On continuity of complex fuzzy functions

Mathematical Theory and Modeling www.iiste.org
ISSN 2224-5804 (Paper) ISSN 2225-0522 (Online)
Vol.3, No.10, 2013
41
On Continuity of Complex Fuzzy Functions
Pishtiwan O. Sabir
Department of Mathematics, Faculty of Science and Science Education, University of Sulaimani, Iraq
pishtiwan.sabir@gmail.com
Abstract
In this paper, Some important theorems on fuzzy type I-continuous and II-continuous of complex fuzzy
functions mapping generalized rectangular valued bounded closed complex complement normalized fuzzy
numbers into itself are proved.
Keywords: Fuzzy Complex Numbers, Fuzzy Complex Functions, Fuzzy Continuity.
1. Introduction
It is well known that fuzzy complex numbers and fuzzy complex analysis were first introduced by (Buckley,
1989; Buckley and Qu, 1991, 1992). Scholars did series research about the properies of fuzzy complex number
from various aspects (Quan, 1996; Ma et al., 2009; Zheng and Ha, 2009). But these achievements were very
abstract, and it did not consummate until today. In view of (Buckley, 1989), Guangquan (1992) discussed the
limit theory of the sequence of fuzzy complex numbers in detail, giving a series of results about limit theory,
which are the counterparts of well-known results valid for real numbers in classical mathematics analysis.
Buckley (1989) suggested that introducing a metric on the space of fuzzy complex numbers provide to study
convergence, continuity and differentiation of fuzzy complex function (Chun and Ma, 1998; Qiu et al., 2000,
2001; Ousmane and Congxin, 2003; Shengquan, 2006; Cai, 2009; Sabir, 2012). On the basis of Buckley’s work,
some authors continued research and have extensively studied the theory of fuzzy complex numbers and fuzzy
complex analysis (Wu and Qiu, 1999; Zengtai and Shengquan, 2006; Qiu and Shu, 2008; Sun and Guo, 2010;
Sabir et al., 2012b). Sabir et al. (2012a) giving the definitions of the complement normalized fuzzy numbers
(CNFNs), bounded closed complex CNFNs (BCCCNFNs), generalized rectangular valued BCCCNFNs
(GRVBCCCNFNs) and discussed some of their basic properties. In section two, we first review the definitions
and characterizations related to fuzzy complex sets. We will also present the notations needed in the rest of the
paper. In the last section, some theorems on the continuity of complex fuzzy functions are proved.
2. Priliminaries
A fuzzy set defined on the universal set is a function , ∶ → 0,1 . Frequently, we will write ( )
instead of , . The family of all fuzzy sets in X is denoted by ℱ( ). The α⎯level of a fuzzy set , denoted
by , is the non-fuzzy set of all elements of the universal set that belongs to the fuzzy set at least to the
degree ∈ 0,1 . The weak α⎯level of a fuzzy set ∈ ℱ( ) is the crisp set that contains all elements of the
universal set whose membership grades in the given set are greater than but do not include the specified value of
α. The largest value of for which the α-level is not empty is called the height of a fuzzy set denoted .
The core of a fuzzy set is the non-fuzzy set of all points in the universal set X at which ! ( ) is
essentially attained.
Let # ∈ ℱ( ). Then the union of fuzzy sets # , denoted $ ## , is defined by $ %%
( ) = ! %
( ) =
⋁ %
( ), the intersection of fuzzy sets #, denoted ( ## , is defined by ( %%
( ) = )*+ %
( ) = ⋀ %
( ),
and the complement of #, denoted ¬ #, is defined by %
( ) + ¬ %
( ) = 1, for all in the universal set .
A fuzzy number a0 is a fuzzy set defined on the set of real numbers 12
characterized by means of a membership
function 0( ): 12
→ 0,1 , which satisfies: (1) 40 is upper semicontinuous, (2) 0( ) = 0 outside some interval
5, 6 , (3) There are real numbers 4, 7 such that 5 ≤ 4 ≤ 7 ≤ 6 and 0( ) is increasing on c, a , 0( ) is
decreasing on 7, 6 , 0( ) = 1, 4 ≤ ≤ 7. We denote the set of all fuzzy numbers by ℱ⋆
. A fuzzy complex
number ;< is defined by its membership functions =>(?) which is a mapping from the set of ordinary complex
numbers into [0,1] if and only if =>(?) is continuous; ;< is open, bounded, and connected; and ;<2
is non-
empty, compact, and arcwise connected. We use ℱ⋆⋆
to the set of all fuzzy complex numbers.
