Rough Continuity Represented by Intuitionistic Fuzzy Sets

Zolt

an Ern

o Csajb

Department of Health Informatics, Faculty of Health, University of Debrecen,

ost

ut 2-4, HU-4406 Ny

ıregyh

aza, Hungary

Keywords:

Rough Set Theory, Pawlak’s Approximation Spaces, Rough Real Functions, Fuzzy Sets, Intuitionistic Fuzzy

Sets.

Abstract:

Studying rough calculus was initiated by Z. Pawlak in his many papers. He originated the concept of rough

real functions. Like the notion of continuity in classical analysis, the rough continuity is also a central notion in

rough calculus. Relying on the Pawlak’s approximation spaces on the real closed bounded intervals, ﬁrst, two

intuitionistic fuzzy sets are established starting from rough functions. Then, based on them, some necessary

and sufﬁcient conditions for the rough continuity in terms of intuitionistic fuzzy set theory will be presented.

1 INTRODUCTION

In 1965, Lotﬁ A. Zadeh initiated the fuzzy set theory

(Zadeh, 1965) as a new mathematical theory to man-

age uncertainty. In the early 1980s, Zdzisław Pawlak

established a new mathematical tool also to manage

uncertainty which is called the rough set theory (RST)

(Pawlak, 1982).

Let U be a nonempty reference set which is com-

monly called the universe. Any set, classical or non-

classical, is formed from the elements of the universe.

They can be represented with more or less similar

tools, called membership functions.

A classical or crisp set S can be represented,

among other things, by its characteristic function

: U → {0,1} (Halmos, 1960; Hayden et al., 1968).

Generalizing this representation, a fuzzy set F is de-

ﬁned by a function µ

: U → [0,1] which is called the

fuzzy membership function. In rough set theory, how-

ever, the deﬁnition of a similar representation tool is

somewhat more complex.

In RST, ﬁrst, it is assumed that a beforehand pre-

deﬁned family of subsets of U is given. Namely,

this set family is a partition of U generated by an

equivalence relation. Any equivalence class can be

viewed as a set of indiscernible objects characterized

by the available information (knowledge) about them.

Accordingly, in RST an equivalence relation is actu-

ally called the indiscernible relation.

The partition is called the base system, and its

elements, i.e., the equivalence classes are the base

https://orcid.org/0000-0002-6357-0233

sets. From the base sets the so–called deﬁnable sets

are formed with the union operation.

Next, with the help of base sets, lower and up-

per approximation sets are formed for any S ⊆ U .

The former is the union of all base sets which are in-

cluded in S, whereas the latter is the union of all base

sets which have a nonempty intersection with S. The

difference of upper and lower approximation sets is

the boundary of S. S is exact if its boundary is the

empty set, otherwise it is rough.

In RST, the rough membership function,

rm–function in short, is deﬁned as follows. Let U be

ﬁnite. Then, the rm–function is commonly deﬁned by

(u) =

|JuK ∩ S|

|JuK|

where |·| denotes the number of elements of a set, and

JuK is the base set (equivalence class) to which u ∈

U belongs (|

0| = 0 by deﬁnition). This rm–function

quantiﬁes the degree of the relative overlap between

the set S and a base set.

Both characteristic and fuzzy membership func-

tions are of a priori nature. They have a wide range of

applications, see, e.g., (Aquino et al., 2020; de Jesus

Rubio, 2009; Chiang et al., 2019; Elias et al., 2020;

Meda-Campa

na, 2018; Hern

andez et al., 2020)

In contrast, in RST, initially some information

about the elements of the universe is necessary to have

at our disposal in order to be able to approximate a set.

Thus, the rm–function is of a posteriori nature.

Still, an rm–function can formally be viewed as a

special type of fuzzy membership function, of course,

264

Csajbók, Z.

Rough Continuity Represented by Intuitionistic Fuzzy Sets.

DOI: 10.5220/0010164302640274

In Proceedings of the 12th International Joint Conference on Computational Intelligence (IJCCI 2020), pages 264-274

ISBN: 978-989-758-475-6

 2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reser ved

with many constraints owing to their derived nature

(Yao and Zhang, 2000). However, the converse is not

true in general ((Biswas, 2000), Example 3.1).

It is broadly accepted that the fuzzy and rough

set theories are related but distinct and complemen-

tary to each other. Nevertheless, they can be com-

bined with each other (Dubois and Prade, 1987;

Dubois and Prade, 1992). Moreover, their common/

distinct features can be outlined based on their fuzzy/

rough membership functions (Chakraborty, 2011;

Csajb

ok and K

odm

on, 2020).

In Section 2 some basic notations are summarized

for the sake of full clarity. Then, the notion of rough

real function and its two possible representations will

be described.

