a function(9)A-=x,1-μAxei2π-αAx,1-νBxei2π-βAx:x∈X
(3)1-=x,(ei2π,0):x∈X and 0-=x,(0,ei2π):x∈X.Example 6.
ConsiderX={a,b,c,d}. Let A,B be a CIF-subset of X, as given by (10)A=0.2ei1.3π,0.4ei0.5πa,1.0ei1.5π,0.0ei0.5πb,0.7ei0.3π,0.2ei1.5πc,0.8ei1.1π,,0.1ei0.7πd,B=0.2ei0.9π,0.3ei0.4πa,0.1ei0.2π,0.3ei0.9πb,0.1ei0.1π,0.2ei0.5πc,0.5ei0.2π,,0.1ei0.5πd.Then (11)A∩B=0.2ei0.9π,0.4ei0.5πa,0.1ei.2π,0.3ei0.9πb,0.1ei0.1π,0.2ei1.5πc,0.5ei0.2π,,0.1ei0.7πdA-=0.8ei0.7π,0.6ei1.5πa,0,1.0ei1.5πb,0.3ei1.7π,0.8ei0.5πc,0.2ei0.9π,,0.9ei1.3πd.A∪B=0.2ei1.3π,0.3ei0.4πa,1.0ei1.5π,0.0ei0.5πb,0.7ei0.3π,0.2ei0.5πc,0.8ei1.1π,,0.1ei05πd.And A⊄B and A≠B.