Let +(?′, ?′′) = A be any mapping from ℂ × ℂ into ℂ. Buckley (1989) extend + to ℱ⋆⋆
× ℱ⋆⋆
into ℱ⋆⋆
and
write + ;′D , ;′′E = FD if GD (A) = ⋁ ( =′D (?′) ∧ =′′E(?′′))I(JK,JKK)LM . One obtains FD = ;′D ⨁;′′E or FD = ;′D ⊙ ;′′E by
using + ;′D , ;′′E = ;′D ⊕ ;′′E or + ;′D , ;′′E = ;′D ⊙ ;′′E, respectively.
3. Properties of Continuous Complex Fuzzy Functions
In this section, we give the continuity of complex fuzzy function mapping GRVBCCCNFNs into itself. Most
results, definitions and standard notations on fuzzy complex analysis which are used in this section can be found

Vol.3, No.10, 2013
42
in Sabir et al. (2012a). Some of the results in this section are without proofs owing to the simplicities.
Definition 3.1. Let ℊR∗
⊆ ℱ¬U
∗∗
, and V be a mapping from ℊR∗
to the set of all GRVBCCCNFNs. If for arbitrary
; ∈ ℊR∗
, there exists unique WFX Y ∈ ℱ¬U
∗∗
, make V ; = WFX Y, we call V a complex fuzzy function defined on
ℊR∗
. Let WFX Y ∈ ℱ¬U
∗∗
, we say V ; is fuzzy continuous at WFX Y if for all Z[ > 0] there exists ^[ > 0] such that
_ `V ; , V WFX Y a ≺ Z[ as _ ; , WFX Y ≺ ^[.
Definition 3.2. We say V ; is fuzzy type I-continuous (resp. II-continuous) at WFX Y ∈ ℱ¬U
∗∗
if for each Z[ > 0]
there is ^[ > 0] such that V ; ≺ V WFX Y + Z[ (resp. V WFX Y ≺ V ; + Z[) whenever ; − WFX Y ≺ ^[.
Theorem 3.3. Let V and d both are fuzzy continuous at GRVBCCCNFN WFX Y then so is V ∗ d for ∗ ∈
e + , − , ∙ , / , ∨ , ∧ i.
Proof: We only prove for ∗ = + , the proof of the rest are similar. By hypothesis, for any Z[ > 0], there exists
^[ > 0], when _ ; , WFX Y ≺ ^[, _ `V ; , V WFX Y a ≺ Z[/2 and Γ `GX Z , GX WWX Y a ≺ ε]/2. Therefore, we have
Γ ` F + GX Z , F WWX Y + GX WWX Y a
= Γ `F Z , F WWX Y + GX WWX Y − GX Z a
≼ Γ `F Z , F WWX Y a + Γ `F WWX Y , F WWX Y + GX WWX Y − GX Z a
= Γ `F Z , F WWX Y a + Γ `GX WWX Y , GX Z a ≺ ε].
Theorem 3.4. Let V be fuzzy continuous at WFX Y ∈ ℱ¬U
∗∗
. Then there exist GRVBCCCNFNs ;r =
0r + ) 02 and ;2 = WsrY + )Ws2Y satisfy 0r
t
< `ReV ; a
t
< WsrY
t
and 02
t
< ÌmV ; a
t
<
Ws2Y
t
when _ ; ), WFX Y ≺ ^[ for ^[ > 0].
Theorem 3.5. Let V be fuzzy continuous function at WFX Y ∈ ℱ¬U
∗∗
, ;r ∈ ℱ¬U
∗∗
, and there exists ^[ > 0] such
that `ReV ; a
t
< Re ;r
t
(resp. `ReV ; a
t
> Re ;r
t
) and ÌmV ; a
t
< Im ;r
t
(resp.