Section 3 contains the material which is required

to establish the connection between the rough

continuity and intuitionistic fuzzy sets. It is the most

extensive section in the paper.

Section 4 presents the main result of the paper. It

provides a necessary and sufﬁcient conditions for the

rough continuity in terms of intuitionistic fuzzy set

theory.

2 ROUGH REAL FUNCTIONS

In the mid 1990s, relying on rough set theory,

Pawlak originated the study of rough calculus in many

papers (Pawlak, 1994; Pawlak, 1996; Pawlak, 1997).

Its basic notion is the rough real function. In

(Csajb

ok, 2020), employing Pawlak’s ideas, some ad-

ditional representations of rough real functions are

given. Two of them, pointwise and blockwise rep-

resentations, will be required in the rest of this paper.

Let U,V be two nonempty sets. A function f

is denoted by f : U → V , u 7→ f (u) with domain

Dom f = U and co-domain Im f = V . In addition,

u 7→ f (u) is the assignment or mapping rule of f . For

any S ⊆ U , f (S) = { f (u) | u ∈ S} ⊆ V is the direct

image of S. V

denotes the set of all such functions.

If f ,g ∈ V

, the operation f  g,  ∈ {+,−, ·, /},

and the relation f  g,  ∈ {=, 6=,≤,<,≥,>} are

understood pointwise.

R is the set of real numbers. R

≥0

denotes the set

of nonnegative real numbers.

Let a,b ∈ R (a ≤ b). [a,b] = {x ∈ R | a ≤ x ≤ b}

and ]a,b[= {x ∈ R | a <x <b} denote the closed and

open bounded intervals, respectively. It is easy to

interpret, then, the open-closed ]a,b] and closed-open

[a,b[ intervals.

By (·,·), we mean an ordered pair.

Let [n] = {0,1,.. . , n} ⊆ N denote a ﬁnite set

of natural numbers. Accordingly, ]n] = {1, . . . ,n},

[n[= {0,1,. . . , n − 1}, and ]n[= {1,...,n − 1}.

Throughout the paper, let I be a closed bounded

interval I = [0,a] (a ∈ R

≥0

, a > 0).

The initial notion of the rough calculus is the fol-

lowing.

Deﬁnition 1. A categorization or discretization of I

is the sequence S

= {x

}

i∈[n]

⊆ R

≥0

, where n ≥ 1 and

0 = x

< x

< ··· < x

= a. 

Let I

denote the equivalence relation generated

by the categorization S

. Let x,y ∈ I. xI

y if

• x = y = x

∈ S

for some i ∈ [n], or

• x,y ∈]x

i+1

[ for some i ∈ [n[.

The partition I/I

generated by I

is the following:

I/I

= {{x

},]x

[,{x

},... , ]x

n−1

[,{x

}},

where {x

} = [x

] (i ∈ [n]).

The block of the partition I/I

containing x ∈ I is

denoted by JxK

. In particular, if x ∈ S

, JxK

= {x}.

If x ∈ JxK

=]x

i+1

[, then JxK

denotes the closed

interval [x

i+1

]. When x ∈ S

, JxK

= JxK

= {x}.

In terms of RST terminology, I

is an indiscerni-

bility relation on I. The members of I/I

are the

base sets. Any union of base sets are referred to as

deﬁnable sets. By deﬁnition,

0 is deﬁnable. Their

collection is D

I/I

In RST, the domain and co–domain of the lower

and upper approximation functions are the power set

of I. In the rough calculus, however, the closed

bounded intervals of the form [0,x] (x ∈ I) will only

be approximated. Therefore, the lower and upper

approximations sets are deﬁned by

([0,x]) = {x

∈I | Jx

⊆ [0,x]}

= ∪{Jx

∈I/I

| Jx

⊆ [0,x]};

([0,x]) = {x

∈I | Jx

∩ [0,x] 6=

= ∪{Jx

∈I/I

| Jx

∩ [0,x] 6=

0}.

PAS(I) = (I, I/I

I/I

) is called the

Pawlak approximation space.

The boundary of [0,x] is

bnd

([0,x]) = u

([0,x]) \ l

([0,x]).

With a slight abuse of the notations, in order to

simplify the above notations, let us deﬁne the follow-

ing numbers:

(x) = max{x

∈ S

| x

≤ x},

(x) = min{x

∈ S

| x

≥ x}.