## 3. Temporal Complex Intuitionistic Fuzzy Set

Definition 7.
LetX be a universe, T be a nonempty set of time moments, and A⊆X. A temporal complex intuitionistic fuzzy set (TCIFS) A defined on a universe of discourse X is an object of the form (12)AT=x,t,μAx,teiαAx,t,νAx,teiβAx,t∣x,t∈X×T,where μA:A×T→[0,1] and νA:A×T→[0,1] such that 0≤μAx,teiαAx,t+νAx,teiβAx,t≤1, μAx,t, and vAx,t being the degrees of membership and nonmembership, respectively, of the element x∈X at the moment t∈T. And αAx,t,βAx,t∈0,2π at the moment t∈T, where i=-1.
The hesitation degree of a TCIFSA is defined by πA:1-μAx,t-vAx,t such that 0≤πA(x,t)≤1 for each x,t∈A×T. For brevity we will write A instead of A(T) when this does not cause confusions.Example 8.
Suppose thatX is a universal, with respect to the time set T1=t∣t=6k,k=0,1,…,100,T2=t∣t=2k,k=0,1,…,100, and αAx,t=βAx,t=ei2π. Then, TCIFSs A,B are defined by (13)AT1x=12,14t=4k,k=1,…,10019,35t=4k+2,k=1,…,100BT2x=23,16t=3k,k=1,…,10038,15t=3k+1,k=1,…,100711,29t=3k+2,k=1,…,100Example 9.
SupposeX={x1,x2,x3} with respect to the time set T={t1,t2,t3}. Then, the details of a TCIFS A are explained in Tables 1, 2, and 3Table 1
TIFSA.
t 1 t 2 t 3 x 1 ( 0.2,0.1 ) ( 0.1,0.6 ) ( 0.3,0.5 )  x 2 ( 0.6,0.1 ) ( 0.1,0.9 ) ( 0.6,0.4 )  x 3 ( 0.7,0.1 ) ( 0.1,0.7 ) ( 0.8,0.5 )Table 2
α A x , t = β A x , t.
t 1 t 2 t 3 x 1 e i π / 2 e i π e i 2 π  x 2 e i 2 π e i π / 2 e i π / 2  x 3 e i 3 π / 2 e i 3 π / 2 e i π / 2Table 3
T C I F S   A.
t 1 t 2 t 3 x 1 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )  x 2 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i )Definition 10.
LetAT1 and BT2 be two TCIFSs. Then (14)AT1∩BT2=(x,min⁡μ-Ax,teiαAx,t,μ-Bx,teiαBx,t,max⁡ν-Ax,teiβAx,t,ν-Bx,teiβBx,t:x,t∈X×(T1∪T2AT1∪BT2=(x,max⁡μ-Ax,teiαAx,t,μ-Bx,teiαBx,t,min⁡ν-Ax,teiβAx,t,ν-Bx,teiβBx,t:x,t∈X×(T1∪T2A-T1=(x,ν-Ax,tei2π-βAx,μ-Ax,tei2π-αAx,:x,t∈X×T1,
where(15)μ-Ax,t=μAx,tt∈T10t∈T2-T1ν-Ax,t=νAx,tt∈T11t∈T2-T1μ-Bx,t=μBx,tt∈T10t∈T2-T1ν-Bx,t=νBx,tt∈T11t∈T2-T1Definition 11.
We define the following two operatorsfandg over a TCIFS A:(16)fAT=(x,x,maxt∈TμAx,teiαAx,t,mint∈TνAx,teiβAx,t:x,t∈X×ThAT=(x,x,maxt∈XμAx,teiαAx,t,mint∈XνAx,teiβAx,t:x,t∈X×TgAT=(x,x,mint∈TμAx,teiαAx,t,maxt∈TνAx,teiβAx,t:x,t∈X×TlAT=(x,x,mint∈XμAx,teiαAx,t,maxt∈XνAx,teiβAx,t:x,t∈X×TTheorem 12.
f A T and gAT are TCIFSs.Proof.
Suppose that(17)maxt∈TμAx,teiαAx,t=μAx,t1eiαAx,t1,for  somet1∈T,and (18)mint∈TνAx,teiαAx,t=νAx,t2eiαAx,t2,for  somet2∈TTherefore, (19)νAx,t2eiβAx,t2≤vAx,t1eiβAx,t1.Andmaxt∈TμAx,teiαAx,t+mint∈TvAx,teiαAx,t=μAx,t1eiαAx,t1+vAx,t2eiαAx,t2≤μAx,t1eiαAx,t1+vAx,t1eiβAx,t1≤1Then fAT is TCIFSs. Also, by the same fashion gAT are TCIFSs.Theorem 13.
For every TCIFSAT,(1)
ffAT=fAT;(2)
ggAT=gAT;(3)
fgAT=gAT;(4)
gfAT=fAT.Proof.
The proof is obvious.Theorem 14.
For every TCIFSAT,(1)
hfAT=fh(AT);(2)
lgAT=glAT.Proof.
(1)(20)hfAT=(x,x,maxt∈Xmaxt∈TμAx,teiαAx,t,mint∈Xmint∈TνAx,teiβAx,t:x,t∈X×T=(x,x,maxt∈Tmaxt∈XμAx,teiαAx,t,mint∈Tmint∈XνAx,teiβAx,t:x,t∈X×T=fhAT
(2) By the same fashion, one has the following.Definition 15.
LetA and B be two TCIFSs defined on the universe of discourse X={x1,x2,x3,…,xn and the time moments T={t1,t2,t3,…,tm}. The correlation coefficient of A and B is given by(21)kA,B=CA,BTATBwhere (22)CA,B=∑i=1n∑j=1mμAxi,tieiαAx,tμBxi,tieiαBx,t+νAxi,tieiβAx,tνBxi,tieiβBx,tis the correlation of two TCIFSs A and B, and (23)TA=∑i=1n∑j=1mμA2xi,tiei2π-αAx+νA2xi,tiei2π-βAx,TB=∑i=1n∑j=1mμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBxare the information temporal complex intuitionistic energies of A and B, respectively.Example 16.
Suppose thatX={x1,x2,x3} with respect to the time set T={t1,t2,t3}. The details of a TCIFSAT are explained in Table 4, Table 5 explained TCIFSBT, and Table 6 explained the correlation coefficient kA,B between TCIFS AT and TCIFS BT.Table 4
TCIFSAT.
t 1 t 2 t 3 x 1 ( 0.2  i , 0.3 i ) ( -  0.1 i , -  0.3 i ) ( 0.3 i , 0.5 i )  x 2 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.8 i , -  0.1 i ) ( 0.8 i , 0.5 i )  x 4 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 5 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i )  x 6 ( 0.6 i , 0.1 i ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )Table 5
TCIFSBT.
t 1 t 2 t 3 x 1 ( 0.1  i , 0.1 i ) ( -  0.1 i , -  0.6 i ) ( 0.1 i , 0.5 i )  x 2 ( 0.4,0.3 ) ( 0.7 i , 0.2 i ) ( 0.9 i , 0.2 i )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.3 i ) ( 0.1 i , 0.1 i )  x 4 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i )  x 5 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 6 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i )Table 6