ÌmV ; a
t
> Im ;r
t
) when _ ; ), WFX Y ≺ ^[ then `ReV WFX Y a
t
< Re ;r
t
(resp.
`ReV WFX Y a
t
> Re ;r
t
) and ÌmV WFX Y a
t
< Im ;r
t
(resp. ÌmV WFX Y a
t
> Im ;r
t
).
Theorem 3.6. Let V and d are both fuzzy type I-continuous (resp. II-continuous) and ∗ ∈ eW∨ , ∧ Y, ∙ i. Then
1. V ∗ d is also fuzzy type I-continuous (resp. II-continuous), such that `ReV ; a
z{
, ÌmV ; a
z{
,
`Red ; a
|{
, Ìmd ; a
|{
, `ReV ; a
|{
, ÌmV ; a
|{
, `Red ; a
z{
,
Ìmd ; a
z{
are all greater than zero for all } ∈ ~ r
2
.
2. V + d is also fuzzy type I-continuous (resp. II-continuous).
3. − d is fuzzy type II-continuous (resp. I-continuous).
4. 1 / d is also fuzzy type I-continuous (resp. II-continuous) such that, `Red ; a
zγ
,
`Red ; a
z
γ
, `Red ; a
|γ
, Ìmd ; a
|γ
, Ìmd ; a
zγ
, Ìmd ; a
z
γ
,
`Red ; a
|γ
, Ìmd ; a
|γ
are all positive for any } ∈ ~ r
2
.
Proof: By hypothesis, for any Z[ > 0] there is a ^′• > 0], ^′′]]] > 0] such that V ; ≺ V WFX Y + Z[ (resp. V WFX Y ≺
V ; + Z[) whenever ; − WFX Y ≺ ^′• and d ; ≺ d WFX Y + Z[ (resp. d WFX Y ≺ d ; + Z[) whenever
; − WFX Y ≺ ^′′]]].
1. Let ^[ = ^′• ∧ ^′′]]] and } ∈ ~ r
2
, we have
`ReV ; a
z{
< `ReV WFX Y a
z{
+ Z €resp. `ReV WFX Y a
z{
< `ReV ; a
z{
+ Z….
`ReV ; a
z
{
< `ReV WFX Y a
z
{
z
{
< `ReV ; a
z
{
+ Z….

Vol.3, No.10, 2013
43
ÌmV ; a
z{
< ÌmV WFX Y a
z{
+ Z €resp. ÌmV WFX Y a
z{
< ÌmV ; a
z{
+ Z….
ÌmV ; a
z
{
< ÌmV WFX Y a
z
{
z
{
< ÌmV ; a
z
{
+ Z….
`ReV ; a
|{
< `ReV WFX Y a
|{
+ Z †resp. `ReV WFX Y a
|{
< `ReV ; a
|{
+ Z‡.
`ReV ; a
|{
< `ReV WFX Y a
|{
|{
< `ReV ; a
|{
+ Z….
ÌmV ; a
|{
< ÌmV WFX Y a
|{
+ Z †resp. ÌmV WFX Y a
|{
< ÌmV ; a
|{
+ Z‡.
ÌmV ; a
|{
< ÌmV WFX Y a
|{
|{
< ÌmV ; a
|{
+ Z….
`Red ; a
z{
< `Red WFX Y a
z{
+ Z €resp. `Red WFX Y a
z{
< `Red ; a
z{
+ Z….
`Red ; a
z
{
< `Red WFX Y a
z
{
z
{
< `Red ; a
z
{
+ Z….
Ìmd ; a
z{
< Ìmd WFX Y a
z{
+ Z €resp. Ìmd WFX Y a
z{
< Ìmd ; a
z{
+ Z….
Ìmd ; a
z
{
< Ìmd WFX Y a
z
{
z
{
< Ìmd ; a
z
{
+ Z….
`Red ; a
|{
< `Red WFX Y a
|{
+ Z †resp. `Red WFX Y a
|{
< `Red ; a
|{
+ Z‡.
`Red ; a
|{
< `Red WFX Y a
|{
|{
< `Red ; a
|{
+ Z….