Then, it is easy to check that

• if x ∈ S

, then l

([0,x]) = [0,l

(x)] = [0,x] and

([0,x]) = [0,u

(x)] = [0,x];

Rough Continuity Represented by Intuitionistic Fuzzy Sets

265

• if x /∈ S

, then l

([0,x]) = [0, l

(x)] $ [0, x], and

([0,x]) = [0,u

(x)[ % [0,x];

• if x ∈ S

, then bnd

([0,x]) =

0; if x /∈ S

, then

bnd

([0,x]) = ]l

(x),u

(x)[ 6=

The number x ∈ I is exact with respect to PAS(I) if

(x) = u

(x), otherwise x is inexact or rough (Pawlak,

1996). Of course, x ∈ I is exact iff x ∈ S

In this context, the members of I/I

are called the

rough numbers with respect to PAS(I). In addition,

the categorization points in S

are called the roughly

isolated points with respect to PAS(I).

Let I = [0,a

] and J = [0,a

] be two closed

bounded intervals with a

, a

∈ R

≥0

, a

> 0. Let S

and P

be the categorizations of I and J, respectively,

where S

= {x

}

i∈[n]

and P

= {y

}

j∈[m]

⊆ R

≥0

in such

a way that m, n ≥ 1, and 0 = x

< x

< ··· < x

= a

0 = y

< y

< · · · < y

= a

. The corresponding

Pawlak approximation spaces are PAS(I), PAS(J).

A Cartesian coordinate system whose x and y

axes equipped with PAS(I) and PAS(J) is called the

)–coordinate system, or rough coordinate sys-

tem in short. Any function f ∈ J

attached to a rough

coordinate system is called the rough real function.

In order to make the rough coordinate system

easier to handle technically, the blocks of the parti-

tion I/I

are enumerated as follows.

: I/I

→ [2n],

JxK

7→



= 2i, if ∃i ∈ [n](JxK

= {x

}),

2i+1

= 2i + 1, if ∃i ∈ [n[(x ∈]x

i+1

[).

The inverse of N

is:

−1

: [2n] → I/I

7→

(

i/2

}, if i ≡ 0 (mod 2)

i−1

i+1

, if i ≡ 1 (mod 2)

The equivalence classes of J/J

can be enumer-

ated in the same way with the help of an enumeration

function N

. They are referred to as C

’s ( j ∈ [2m]).

Example 1. Figure 1 (a) depicts a rough coordinate

system with S

= {x

= 0, x

} and P

[0,1]

= 0, y

= 1}. Figure 1 (b) presents a

rough real function attached to this rough coordinate

system. 

Deﬁnition 2 ((Pawlak, 1994)). Let f ∈ J

. The

pointwise (S

)–lower and (S

)–upper approx-

imations of f are the functions

: I → P

, x 7→ l

( f (x)) = max{y ∈ P

| y ≤ f (x)},

f : I → P

, x 7→ u

( f (x))= min{y ∈ P

| y ≥ f (x)}.

f is pointwise exact at x if f (x) = f (x), otherwise

f is pointwise inexact or rough at x.

f is pointwise exact on I

⊆ I if f (x) = f (x) for all

x∈I

, otherwise f is pointwise inexact (rough) on I

.

Deﬁnition 3. Let f ∈ J

. The block by block,

blockwise in short, (S

)–lower and (S

)–upper

approximations of f are the functions

←→

: I → P

, x 7→ l

(inf f (JxK

)),

←→

f : I → P

, x 7→ u

(sup f (JxK

)).

The function f is blockwise exact on B

for some

i ∈ [2n] if f

←→

) =

←→

f (B

), that is, the direct images

of B

with respect to f

←→

and

←→

f are equal; otherwise

f is blockwise inexact (rough) on B

The function f is blockwise exact on I if f is

blockwise exact on all B

∈ I/I

, otherwise f is block-

wise inexact (rough) on I. 

Owing to the fact that inf f (JxK

) and sup f (JxK

)

are constant on every B

, the functions f

←→

and

←→

f are

constant on every B

(i ∈ [2n]). Accordingly, using the

word “blockwise” is appropriate.

Example 2. Figure 2 (a) depicts the pointwise lower

and upper approximations of f . f is pointwise exact

at x

iii

, and pointwise rough at all other

points.

Figure 2 (b) shows the blockwise lower and upper

approximations of f . f is blockwise exact only on

= {x

}, and blockwise rough on all other blocks.

It is easy to check the following simple but impor-

tant statement.

Lemma 1. Let f ∈ J

be a rough real function. Then,

←→

≤ f ≤ f ≤ f ≤

←→

holds on I.

3 DERIVING INTUITIONISTIC

FUZZY SETS FROM ROUGH

REAL FUNCTIONS

Let U be a nonempty set.

According to (Zadeh, 1965) a fuzzy set (FS) on U

is the function µ ∈ [0, 1]

; see also, (Klir and Yuan,

1995; Dubois and Prade, 2000; Zimmermann, 2001;

Ross, 2010). µ is also called the membership function.