Thenk(A,B).
t 1 t 2 t 3 x 1 k A , B = . 0509 k A , B = . 1923 k A , B = . 2973  x 2 k A , B =.3041 k A , B = . 6592 k A , B =.6648  x 3 k A , B =.5000 k A , B =.2544 k A , B = . 1334  x 4 k A , B =.4301 k A , B =.6403 k A , B =.6802  x 5 k A , B = . 1360 k A , B =.5508 k A , B =.4204  x 6 k A , B = . 1360 k A , B =.5581 k A , B = . 4204Proposition 17.
LetAT1 and BT2 be two TCIFS. Then (1)
TA=TA-;(2)
IfαAx,t=2π and βAx,t=2π, then, CA,A=TA;(3)
CA,B=CB,A.Proof.
Lett∈T1. From Definition 10,μ-Ax,t=μAx,t, ν-Ax,t=νAx,t, μ-Bx,t=μBx,t, ν-Bx,t=νBx,t, and then
(1)(24)TA=∑i=1n∑j=1mμA2xi,tiei2π-αAx+νA2xi,tiei2π-βAx,=∑i=1n∑j=1mνA2xi,tiei2π-βAx+μA2xi,tiei2π-αAx=TA-
(2) IfαAx,t=2π and βAx,t=2π(25)CA,A=∑i=1n∑j=1mμAxi,tieiαAx,tμAxi,tieiαAx,t+νAxi,tieiβAx,tνAxi,tieiβAx,t=∑i=1n∑j=1mμA2xi,tie2iαAx,t+νA2xi,tie2iβAx,t=∑i=1n∑j=1mμA2xi,tiei2π-αAx+νA2xi,tiei2π-βAx,=TA
(3)(26)CA,B=∑i=1n∑j=1mμAxi,tieiαAx,tμBxi,tieiαBx,t+νAxi,tieiβAx,tνBxi,tieiβBx,t=∑i=1n∑j=1mνBxi,tieiβBx,tνAxi,tieiβAx,t+μBxi,tieiαBx,tμAxi,tieiαAx,t=CB,ATheorem 18.
Suppose thatAT and BT are TCIFSs in the universal X with respect to the time set T. Then(1)
ifαAx,t=2π and βAx,t=2π, then k(A,B)=1;(2)
k(A,B)=k(B,A);(3)
0<kA,B<1.Proof.
(1) LetA and B be two TCIFSs defined on the universe of discourse X={x1,x2,x3,…,xn and the time moments T={t1,t2,t3,…,tm}. The correlation coefficient of A and B is given by(27)kA,B=CA,BTATBIf A=B, then TAT(B)=TA2=TA. Then from Proposition 17 (2), one has the following.
IfαAx,t=2π and βAx,t=2π, then CA,A=TA; then (28)kA,B=TATA=1
(2) From Proposition17 (3), CA,B=CB,A. Then (29)kA,B=CB,ATBTA=kB,A
(3) We will prove thatkA,B<1 such that it is evident 0<kA,B, so suppose that (30)∑i=1n∑j=1mμA2xi,tiei2π-βAx=α1,∑i=1n∑j=1mμB2xi,tiei2π-βBx=α2,∑i=1n∑j=1mνA2xi,tiei2π-βAx=α3,∑i=1n∑j=1mνB2xi,tiei2π-βBx=α4.Then (31)TATB=∑i=1n∑j=1mμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAx×∑i=1n∑j=1mμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBx​,=∑i=1n∑j=1mμA2xi,tiei2π-βAx+∑i=1n∑j=1mνA2xi,tiei2π-βAx×∑i=1n∑j=1mμB2xi,tiei2π-αBx,t+∑i=1n∑j=1mνB2xi,tiei2π-βBxThen(32)TATB=α1+α3×α2+α41/2=α1+α31/2×α2+α41/2CA,B=∑i=1n∑j=1mμAxi,tieiαAx,tμBxi,tieiαBx,t+∑i=1n∑j=1mνAxi,tieiβAx,tνBxi,tieiβBx,t,Then (33)CA,B=α1×α21/2+α3×α41/2Then (34)k2A,B≤α1×α2+2α1×α2×α3×α41/2+α3×α4α1+α3×α2+α4But (35)k2A,B-1≤α1×α2+2α1×α2×α3×α41/2+α3×α4α1+α3×α2+α4-1=α1×α2+2α1×α2×α3×α41/2+α3×α4-α1+α3×α2+α4α1+α3×α2+α4=α1×α41/2-α2×α31/22α1+α3×α2+α4≤0Hence k2A,B-1≤0, and then kA,B≤1.Definition 19.
LetS:TICFSsX,T×TICFSsX,T→[0,1] be a function, and let AT,BTandCTbe TCIFSs in the universal X={x1,x2,x3,…,xn} with respect to the time set T={t1,t2,t3,…,tm}. Then S(A,B) is said to be the similarity degree between TICFSsA and B, satisfying the following statements:(1)
0≤S(A,B)≤1.(2)
SA,B=1 if A=B.(3)
SA,B=S(B,A).(4)
IfA⊆B⊆C. Then SA,C≤SA,B,SA,C≤SB,C.
Now we can have the following degrees of the similarity betweenA,B and satisfy the conditions from (1) to (4). Let (36)SAi,j=μAxi,tjeiαAxi,tj-νAxi,tieiβAxi,tj,SBi,j=μBxi,tjeiαBxi,tj-νBxi,tieiβBxi,tj,ψAi,j=μAxi,tjeiαAxi,tj+1-μBxi,tjeiαBxi,tj2,ψBi,j=νAxi,tieiβAxi,tj+1-νBxi,tieiβBxi,tj2where i=1,2,3,…,n; j=1,2,3,…,m. Then(37)S1A,B=1-12mn∑i=1n∑j=1mSAi,j-SBi,j,S2A,B=1-12mn∑i=1n∑j=1mμAxi,tjeiαAxi,tj-μBxi,tjeiαBxi,tj+νAxi,tieiβAxi,tj-νBxi,tieiβBxi,tj,S0A,B=1-12mn∑i=1n∑j=1mμAxi,tjeiαAxi,tj-μBxi,tjeiαBxi,tj2+νAxi,tieiβAxi,tj-νBxi,tieiβBxi,tj2S3A,B=1-12mny∑i=1n∑j=1mψAi,j-ψBi,jy,1≤y<+∞,S4A,B=∑i=1n∑j=1min⁡μAxi,tjeiαAxi,tj,μBxi,tjeiαBxi,tj+min⁡1-νAxi,tieiβAxi,tj,1-νAxi,tieiβAxi,tjm​∑i=1n∑j=1mmax⁡μAxi,tjeiαAxi,tj,μBxi,tjeiαBxi,tj+max⁡1-νAxi,tieiβAxi,tj,1-νBxi,tieiβBxi,tjFrom a comparison between similarity measures S0A,B,S1A,B,S2A,B,S3A,B,S4A,B, we give the following example.Example 20.
Suppose thatATandBT is TCIFSs defined on X={x1,x2,x3,x4,x5,x6} with respect to the time set T={t1,t2,t3,t4,t5,t6}. The details of a TCIFSAT are explained in Table 7, Table 8 explained TCIFSBT, and Table 9 explained a comparison between similarity measures S0A,B,S1A,B,S2A,B,S3A,B, S4A,BTCIFSAT, and TCIFSBT.Table 7
TCIFSAT.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 ( 0.2  i , 0.3 i ) ( -  0.1 i , -  0.3 i ) ( 0.3 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )  x 2 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.1  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.8 i , -  0.1 i ) ( 0.8 i , 0.5 i ) ( 0.2,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 4 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i )  x 5 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )  x 6 ( 0.6 i , 0.1 i ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.1  i , 0.2 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )Table 8