Ìmd ; a
|{
< Ìmd WFX Y a
|{
+ Z †resp. Ìmd WFX Y a
|{
< Ìmd ; a
|{
+ Z‡.
Ìmd ; a
|{
< Ìmd WFX Y a
|{
|{
< Ìmd ; a
|{
+ Z…. Hence,
`ReV ; a
z{
∗ `Red ; a
z{
< `ReV WFX Y a
z{
∗ `Red WFX Y a
z{
+ € `ReV WFX Y a
z{
+ `Red WFX Y a
z{
+ Z… ∗ Z
< `ReV WFX Y a
z{
∗ `Red WFX Y a
z{
+ € `ReV WFX Y a
zr
+ `Red WFX Y a
zr
+ Z… ∗ Z
= `ReV WFX Y a
z{
∗ `Red WFX Y a
z{
+ Z′
€resp. `ReV WFX Y a
z{
∗ `Red WFX Y a
z{
ˆ
< `ReV ; a
z{
∗ `Red ; a
z{
+ € `ReV ; a
z{
+ `Red ; a
z{
+ Z… ∗ Z
< `ReV ; a
z{
∗ `Red ; a
z{
+ € `ReV ; a
zr
+ `Red ; a
zr
+ Z… ∗ Z
= ˆ `ReV ; a
z{
∗ `Red ; a
z{
+ Z′′….
`ReV ; a
z
{
∗ `Red ; a
z
{
< `ReV WFX Y a
z
{
∗ `Red WFX Y a
z
{
+ Z′ €resp. `ReV WFX Y a
z
{
∗
γ+RedFŠ+<γ+ReV;Š+∗γ+Red;Š++Z′′.

Vol.3, No.10, 2013
44
ÌmV ; a
z{
∗ Ìmd ; a
z{
< ÌmV WFX Y a
z{
∗ Ìmd WFX Y a
z{
+ Z′ €resp. ÌmV WFX Y a
z{
∗
γ+ImdFŠ−<γ+ImV;Š−∗γ+Imd;Š−+Z′′.
ÌmV ; a
z
{
∗ Ìmd ; a
z
{
< ÌmV WFX Y a
z
{
∗ Ìmd WFX Y a
z
{
+ Z′ €resp. ÌmV WFX Y a
z
{
∗
γ+ImdFŠ+<γ+ImV;Š+∗γ+Imd;Š++Z′′
`ReV ; a
|{
∗ `Red ; a
|{
< `ReV WFX Y a
|{
∗ `Red WFX Y a
‹{
+ Z′ †resp. `ReV WFX Y a
|{
∗
γ+RedF −<γ+ReV; −∗γ+Red; −+Z′′.
`ReV ; a
|{
∗ `Red ; a
|{
< `ReV WFX Y a
|{
∗ `Red WFX Y a
|{
+ Z′ €resp. `ReV WFX Y a
|{
∗
γ+RedF +<γ+ReV; +∗γ+Red; ++Z′′.
ÌmV ; a
|{
∗ Ìmd ; a
|{
< ÌmV WFX Y a
|{
∗ Ìmd WFX Y a
|{
+ Z′ †resp. ÌmV WFX Y a
|{
∗
γ+ImdF −<γ+ImV; −∗γ+Imd; −+Z′′.
ÌmV ; a
|{
∗ Ìmd ; a
|{
< ÌmV WFX Y a
|{
∗ Ìmd WFX Y a
|{
+ Z′
€resp. ÌmV WFX Y a
|{
∗ Ìmd WFX Y a
|{
< ÌmV ; a
|{
∗ Ìmd ; a
|{
+ Z′′….
2. For any } ∈ ~ r
2
, we have
`ReV ; a
z{
+ `Red ; a
z{
< `ReV WFX Y a
z{
+ `Red WFX Y a
z{
+ 2Z €resp. `ReV WFX Y a
z{
+
γ+RedFŠ−<γ+ReV;Š−+γ+Red;Š−+2Z.
`ReV ; a
z
{
+ `Red ; a
z
{
< `ReV WFX Y a
z
{
+ `Red WFX Y a
z
{
+ 2Z €resp. `ReV WFX Y a
z
{
+
γ+RedFŠ+<γ+ReV;Š++γ+Red;Š++2Z.