F S (U) denotes the family of all fuzzy sets on U.

FCTA 2020 - 12th International Conference on Fuzzy Computation Theory and Applications

266

(a) (b)

Figure 1: A rough coordinate system and a rough real function.

(a) (b)

Figure 2: Pointwise and blockwise lower/upper approximations of f .

Let I = {[a,b] | 0 ≤ a ≤ b ≤ 1}.

Let µ

,ν

∈ F S (U) with µ

≤ ν

. An interval-

valued fuzzy set (IVFS) on U is the function

IV FS

: U → I, u 7→ [µ

(u),ν

(u)] (Gorzałczany,

1987). µ

IV FS

is also denoted simply by [µ

,ν

Let µ

,ν

∈ F S (U) with 0 ≤ µ

+ ν

≤ 1. An

intuitionistic fuzzy set (IFS) on U is deﬁned by

the function pair µ

IFS

= (µ

,ν

) (Atanassov, 1986;

Atanassov, 1999; Atanassov, 2012). µ

and ν

are the

IFS membership and IFS nonmembership functions,

respectively. π

= 1 − µ

− ν

∈ F S (U) is the IFS

indeterminacy function. The family of all intuitionis-

tic fuzzy sets on U is denoted by I F S (U).

Let µ

IFS

,µ

IFS

∈ I F S (U). Then,

• µ

IFS

= µ

IFS

if µ

= µ

and ν

= ν

;

• µ

IFS

⊆ µ

IFS

if µ

≤ µ

and ν

≥ ν

It is well known that every IVFS [µ

,ν

] cor-

responds to an IFS (µ

,1 − ν

), while every IFS

(µ

,ν

) corresponds to an IVFS [µ

,1 − ν

]

(Atanassov and Gargov, 1989; Bustince and Burillo,

1996).

There are many papers dealing with the interre-

lationship between rough set and intuitionistic fuzzy

set theory (Rizvi et al., 2002; Cornelis et al., 2003;

Zhou and Wu, 2011; Xu et al., 2014). In this paper,

the starting point is the rough real functions, i.e., real

functions managing them in rough coordinate sys-

tems. Thereafter, intuitionistic fuzzy sets are derived

from their pointwise and blockwise representations.

For the rest of this section, let PAS(I) and

PAS([0, 1]) be two Pawlak approximation spaces

deﬁned on the intervals I and [0,1] with the catego-

rizations S

= {x

= 0, x

,... , x

} and P

[0,1]

= {y

= 0,

,... , y

= 1}. In addition, let f ∈ [0, 1]

be a rough

function attached to the (S

[0,1]

)–coordinate system.

According to Deﬁnition 2 , f , f ∈ [0,1]

, that is,

the pointwise (S

[0,1]

)–lower and upper approxi-

mations of f are fuzzy sets. Moreover, f ≤ f also

holds. Hence, f

IV FS

= [ f , f ] forms an interval–valued

fuzzy set. Then, the function pair f

IFS

= ( f ,1 − f )

forms an intuitionistic fuzzy set. (The subscript “pw”

refers to “pointwise”.)

In the intuitionistic fuzzy set theory context, f and

1 − f are the IFS membership and nonmembership

functions, respectively, and π

−

= 1 − f − (1 − f ) =

Rough Continuity Represented by Intuitionistic Fuzzy Sets

267

f − f is the IFS indeterminacy function.

Similarly, according to Deﬁnition 3 , f

←→

f ∈

[0,1]

, that is, the blockwise (S

[0,1]

)–lower and

upper approximations of f are also fuzzy sets, and

←→

≤

←→

f holds, too. Hence, f

IV FS



←→



forms

an interval–valued fuzzy set, and so the function pair

IFS



←→

,1 −

←→



is an intuitionistic fuzzy set.

(The subscript “bw” refers to “blockwise”.)

In terms of intuitionistic fuzzy set theory, f

←→

and

1 −

←→

f are the IFS membership and nonmembership

functions, respectively, and π

←→

= 1 − f

←→

− (1 −

←→

f )

←→

f − f

←→

is the IFS indeterminacy function.

Intuitionistic fuzzy sets f

IFS

and f

IFS

are derived

from f with respect to a (S

[0,1]

)–coordinate sys-

tem. They are called the pointwise and blockwise

roughly derived intuitionistic fuzzy sets.

There are many different geometric interpreta-

tions of intuitionistic fuzzy sets. For our purposes,

the so–called “unit segments” representation will be

appropriate (cf. (Atanassov, 1999), Figure 1.3.).