TCIFSBT.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 ( 0.1  i , 0.1 i ) ( -  0.1 i , -  0.6 i ) ( 0.1 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2,0.1 ) ( 0.2  i , 0.1 i )  x 2 ( 0.4,0.3 ) ( 0.7 i , 0.2 i ) ( 0.9 i , 0.2 i ) ( 0.1,0.1 i ) (  0.1 i , 0.6 i ) ( 0.1,0.2 )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.3 i ) ( 0.1 i , 0.1 i ) ( 0.1 i , 0.1 ) ( 0.1 i , 0.8 i ) ( 0.1 i , 0.4 i )  x 4 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i ) ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 5 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.4  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.1  i , 0.1 i )  x 6 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i ) ( 0.3  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )Table 9
A comparison between similarity measuresS0A,B,S1A,B,S2A,B,S3A,B,S4A,B.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 S 0 A , B = 0.9795 S 0 A , B = . 9646 S 0 A , B = 1 - . 0235 i S 0 A , B = 1 . S 0 A , B = 0.9795 S 0 A , B = 1 . S 1 A , B = . 9986 S 1 A , B = . 9958 S 1 A , B = . 9972 S 1 A , B = 1 . S 1 A , B = . 9980 S 1 A , B = 1 . S 2 A , B = 1.00 S 2 A , B = 1 . S 2 A , B = . 9972 S 2 A , B = 1 . S 2 A , B = . 9980 S 2 A , B = 1 . S 3 A , B = . 9941 S 3 A , B = . 9823 S 3 A , B = . 9882 S 3 A , B = 1 . S 3 A , B = . 9916 S 3 A , B = 1 . S 4 A , B = . 8882 S 4 A , B = . 9035 S 4 A , B = . 8589 S 4 A , B = 1 . S 4 A , B = . 9128 S 4 A , B = 1 .  x 2 S 0 A , B = 1 . S 0 A , B = . 9575 S 0 A , B = 1 - . 0263i S 0 A , B = . 9882 S 0 A , B = 0.9302 S 0 A , B = 1 - . 0263 i S 1 A , B = . 9944 S 1 A , B = . 9819 S 1 A , B = . 9930 S 1 A , B = .9980 S 1 A , B = . 9901 S 1 A , B = .9986 S 2 A , B = 1 . S 2 A , B = 1 . S 2 A , B = . 9986 S 2 A , B = . 9980 S 2 A , B = 1 . S 2 A , B = 1 . S 3 A , B = . 9764 S 3 A , B = . 9233 S 3 A , B = . 9705 S 3 A , B = .9916 S 3 A , B = . 9583 S 3 A , B = .9941 S 4 A , B = . 7333 S 4 A , B = . 5474 S 4 A , B = . 8193 S 4 A , B = 1 . S 4 A , B = . 7448 S 4 A , B = . 5454  x 3 S 0 A , B = 1 . S 0 A , B = 1 . - . 0790 i S 0 A , B = 1 - . 0677 i S 0 A , B = . 9764 S 0 A , B = . 9882 S 0 A , B = 1 - . 5892 i S 1 A , B = 1 . S 1 A , B = . 9875 S 1 A , B = . 9958 S 1 A , B = . 9968 S 1 A , B = . 9986 S 1 A , B = .9930 S 2 A , B = 1 . S 2 A , B = . 9930 S 2 A , B = . 9958 S 2 A , B = . 9968 S 2 A , B = 1 . S 2 A , B = . 9930 S 3 A , B = 1 . S 3 A , B = . 9469 S 3 A , B = . 9823 S 3 A , B = . 9868 S 3 A , B = . 9941 S 3 A , B = .9705 S 4 A , B = 1 . S 4 A , B = . 5992 S 4 A , B = . 5761 S 4 A , B = . 9090 S 4 A , B = . 9552 S 4 A , B = . 7018  x 4 S 0 A , B = . 9302 - . 0816 i S 0 A , B = . 8129 S 0 A , B = 1 - . 0204 i S 0 A , B = . 9302 S 0 A , B = . 8129 S 0 A , B = 1 - . 0204 i S 1 A , B = . 9891 S 1 A , B = . 9805 S 1 A , B = . 9986 S 1 A , B = . 9891 S 1 A , B = . 9805 S 1 A , B = .9986 S 2 A , B = . 9891 S 2 A , B = 1 . S 2 A , B = . 9986 S 2 A , B = . 99891 S 2 A , B = 1 . S 2 A , B = . 9986 S 3 A , B = . 9539 S 3 A , B = . 9175 S 3 A , B = . 9941 S 3 A , B = . 9539 S 3 A , B = . 9175 S 3 A , B = .9941 S 4 A , B = . 87977 S 4 A , B = . 9137 S 4 A , B = . 8743 S 4 A , B = . 8797 S 4 A , B = . 9137 S 4 A , B = . 8743  x 5 S 0 A , B = . 9298 + . 0217 i S 0 A , B = . 9845 + . 0697 i S 0 A , B = 1 - . 0333 i S 0 A , B = 1 - . 0236 i S 0 A , B = 1 . S 0 A , B = 1 - . 0117 i S 1 A , B = . 9929 S 1 A , B = . 9868 S 1 A , B = . 9944 S 1 A , B = . 9972 S 1 A , B = 1 . S 1 A , B = .9986 S 2 A , B = . 9931 S 2 A , B = 1 . S 2 A , B = . 9972 S 2 A , B = . 9972 S 2 A , B = 1 . S 2 A , B = 1 . S 3 A , B = . 9899 S 3 A , B = . 9444 S 3 A , B = . 9764 S 3 A , B = . 9882 S 3 A , B = 1 . S 3 A , B = .9941 S 4 A , B = . 6853 S 4 A , B = . 8502 S 4 A , B = . 8015 S 4 A , B = . 8576 S 4 A , B = 1 . S 4 A , B = . 9170  x 6 S 0 A , B = 1 . - . 0471 i S 0 A , B = . 8945 + . 0697 i S 0 A , B = 1 - . 0333 i S 0 A , B = 1 - . 0204 i S 0 A , B = . 9923 S 0 A , B = . 9973 S 1 A , B = . 9944 S 1 A , B = . 9868 S 1 A , B = . 9944 S 1 A , B = . 9958 S 1 A , B = .9929 S 1 A , B = .9968 S 2 A , B = . 9944 S 2 A , B = 1 . S 2 A , B = . 9972 S 2 A , B = . 9986 S 2 A , B = 1 . S 2 A , B = 1 . S 3 A , B = . 9764 S 3 A , B = . 9444 S 3 A , B = . 9764 S 3 A , B = . 9823 S 3 A , B = .9699 S 3 A , B = .9868 S 4 A , B = . 7507 S 4 A , B = . 8502 S 4 A , B = . 8015 S 4 A , B = . 8372 S 4 A , B = . 6138 S 4 A , B = . 5364