ÌmV ; a
z{
+ Ìmd ; a
z{
< ÌmV WFX Y a
z{
+ Ìmd WFX Y a
z{
+ 2Z €resp. ÌmV WFX Y a
z{
+
γ+ImdFŠ−<γ+ImV;Š−+γ+Imd;Š−+2Z.
ÌmV ; a
z
{
+ Ìmd ; a
z
{
< ÌmV WFX Y a
z
{
+ Ìmd WFX Y a
z
{
+ 2Z €resp. ÌmV WFX Y a
z
{
+
γ+ImdFŠ+<γ+ImV;Š++γ+Imd;Š++2Z
`ReV ; a
|{
+ `Red ; a
|{
< `ReV WFX Y a
|{
+ `Red WFX Y a
‹{
+ 2Z
†resp. `ReV WFX Y a
|{
+ `Red WFX Y a
|{
< `ReV ; a
|{
+ `Red ; a
|{
+ 2Z‡.
`ReV ; a
|{
+ `Red ; a
|{
< `ReV WFX Y a
|{
+ `Red WFX Y a
|{
+ 2Z
€resp. `ReV WFX Y a
|{
+ `Red WFX Y a
|{
< `ReV ; a
|{
+ `Red ; a
|{
+ 2Z….

Vol.3, No.10, 2013
45
ÌmV ; a
|{
+ Ìmd ; a
|{
< ÌmV WFX Y a
|{
+ Ìmd WFX Y a
|{
+ 2Z
†resp. ÌmV WFX Y a
|{
+ Ìmd WFX Y a
|{
< ÌmV ; a
|{
+ Ìmd ; a
|{
+ 2Z‡.
ÌmV ; a
|{
+ Ìmd ; a
|{
< ÌmV WFX Y a
|{
+ Ìmd WFX Y a
|{
+ 2Z
€resp. ÌmV WFX Y a
|{
+ Ìmd WFX Y a
|{
< ÌmV ; a
|{
+ Ìmd ; a
|{
+ 2Z….
3. For any } ∈ ~ r
2
, we have
− `Red ; a
z{
> − `Red WFX Y a
z{
+ Z €resp. − `Red WFX Y a
z{
> − `Red ; a
z{
+ Z….
− `Red ; a
z
{
> − `Red WFX Y a
z
{
z
{
> − `Red ; a
z
{
+ Z….
− Ìmd ; a
z{
> − Ìmd WFX Y a
z{
+ Z €resp. − Ìmd WFX Y a
z{
> − Ìmd ; a
z{
+ Z….
− Ìmd ; a
z
{
> − Ìmd WFX Y a
z
{
z
{
> − Ìmd ; a
z
{
+ Z….
− `Red ; a
|{
> − `Red WFX Y a
|{
+ Z †resp. − `Red WFX Y a
|{
> − `Red ; a
|{
+ Z‡.
− `Red ; a
|{
> − `Red WFX Y a
|{
|{
> − `Red ; a
|{
+ Z….
− Ìmd ; a
|{
> − Ìmd WFX Y a
|{
+ Z †resp. − Ìmd WFX Y a
|{
> − Ìmd ; a
|{
+ Z‡.
− Ìmd ; a
|{
> − Ìmd WFX Y a
|{
|{
> − Ìmd ; a
|{
+ Z….
4. Obvious.
Theorem 3.7. If V and d are a fuzzy type I-continuous and II-continuous such that `ReV ; a
γ
≤
`Red ; a
γ
and ÌmV ; a
γ
≤ Ìmd ; a
γ
for every ; ∈ ℱ¬U
∗∗
and } ∈ ~ r
2
, then there is a fuzzy
continuous function ŒX satisfy `ReV ; a
γ
≤ `ReŒX ; a
γ
≤ `Red ; a
γ
and ÌmV ; a
γ
≤
ÌmŒX ; a
γ
≤ Ìmd ; a
γ
.
Here, an open problem is presented for further investivigations: One can study that V is both fuzzy type I-
continuous and II-continuous if and only if V is fuzzy continuous.
References
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