Figure 3 in this way depicts the geometric interpre-

tations of f

IFS

= ( f , 1 − f ) and f

IFS

= ( f

←→

,1 −

←→

f ) .

Accordingly, in Figure 3 (a), unit segments f (x),

f (x) − f (x), and 1 − f (x) are assigned to every

x ∈ I; correspondingly, in Figure 3 (b), unit segments

←→

(x),

←→

f (x) − f

←→

(x), and 1 −

←→

f (x) are assigned to

every x ∈ I.

Proposition 1. The both inclusion relations

IFS

= ( f ,1 − f ) ⊆ f

IFS

= ( f

←→

,1 −

←→

f ) (1)

IFS

= ( f

←→

,1 −

←→

f ) ⊆ f

IFS

= ( f ,1 − f ) (2)

fail in general.

Proof. Due to Lemma 1 , in general,

• f ≤ f

←→

fails in the case of Equation 1 , and

• 1 −

←→

f ≥ 1 − f fails in the case of Equation 2 . 

Proposition 2. Let f

IFS

and f

IFS

be the pointwise

and blockwise roughly derived intuitionistic fuzzy

sets. Then, f

IFS

) = f

IFS

) for every x

∈ S

(i ∈ [n]) categorization point.

Proof. f

IFS

= f

IFS

⇔ ( f , 1 − f ) = ( f

←→

,1 −

←→

f ) ⇔

f = f

←→

and f =

←→

f . Then, the statement follows from

the fact that f (x

) = f

←→

) and f (x

) =

←→

f (x

) for

every x

∈ S

(i ∈ [n]) categorization point. 

Example 3. According to Figures 2 (a) and 2 (b),

• f (x

) = f

←→

) = y

= 0, f (x

) = f

←→

) = y

f (x

) = f

←→

) = y

, f (x

) = f

←→

) = y

f (x

) = f

←→

) = y

, f (x

) = f

←→

) = y

• f (x

) =

←→

f (x

) = y

, f (x

) =

←→

f (x

) = y

f (x

) =

←→

f (x

) = y

, f (x

) =

←→

f (x

) = y

f (x

) =

←→

f (x

) = y

, f (x

) =

←→

f (x

) = y

. 

Linked to the indeterminacy functions π

−

and

←→

, pointwise and blockwise indeterminacy regions

−

and Π

←→

are deﬁned by

−

= {(x,y) | x ∈ I, y = f (x) = f (x) or

y ∈] f (x), f (x)[ if f (x) 6= f (x)};

←→

(x,y)

x ∈ I, y = f

←→

(x) =

←→

f (x) or

y ∈

←→

(x),

←→

f (x)

if f

←→

(x) 6=

←→

f (x)

Example 4. In Figures 3 (a) and 3 (b), the areas ﬁlled

with grid pattern and solid circles depict the indeter-

minacy regions Π

−

and Π

←→

, respectively. 

Having deﬁned the indeterminacy regions, let us

deﬁne two families of functions with the help of them:

−

= {g | g : I → [0, 1],(x,g(x)) ∈ Π

−

←→

= {g | g : I → [0, 1],(x,g(x)) ∈ Π

←→

Example 5. In Figures 4 (a) and 4 (b) show a function

g from G

−

and a function g

from G

←→

, respectively.

Proposition 3. Let f be a rough real function. Then,

1. Π

−

⊆ Π

←→

2. Π

−

= Π

←→

if and only if f

IFS

= f

IFS

Proof. 1. Case x ∈ I, f (x) 6= f (x). Applying Lemma

1 , ] f (x), f (x)[ ⊆

←→

(x),

←→

f (x)

, and so

{(x,y) | x ∈ I, y ∈] f (x), f (x)[}

⊆

(x,y) | x ∈ I, y ∈

←→

(x),

←→

f (x)

Case x ∈ I, f (x) = f (x). If x

∈ S

, then

f (x

) = f (x

) = f

←→

) =

←→

f (x

Thus, (x

, f (x

)) ∈ {(x,y) | x ∈ I,y = f (x) = f (x)},

and (x

, f (x

)) ∈

(x,y) | x ∈ I, y = f

←→

(x) =

←→

f (x)

also holds.

FCTA 2020 - 12th International Conference on Fuzzy Computation Theory and Applications

268

(a) (b)

Figure 3: Geometric interpretations of f

IFS

= ( f , 1 − f ) and f

IFS

= ( f

←→

,1 −

←→

f ).

(a) (b)

Figure 4: A function g from G

−

, and a function g

from G

←→

Let x

/∈ S

. It may occur that

f (x

) = f (x

) = f

←→

) =

←→

f (x

Then, (x

, f (x

)) ∈ {(x, y) | x ∈ I,y = f (x) = f (x)} and

, f (x

)) ∈

(x,y) | x ∈ I,y = f

←→

(x) =

←→

f (x)

holds

at the same time.