## 4. Similarity Measures between Other Extensions of Temporal Complex Intuitionistic Fuzzy Set

The following definition extend the method proposed by Chaira [12] for intuitionistic fuzzy set based on the Sugeno [13] and Omar [10] intuitionistic fuzzy generator.Definition 21.
IfμAx,teiαAx,t is the degrees of membership function of the element x∈X at the moment t∈T, then nonmembership function νAxeiβA(x)=G(μAx,teiαAx,t), where (38)GμAx,teiαAx,t=1-μAx,teiαAx,t1+αμAx,teiαAx,t,α>0And G1=0, G0=1, and by help of the Sugeno [6] intuitionistic fuzzy generator, TCIFS A is given by (39)AαT=x,t,μAx,teiαAx,t,1-μAx,teiαAx,t1+αμAx,teiαAx,t∣x,t∈X×TThe hesitation degree of a TCIFS A is (40)πAαx,t=1-μAx,t-1-μAx,teiαAx,t1+αμAx,teiαAx,tExample 22.
Suppose thatAT is TCIFS defined on X={x1,x2,x3} with respect to the time set T={t1,t2,t3}. The details of a TCIFSAT are explained in Tables 10, 11, and 12, Table 13 explained TCIFSA1 when α=1, and Table 14 explained the hesitation degree of a TCIFS A.
Ifα=1, then one has the following (see Tables 13 and 14).Table 10
TIFSA.
t 1 t 2 t 3 x 1 ( 0.2,0.1 ) ( 0.1,0.6 ) ( 0.3,0.5 )  x 2 ( 0.6,0.1 ) ( 0.1,0.9 ) ( 0.6,0.4 )  x 3 ( 0.7,0.1 ) ( 0.1,0.7 ) ( 0.8,0.5 )Table 11
α A x , t = β A x , t.
t 1 t 2 t 3 x 1 e i π / 2 e i π e i 2 π  x 2 e i 2 π e i π / 2 e i π / 2  x 3 e i 3 π / 2 e i 3 π / 2 e i π / 2Table 12
TCIFSA.
t 1 t 2 t 3 x 1 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )  x 2 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i )Table 13
TCIFSA1.
t 1 t 2 t 3  x 1 0.2  i , 1 - 0.2 i 1 + 0.2 i -  0.1,1.2 0.3,0.53  x 2 0.6,0.25 0.1 i , 1 - 0.1 i 1 + 0.1 i 0.6 i , 1 - 0.6 i 1 + 0.6 i  x 3 -  0.7 i , 1 + 0.7 i 1 - 0.7 i ( -  0.1 i , 1 + 0.1 i 1 - 0.1 i 0.8 i , 1 - 0.8 i 1 + 0.8 iTable 14
The hesitation degree of a TCIFSA.
t 1 t 2 t 3 x 1 0.133 - 1.2 0.16  x 2 0.15 1 - 0.1 i - 1 - 0.1 i 1 + 0.1 i 1 - 0.6 i - 1 - 0.6 i 1 + 0.6 i  x 3 1 -  0.7 i - 1 + 0.7 i 1 - 0.7 i 1 -  0.1 i - 1 - 0.1 i 1 + 0.1 i 1 - 0.8 i - 1 - 0.8 i 1 + 0.8 iDefinition 23.
Suppose thatATandBT is TCIFSs in the universal X={x1,x2,x3,…,xn} with respect to the time set T=t1,t2,t3,…,tm. Then a cosine similarity measure between ATandBT is proposed as follows:(41)CTAxi,tj,Bxi,tj=1mn∑i=1n∑j=1mμAxi,tieiαAxi,tjμBxi,tieiαBxi,tj+νAxi,tieiβAxi,tjνBxi,tieiβBxi,tjμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBxwhere i=1,2,3,…,n; j=1,2,3,…,mTheorem 24.
Suppose thatAT and BT are TCIFSs in the universal X with respect to the time set T. Then(1)
CTA,B=1 if A=B and αAx,t=βAx,t=2π,(2)
CTA,B=CTB,A,(3)
-1≤CTA,B≤1,(4)
ifn=m=1, then CTA,B=kA,B.Proof.
(1) LetA and B be two TCIFSs defined on the universe of discourse X={x1,x2,x3,…,xn} and the time moments T={t1,t2,t3,…,tm}. The cosine similarity measure between ATandBT is given by(42)CTA,B=1mn∑i=1n∑j=1mμAxi,tieiαAx,tμBxi,tieiαBx,t+νAxi,tieiβAx,tνBxi,tieiβBx,tμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBxIf A=B and αAx,t=βAx,t=2π, then (43)CTA,B=1mn∑i=1n∑j=1mμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAx=1
(2)(44)CTA,B=1mn∑i=1n∑j=1mμAxi,tieiαAx,tμBxi,tieiαBx,t+νAxi,tieiβAx,tνBxi,tieiβBx,tμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBxCTA,B=1nm∑j=1m∑i=1nμBxi,tieiαBx,tμAxi,tieiαAx,t+νBxi,tieiβBx,tνAxi,tieiβAx,tμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBxμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAx=CTB,A
(3) By the same way in (3) in Theorem 2.1, one has the following.
(4) Ifn=m=1, then (45)CTA,B=1mn∑i=1n∑j=1mμAxi,tieiαAx,tμBxi,tieiαAx,t+νAxi,tieiβAx,tνBxi,tieiβAx,tμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBxCTA,B=∑i=1n∑j=1mμAxi,tieiαAx,tμBxi,tieiαBx,t+νAxi,tieiβAx,tνBxi,tieiβBx,tμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBx=kA,BDefinition 25.
Suppose thatATandBT is TCIFSs in the universal X={x1,x2,x3,…,xn} with respect to the time set T=t1,t2,t3,…,tm. Then, the distance measure of the angle is proposed as follows: (46)dA,B=cos-1⁡CTA,BTheorem 26.
Suppose thatAT and BT are TCIFSs in the universal X with respect to the time set T. Then(1)
CTA,B=1 and αAx,t=βAx,t=2π; then dA,B=0,(2)
CTA,B=CT(B,A), then dA,B=dB,A,(3)
if-1≤CTA,B≤1, then dA,B≥0,(4)
ifA⊆B⊆C, then dA,C≤dA,B+dB,C.Proof.
(1), (2), and (3) are simple proof.
(4) LetAT, BTandCTbe TCIFSs in the universal X={x1,x2,x3,…,xn} with respect to the time set T={t1,t2,t3,…,tm}. Then, the distance measure of the angle is proposed as follows: (47)di,jAxi,tj,Bxi,tj=cos-1⁡CTAxi,tj,Bxi,tjdi,jAxi,tj,Cxi,tj=cos-1⁡CTAxi,tj,Cxi,tjdi,jBxi,tj,Cxi,tj=cos-1⁡CTBxi,tj,Cxi,tjwhere i=1,2,3,…,n, j=1,2,3,…,m, and (48)CTAxi,tj,Bxi,tj=1mn∑i=1n∑j=1mμAxi,tieiαAxi,tjμBxi,tieiαBxi,tj+νAxi,tieiβAxi,tjνBxi,tieiβBxi,tjμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBxCTBxi,tj,Cxi,tj=1mn∑i=1n∑j=1mμBxi,tieiαAxi,tjμCxi,tieiαCxi,tj+νBxi,tieiβBxi,tjνCxi,tieiβCxi,tjμB2xi,tiei2π-βAx+νB2xi,tiei2π-βAxμC2xi,tiei2π-αBx,t+νC2xi,tiei2π-βBxCTAxi,tj,Cxi,tj=1mn∑i=1n∑j=1mμAxi,tieiαAxi,tjμCxi,tieiαCxi,tj+νAxi,tieiβAxi,tjνCxi,tieiβCxi,tjμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμC2xi,tiei2π-αCx,t+νC2xi,tiei2π-βCxIf Axi,tj⊆Bxi,tj⊆Cxi,tj, for each i=1,2,3,…,n, j=1,2,3,…,m, then (49)di,jAxi,tj,Bxi,tj+di,jBxi,tj,Cxi,tj≥di,jAxi,tj,Cxi,tj.Definition 27.
LetA and B be two TCIFSs defined on the universe of discourse X=x1,x2,x3,…,xn} and the time moments T=t1,t2,t3,…,tm. Suppose that k(A,B) is correlation coefficient of A and B. Then a weight similarity measure between TCIFSs ATandBT is proposed as follows:(50)ρTAxi,tj,Bxi,tj=∑i=1n∑j=1mkAxi,tj,Bxi,tjμAxi,tieiαAxi,tjμBxi,tieiαBxi,tj+νAxi,tieiβAxi,tjνBxi,tieiβBxi,tjμA2xi,tiei2π-βAx+νA2xi,tiei2π-βAxμB2xi,tiei2π-αBx,t+νB2xi,tiei2π-βBxAnd we have the following properties: (1)
ρTA,B=1 then A=B(2)
ρTA,B=ρT(B,A)(3)
-1≤ρTA,B≤1Remark 28.
ρ T A x i , t j , B x i , t j = C T A x i , t j , B x i , t j if(51)∑i=1n∑j=1mkAxi,tj,Bxi,tj=1mnFrom a comparison between similarity measures CTA,B, ρTA,B we give the following example (the same data in Example 16).Example 29.
Suppose thatATandBT is TCIFSs defined on X={x1,x2,x3,x4,x5,x6} with respect to the time set T={t1,t2,t3,t4,t5,t6}. The details of a TCIFSAT are explained in Table 15, Table 16 explained TCIFSBT, and Table 17 explained a comparison between similarity measures between CTA,B, ρTA,B.Table 15
TCIFSAT.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 ( 0.2  i , 0.3 i ) ( -  0.1 i , -  0.3 i ) ( 0.3 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )  x 2 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.1  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.8 i , -  0.1 i ) ( 0.8 i , 0.5 i ) ( 0.2,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 4 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i )  x 5 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )  x 6 ( 0.6 i , 0.1 i ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.1  i , 0.2 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )Table 16
TCIFSBT.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 ( 0.1  i , 0.1 i ) ( -  0.1 i , -  0.6 i ) ( 0.1 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2,0.1 ) ( 0.2  i , 0.1 i )  x 2 ( 0.4,0.3 ) ( 0.7 i , 0.2 i ) ( 0.9 i , 0.2 i ) ( 0.1,0.1 i ) (  0.1 i , 0.6 i ) ( 0.1,0.2 )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.3 i ) ( 0.1 i , 0.1 i ) ( 0.1 i , 0.1 ) ( 0.1 i , 0.8 i ) ( 0.1 i , 0.4 i )  x 4 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i ) ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 5 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.4  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.1  i , 0.1 i )  x 6 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i ) ( 0.3  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )Table 17
Then the similarity measuresCTA,B and ρTA,B.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 C T A , B = - . 02723 C T A , B = - . 0010 C T A , B = - . 0023 C T A , B = - . 0006 C T A , B = - .006 C T A , B = - . 006 ρ T A , B = - . 1278 ρ T A , B = - . 1900 ρ T A , B = - . 2800 ρ T A , B = - 1.000 ρ T A , B = . 0707 ρ T A , B = - 1.000  x 2 C T A , B = . 0246 C T A , B = - . 0045 C T A , B = - . 1144 C T A , B = 0.00 C T A , B = . 003 C T A , B = . 004 ρ T A , B = . 3818 ρ T A , B = - . 0729 ρ T A , B = - . 2772 ρ T A , B = - 1.00 ρ T A , B - . 0500 ρ T A , B = . 1356  x 3 C T A , B = - . 0277 C T A , B = - . 0007 C T A , B = - . 0004 C T A , B = 0.00 C T A , B = - . 014 C T A , B = - . 001 ρ T A , B = - . 3041 ρ T A , B = - . 2844 ρ T A , B = - . 6478 ρ T A , B = - 1.00 ρ T A , B - . 7300 ρ T A , B = - . 0964  x 4 C T A , B = - . 0277 C T A , B = . 0113 C T A , B = - . 0128 C T A , B = . 0051 C T A , B = . 011 C T A , B = - . 012 ρ T A , B = - . 3040 ρ T A , B = . 2548 ρ T A , B = - . 1333 ρ T A , B = - . 9997 ρ T A , B = . 7297 ρ T A , B = - . 2971  x 5 C T A , B = . 0265 C T A , B = . 0048 C T A , B = - . 0044 C T A , B = - . 0008 C T A , B - . 006 C T A , B = - . 002 ρ T A , B = . 4110 ρ T A , B = - . 6393 ρ T A , B = - . 6148 ρ T A , B = - . 4198 ρ T A , B = - 1.00 ρ T A , B = - . 6453  x 6 C T A , B = - . 0265 C T A , B = . 0048 C T A , B = - . 0044 C T A , B = - . 0008 C T A , B = . 003 C T A , B = - . 003 ρ T A , B = - . 1300 ρ T A , B = - . 5500 ρ T A , B = - . 3800 ρ T A , B = - . 6519 ρ T A , B = - . 588 ρ T A , B = . 0266


### 4.1. Application in Pattern Recognition and Medical Diagnosis

LetL=x1,x2,x3,x4,x5,x6 be the set of symptoms of the diseases with respect to the time set T={t1,t2,t3,t4,t5,t6} and L1 be the set of diagnoses. By using the similarity measures CTL,L1 we try to discover that the patient may suffer from one from diseases L which have symptoms x1 at the time t1, and we let L be standard case symptoms of one of diseases (Table 18) and L1 be any case (Table 19); Table 20 explained the similarity measures CTL,L1 between a standard case L and any case L1.Table 18
Standard case symptoms of one of diseasesL.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 ( 0.2  i , 0.3 i ) ( -  0.1 i , -  0.3 i ) ( 0.3 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )  x 2 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.1  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.8 i , -  0.1 i ) ( 0.8 i , 0.5 i ) ( 0.2,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 4 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i )  x 5 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )  x 6 ( 0.6 i , 0.1 i ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.1  i , 0.2 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )Table 19
The symptoms of caseL1.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 ( 0.1  i , 0.1 i ) ( -  0.1 i , -  0.6 i ) ( 0.1 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2,0.1 ) ( 0.2  i , 0.1 i )  x 2 ( 0.4,0.3 ) ( 0.7 i , 0.2 i ) ( 0.9 i , 0.2 i ) ( 0.1,0.1 i ) (  0.1 i , 0.6 i ) ( 0.1,0.2 )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.3 i ) ( 0.1 i , 0.1 i ) ( 0.1 i , 0.1 ) ( 0.1 i , 0.8 i ) ( 0.1 i , 0.4 i )  x 4 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i ) ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 5 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.4  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.1  i , 0.1 i )  x 6 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i ) ( 0.3  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )Table 20
Similarity measuresCTL,L1.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 C T L , L 1 = - . 02723 C T L , L 1 = - . 0010 C T L , L 1 = - . 0023 C T L , L 1 = - . 0006 C T L , L 1 = - .006 C T L , L 1 = - . 006  x 2 C T L , L 1 = . 0246 C T L , L 1 = - . 0045 C T L , L 1 = - . 1144 C T L , L 1 = 0.00 C T L , L 1 = . 003 C T L , L 1 = . 004  x 3 C T L , L 1 = - . 0277 C T L , L 1 = - . 0007 C T L , L 1 = - . 0004 C T L , L 1 = 0.00 C T L , L 1 = - . 014 C T L , L 1 = - . 001  x 4 C T L , L 1 = - . 0277 C T L , L 1 = . 0113 C T L , L 1 = - . 0128 C T L , L 1 = . 0051 C T L , L 1 = . 011 C T L , L 1 = - . 012  x 5 C T L , L 1 = . 0265 C T L , L 1 = . 0048 C T L , L 1 = - . 0044 C T L , L 1 = - . 0008 C T L , L 1 - . 006 C T L , L 1 = - . 002  x 6 C T L , L 1 = - . 0265 C T L , L 1 = . 0048 C T L , L 1 = - . 0044 C T L , L 1 = - . 0008 C T L , L 1 = . 003 C T L , L 1 = - . 003And we define the symptoms of case by Table19.Then Table20 explained the similarity measures CTL,L1 between a standard case L and any case L1.When the similarity measures-1≤CTL,L1≤1 are small, then probability that the patient is suffering from the disease x at the time t is big and the conversely is true.