If f

←→

)6=

←→

f (x

), then (x

, f (x

))∈ {(x, y) | x ∈I,

y = f (x) = f (x)}, and, due to Lemma 1 , (x

, f (x

)) ∈

(x,y) | x ∈ I, y ∈

←→

(x),

←→

f (x)

also holds.

2. (⇒) It should be proved that “Π

−

= Π

←→

implies f

IFS

= f

IFS

”. Instead, its contrapositive form

“ f

IFS

6= f

IFS

implies Π

−

6= Π

←→

” will be proved.

On one hand, f

IFS

6= f

IFS

⇔ ( f ,1 − f ) 6=

( f

←→

,1−

←→

f ) ⇔ f 6= f

←→

or f 6=

←→

f . On the other hand,

according to point 1. of this Proposition, Π

−

6= Π

←→

⇔ Π

←→

6⊆ Π

−

Let us assume that f 6= f

←→

(the case f 6=

←→

f can be

proved similarly). Then, there is an x

∈ I in such

a way that f

←→

) < f

). Hence, there is an

h ∈ G

←→

in such a way that f

←→

) < h(x

) < f (x

If f

←→

) < h(x

) < f (x

) ≤

←→

f (x

), then it

is straightforward that (x

,h(x

)) ∈

←→

f (x

)

but (x

,h(x

)) /∈] f (x

), f (x

)[. In other words,

,h(x

)) ∈ Π

←→

, but (x

,h(x

)) /∈ Π

−

, i.e., Π

←→

6⊆ Π

−

satisﬁes.

If f

←→

) < h(x

) < f (x

) = f (x

) ≤

←→

f (x

of course, (x

,h(x

)) ∈

←→

f (x

)

also holds.

In addition, h(x

) 6= f (x

) = f (x

). Therefore, in

this case, (x

,h(x

)) ∈ Π

←→

, but (x

,h(x

)) /∈ Π

−

, i.e.,

←→

6⊆ Π

−

also satisﬁes.

(⇐) It is straightforward. 

Example 6. In Figures 3 (a), (b), it can be observed

that the area of the pointwise indeterminacy region

−

is included in the area of the blockwise indetermi-

nacy region Π

←→

in accordance with Proposition 3 1.

As shown in Figure 3, f

IFS

6= f

IFS

, in particular,

Rough Continuity Represented by Intuitionistic Fuzzy Sets

269

(a) (b)

Figure 5: Π

−

6= Π

←→

both f

←→

≤ f and f ≤

←→

f satisfy. It can be seen that

−

⊆ Π

←→

but Π

−

6= Π

←→

in accordance with Propo-

sition 3 2. 

Example 7. Figure 2 (a) shows that the graph of f in-

tersects the horizontal line segments y = y

at x

, and

y = y

at x

. In Figure 5 (a), x

∈ ]x

[ in such a

way that y

= f

←→

) < h(x

) < f (x

) = y

. Moreover,

h(x

) ∈

←→

), f (x

)

= ]y

[ and f (x

) ∈

] f (x

), f (x

)[=]y

In other words, the points (x

, f (x

)) and

,h(x

)) are on the vertical line segment x = x

More precisely, (x

,h(x

)) is between the points

) and (x

), and (x

, f (x

)) is between the

points (x

) and (x

Moving on for any x ∈ ]x

[ and suitable func-

tions h ∈ G

←→

, the points (x, h(x))’s form the rectan-

gular area ]x

[ × ]y

[ .

This area is ﬁlled with

diagonal up pattern in Figure 5 (b). It is belongs to

←→

but does not belong to Π

−

. The area ﬁlled with

diagonal down pattern can be derived similarly. 

Corollary 1. Let f be a rough real function. Then,

1. G

−

⊆ G

←→

2. G

−

= G

←→

if and only if f

IFS

= f

IFS

Proof. These statements immediately follow from

Proposition 3 . 

In Figure 4 (b), it can be observed that g

∈G

←→

, but

←→

6= g

←→

f 6=

←→

. This is because f

←→

< y

= g

←→

on B

=]x

[, and

←→

f = y

> y

←→

on B

[. In other words, f

IFS

6= g

0IFS

. This observa-

tion motivates the following deﬁnition.

Here, × denotes the Cartesian product operation.

Deﬁnition 4. Let f

IFS

and f

IFS

be pointwise and

blockwise roughly derived IFSs.

• f

IFS

is roughly strong if f

IFS

= g

IFS

for all

g ∈ G

, otherwise f

IFS

is roughly weak.

• f

IFS

is roughly strong if f

IFS

= g

IFS

for all

g ∈ G

←→

, otherwise f

IFS

is roughly weak. 

In the case of f

IFS

, Figure 4 (a) suggests that f

IFS

is always roughly strong. That is what the following

proposition is about.

Proposition 4. For any pointwise roughly derived

IFS f

IFS

, f

IFS

is roughly strong, that is, f

IFS

= g

IFS

for all g ∈ G

−

Proof. Let g ∈ G

−

. Then, (x,g(x)) ∈ Π

−

Case x ∈ I, g(x) ∈ ] f (x), f (x)[ . In this case,

g(x) ∈ ] f (x), f (x)[= ]l

( f (x)),u

( f (x))[

= ] max{y ∈ P

[0,1]

| y ≤ f (x)},

min{y∈P

[0,1]

| y ≥ f (x)}[

= ]y

j+1

[

= ] max{y ∈ P

[0,1]

| y ≤ g(x)},

min{y∈P

[0,1]

| y ≥ g(x)}[

= ]g(x),g(x)[,

where y

j+1

∈ P

[0,1]

for some j ∈ [m[ .

That is,

IFS

(x) = ( f (x),1− f (x)) = (g(x),1−g(x)) = g

IFS

(x)

satisﬁes for all such x ∈ I that f (x) 6= f (x).

Case x ∈ I, g(x) = f (x) = f (x). Then, g(x) =

f (x) = f (x) = y

, where y

∈ P

[0,1]

for some j ∈ [m].

And so g(x) = y

= g(x) = g(x).

That is, f

IFS

(x) = g

IFS

(x) also holds for all such

x ∈ I that f (x) = f (x). 

FCTA 2020 - 12th International Conference on Fuzzy Computation Theory and Applications

270

Proposition 5. Let f

IFS

be a blockwise roughly

derived IFS.

IFS

is roughly strong if and only if f

IFS

= f

IFS

Proof. (⇒) On the contrary, let us assume that

IFS

6= f

IFS

. Then, such a function g ∈ G

←→

will be

constructed for which g

IFS

6= f

IFS

holds. However, it

contradicts the condition that f

IFS

is roughly strong.

IFS

6= f

IFS

⇔ ( f , 1 − f ) 6= ( f

←→

,1 −

←→

f ) ⇔

f 6= f

←→

or f 6=

←→

f . It us assumed that f 6= f

←→

, the case

f 6=

←→

f can be proved similarly. However, due to

Proposition 2 , f (x

) = f

←→

) satisﬁes for every x

∈

(i ∈ [n]) categorization point. Then, there should

be an open interval B

i−1

i+1

∈ I/I

, where

i ≡ 1 (mod 2) (i ∈ [2n[) in such a way that f 6= f

←→

i.e., f

←→

< f on B

The case f = f on B

is not possible, because it

would imply that f = f = f

←→

f on B

. However, it

contradicts the condition that f

←→

< f on B

When f 6= f on B

, let g ∈ G

←→

with the constraint

that g(x) ∈ ] f , f [ on B

. It is possible, because ] f , f [⊆

←→

on B

. Then, f

←→

< f = g

←→

on B

, and so

IFS

6= g

IFS

which is the requested contradiction.

(⇐) According to Proposition 4 , f

IFS

is roughly

strong, and so f

IFS

is roughly strong as well. 

4 ROUGH CONTINUITY AND

ROUGHLY DERIVED

INTUITIONISTIC FUZZY SETS

Rough continuity is a central notion in rough calculus

like the continuity in the classical real analysis.

Let I and J two real intervals with categorizations

and P

as they are given above.

Deﬁnition 5 ((Pawlak, 1996)). A rough real

function f ∈ J

is (S

)–continuous or roughly

continuous at x if

f (JxK

) ⊆ J f (x)K

Otherwise, f is (S

)–discontinuous or roughly

discontinuous at x ∈ I.

f is (S

)–continuous (roughly continuous)

on I

⊆ I if f is (S

)–continuous at every point of

. Otherwise, f is not roughly continuous on I

. 

Proposition 6 ((Csajb

ok, 2019)). A rough real

function f ∈ J

is (S

)–continuous at every x ∈ S

roughly isolated point.

Deﬁnition 6 ((Csajb

ok, 2019)). The (S

)-discon-

tinuity types of f ∈ J

are deﬁned as follows.

The rough discontinuity of f is called

(1) the rough jump discontinuity of the ﬁrst kind if it

is derived from touching a straight line y = y

for

some j ∈ [m];

(2) the rough jump discontinuity of the second kind if

it is derived from intersecting a straight line y = y

for some j ∈]m[;

(3) any other type of discontinuity is called the rough

jump discontinuity of the third kind. 