### 4.2. Complex Intuitionistic Fuzzy Topology

Definition 30.
An intuitionistic complex fuzzy topology onX is a family τ of CIF-sets in X which satisfies the following properties: (1)
1-, 0-∈τ,(2)
ifA,B∈τ, then A∩B∈τ,(3)
ifBi∈τ for each i∈Γ, then ⋃i∈ΓBi∈τ.
Then(X,τ) is called complex intuitionistic fuzzy topological space. The elements of τ are called CIFO-sets and the complement of the CIO-sets is called CIFC-sets.Example 31.
ConsiderX={a,b,c,d}. Let A,B be a CIO-subset of X, as given by (52)A=0.2ei1.3π,0.4ei0.5πa,1.0ei1.5π,0.0ei0.5πb,0.7ei0.3π,0.2ei1.5πc,0.8ei1.1π,,0.1ei0.7πd,B=0.2ei0.9π,0.3ei0.4πa,0.1ei0.2π,0.3ei0.9πb,0.1ei0.1π,0.2ei0.5πc,0.5ei0.2π,,0.1ei0.5πd.Then τ={0-,1-,A,B,A∩B,A∪B} is an complex intuitionistic fuzzy topology on X.Definition 32.
If(X,τ) is called complex intuitionistic fuzzy topological space, A⊆X, then the interior of A is defined as the union of all CIFO-subsets of A and it is denoted by A∘. That is, A∘ is the largest CIFO-subset of A. The closure of A is defined as the intersection of all CIFC sets containing A and it is denoted by A-. That is, A- is the smallest CIFC-set containing A.Example 33.
ConsiderX={a,b}. Let A,B be a CIF-subset of X, as given by (53)A=0.2ei0.3π,0.4ei0.5πa,0.2ei0.1π,,0.1ei0.7πb,B=0.2ei0.9π,0.3ei0.4πa,0.5ei0.2π,,0.1ei0.5πb.C=0.1ei0.7π,0.2ei0.1πa,0.4ei0.1π,,0.1ei0.1πbThen τ={0-,1-,A,B} is an intuitionistic complex fuzzy topology on X, C∘=0-=x,(0,ei(2π)):x∈X and C-=1-=x,(ei(2π),0):x∈X.Definition 34.
An intuitionistic complex fuzzy topological space(X,τ) is said to be extremely disconnected, if the closure of each CIFO-set is CIFO-set.Definition 35.
Let(X,τ) be an intuitionistic complex fuzzy topological space. A subset A of X is said to be CIF semiopen set (by short CIFSO-set) (resp., CIF preopen (by short CIPO-set), CIFα-open (by short CIFαO-set), CIFβ-open (by short CIβO-set), and CIb-open (by short CIbO-set). If A⊆A∘- (resp., A⊆A-∘, A⊆A∘-∘, A⊆A-∘-, and ⊆A∘-∪A-∘), the family of all CISO-set (resp., CIPO-set, CIαO-set, CIβO-set, and CIbO-set) in X is denoted by CISO(X) (resp., CIPO(X), CIαO(X), and CIbO(X)).The implications between these concepts in the following diagram and the converse are not true in general,(54)CIFO⇓⇓CISFOCIPFO⇓⇓CIFαO⇓CIbO⇓CIFβ-O


## 4.1. Application in Pattern Recognition and Medical Diagnosis

LetL=x1,x2,x3,x4,x5,x6 be the set of symptoms of the diseases with respect to the time set T={t1,t2,t3,t4,t5,t6} and L1 be the set of diagnoses. By using the similarity measures CTL,L1 we try to discover that the patient may suffer from one from diseases L which have symptoms x1 at the time t1, and we let L be standard case symptoms of one of diseases (Table 18) and L1 be any case (Table 19); Table 20 explained the similarity measures CTL,L1 between a standard case L and any case L1.Table 18
Standard case symptoms of one of diseasesL.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 ( 0.2  i , 0.3 i ) ( -  0.1 i , -  0.3 i ) ( 0.3 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )  x 2 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.1  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.8 i , -  0.1 i ) ( 0.8 i , 0.5 i ) ( 0.2,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 4 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i )  x 5 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )  x 6 ( 0.6 i , 0.1 i ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.1  i , 0.2 i ) ( -  0.1 , -  0.6 ) ( 0.3,0.5 )Table 19
The symptoms of caseL1.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 ( 0.1  i , 0.1 i ) ( -  0.1 i , -  0.6 i ) ( 0.1 i , 0.5 i ) ( 0.2  i , 0.1 i ) ( 0.2,0.1 ) ( 0.2  i , 0.1 i )  x 2 ( 0.4,0.3 ) ( 0.7 i , 0.2 i ) ( 0.9 i , 0.2 i ) ( 0.1,0.1 i ) (  0.1 i , 0.6 i ) ( 0.1,0.2 )  x 3 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.3 i ) ( 0.1 i , 0.1 i ) ( 0.1 i , 0.1 ) ( 0.1 i , 0.8 i ) ( 0.1 i , 0.4 i )  x 4 ( -  0.7 i , -  0.1 i ) ( -  0.1 i , -  0.7 i ) ( 0.8 i , 0.5 i ) ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i )  x 5 ( 0.6,0.1 ) ( 0.1 i , 0.9 i ) ( 0.6 i , 0.4 i ) ( 0.4  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.1  i , 0.1 i )  x 6 ( 0.2  i , 0.1 i ) ( -  0.1 , -  0.6 ) ( 0.3 i , 0.5 i ) ( 0.3  i , 0.1 i ) ( 0.2  i , 0.1 i ) ( 0.2  i , 0.1 i )Table 20
Similarity measuresCTL,L1.
t 1 t 2 t 3 t 4 t 5 t 6 x 1 C T L , L 1 = - . 02723 C T L , L 1 = - . 0010 C T L , L 1 = - . 0023 C T L , L 1 = - . 0006 C T L , L 1 = - .006 C T L , L 1 = - . 006  x 2 C T L , L 1 = . 0246 C T L , L 1 = - . 0045 C T L , L 1 = - . 1144 C T L , L 1 = 0.00 C T L , L 1 = . 003 C T L , L 1 = . 004  x 3 C T L , L 1 = - . 0277 C T L , L 1 = - . 0007 C T L , L 1 = - . 0004 C T L , L 1 = 0.00 C T L , L 1 = - . 014 C T L , L 1 = - . 001  x 4 C T L , L 1 = - . 0277 C T L , L 1 = . 0113 C T L , L 1 = - . 0128 C T L , L 1 = . 0051 C T L , L 1 = . 011 C T L , L 1 = - . 012  x 5 C T L , L 1 = . 0265 C T L , L 1 = . 0048 C T L , L 1 = - . 0044 C T L , L 1 = - . 0008 C T L , L 1 - . 006 C T L , L 1 = - . 002  x 6 C T L , L 1 = - . 0265 C T L , L 1 = . 0048 C T L , L 1 = - . 0044 C T L , L 1 = - . 0008 C T L , L 1 = . 003 C T L , L 1 = - . 003And we define the symptoms of case by Table19.Then Table20 explained the similarity measures CTL,L1 between a standard case L and any case L1.When the similarity measures-1≤CTL,L1≤1 are small, then probability that the patient is suffering from the disease x at the time t is big and the conversely is true.


## 4.2. Complex Intuitionistic Fuzzy Topology

Definition 30.
An intuitionistic complex fuzzy topology onX is a family τ of CIF-sets in X which satisfies the following properties: (1)
1-, 0-∈τ,(2)
ifA,B∈τ, then A∩B∈τ,(3)
ifBi∈τ for each i∈Γ, then ⋃i∈ΓBi∈τ.
Then(X,τ) is called complex intuitionistic fuzzy topological space. The elements of τ are called CIFO-sets and the complement of the CIO-sets is called CIFC-sets.Example 31.
ConsiderX={a,b,c,d}. Let A,B be a CIO-subset of X, as given by (52)A=0.2ei1.3π,0.4ei0.5πa,1.0ei1.5π,0.0ei0.5πb,0.7ei0.3π,0.2ei1.5πc,0.8ei1.1π,,0.1ei0.7πd,B=0.2ei0.9π,0.3ei0.4πa,0.1ei0.2π,0.3ei0.9πb,0.1ei0.1π,0.2ei0.5πc,0.5ei0.2π,,0.1ei0.5πd.Then τ={0-,1-,A,B,A∩B,A∪B} is an complex intuitionistic fuzzy topology on X.Definition 32.
If(X,τ) is called complex intuitionistic fuzzy topological space, A⊆X, then the interior of A is defined as the union of all CIFO-subsets of A and it is denoted by A∘. That is, A∘ is the largest CIFO-subset of A. The closure of A is defined as the intersection of all CIFC sets containing A and it is denoted by A-. That is, A- is the smallest CIFC-set containing A.Example 33.
ConsiderX={a,b}. Let A,B be a CIF-subset of X, as given by (53)A=0.2ei0.3π,0.4ei0.5πa,0.2ei0.1π,,0.1ei0.7πb,B=0.2ei0.9π,0.3ei0.4πa,0.5ei0.2π,,0.1ei0.5πb.C=0.1ei0.7π,0.2ei0.1πa,0.4ei0.1π,,0.1ei0.1πbThen τ={0-,1-,A,B} is an intuitionistic complex fuzzy topology on X, C∘=0-=x,(0,ei(2π)):x∈X and C-=1-=x,(ei(2π),0):x∈X.Definition 34.
An intuitionistic complex fuzzy topological space(X,τ) is said to be extremely disconnected, if the closure of each CIFO-set is CIFO-set.Definition 35.
Let(X,τ) be an intuitionistic complex fuzzy topological space. A subset A of X is said to be CIF semiopen set (by short CIFSO-set) (resp., CIF preopen (by short CIPO-set), CIFα-open (by short CIFαO-set), CIFβ-open (by short CIβO-set), and CIb-open (by short CIbO-set). If A⊆A∘- (resp., A⊆A-∘, A⊆A∘-∘, A⊆A-∘-, and ⊆A∘-∪A-∘), the family of all CISO-set (resp., CIPO-set, CIαO-set, CIβO-set, and CIbO-set) in X is denoted by CISO(X) (resp., CIPO(X), CIαO(X), and CIbO(X)).The implications between these concepts in the following diagram and the converse are not true in general,(54)CIFO⇓⇓CISFOCIPFO⇓⇓CIFαO⇓CIbO⇓CIFβ-O


## 5. Conclusion

In this paper we introduced and studied a temporal complex intuitionistic fuzzy sets as generalization of complex Atanassov’s intuitionistic fuzzy sets by taking the time in the moving of the point; a correlation between two temporal complex intuitionistic fuzzy sets is discussed. A similarity between temporal complex intuitionistic fuzzy sets is main points in the paper as a generalization of the similarity introduced by Omar [10] and Sugeno [13]. We calculate the results by the program Maple 7. Finally we give an applications to know if the patient is suffering from the diseases or not and introduce the main building in a topology by using the same the set. In future research, similarity measures between temporal complex multifuzzy soft and applications in engineering, medical, physics, and automobiles will be studied.



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