Example 8. Figure 6 (a) depicts rough jump disconti-

nuities of the ﬁrst and second type.

• f has the rough jump discontinuity of the ﬁrst

kind at x

because it is derived from touching the

straight line y = y

at x

f (Jx

) = f ([x

]) ⊆ ]y

]

6⊆ {y

} = J f (x

• f has the rough jump discontinuities of the

second kind at x

, x

, and x

because they are

derived from intersecting the line segments y = y

y = y

, and y = y

, respectively. For instance,

f (Jx

) = f ([x

]) ⊆ ]y

]

6⊆ {y

} = J f (x

The discontinuities at x

and x

can be showed in

similar way.

It should be noted that f touches at x

and in-

tersects at x

the line segment y = y

but f is still

continuous at both points. It can be seen that the

contact point is (x

) and the intersection point is

), that is, their both coordinates are categoriza-

tions points.

In Figure 6 (b), f has rough jump discontinuities

of the third kind at x

iii

and x

f (Jx

iii

) = f ([x

]) ⊆ ]y

[

6⊆ [y

] = J f (x

iii

;

f (Jx

) = f ([x

]) ⊆ ]y

[

6⊆ [y

] = J f (x

iii

Although, f is roughly continuous at x

pursuant

to Proposition 6 , it may cause rough jump discontinu-

ities of the third kind in blocks ]x

[ and/or ]x

specially, in ]x

[ at x

. 

Rough Continuity Represented by Intuitionistic Fuzzy Sets

271

(a) (b)

Figure 6: Rough discontinuities.

Proposition 7 ((Csajb

ok, 2019)). A rough real func-

tion f ∈ J

is (S

)–continuous on I if and only if f

does not have rough jump discontinuity of any kind.

Proposition 8. A rough real function f ∈ J

)–continuous on I if and only if the blockwise

roughly derived IFS f

IFS

is roughly strong.

Proof. (⇒) Since f is roughly continuous on I, then

f (JxK

) ⊆ J f (x)K

for all x ∈ I.

It is straightforward that on C

(∈ J/J

, j ∈ [2m]),

←→

f = y

j/2

if j ≡ 0 (mod 2), and f

←→

= y

j−1

←→

f = y

j+1

if j ≡ 1 (mod 2). Thus,

• f (x) = y

j/2

on JxK

if f (JxK

) = J f (x)K

= C

for some j ∈ [2m], j ≡ 0 (mod 2);

• y

j−1

≤ f (x) ≤ y

j+1

on JxK

if f (JxK

) ⊆

J f (x)K

= C

for some j ∈ [2m], j ≡ 1 (mod 2).

It means, considering the deﬁnition of Π

←→

and

←→

, that for all g ∈ G

←→

, f

IFS

= g

IFS

, i.e., f

IFS

roughly strong.

(⇐) On the the contrary, let us assume that f is

roughly discontinuous for some x ∈ I. Since f is

roughly continuous in every roughly isolated point,

see, Proposition 6 , x belongs to an open interval

= JxK

i−1

i+1

∈ I/I

for some i ∈ [2n[,

i ≡ 1 (mod 2).

First, it can be stated that f (JxK

) ∩ J f (x)K

because x ∈ JxK

, and so f (x) ∈ f (JxK

),J f (x)K

Moreover, there must be an x 6= x

∈ JxK

in such

a way that f (x) 6= f (x

) and f (x

) ∈ f (JxK

) but

f (x

) /∈ J f (x)K

, otherwise f (JxK

) ⊆ J f (x)K

would

be, which contradicts the assumption that f discontin-

uous at x.

It may occur that |J f (x)K

| = 1. It happens when

f (x) = f (x) = f (x), that is when f touches or inter-

sects a horizontal line segment y = y

at x.

Case |J f (x)K

=1. Then, J f (x)K

= {y

} for

some y

∈ P

, j ∈ [m]. Moreover, let us recall that

f (x

) 6= f (x) = y

First, let us assume that f touches the line segment

y = y

at x, i.e., f (x)= f (x)= f (x)=y

Let g ∈ G

←→

with the constraint that g(x) = y

JxK

. Such a function g exists, because f (x) = y

f (x

), and so

• f

←→

< y

≤

←→

f if f (x

) < y

, or

• f

←→

≤ y

←→

f if f (x

) > y

hold on JxK

Then, f

←→

< g

←→

= y

on JxK

if f (x

) < y

, while

←→

g = y

←→

f on JxK

if f (x

) > y

, i.e., f

IFS

6= g

IFS

which contradicts the condition that f

IFS

is roughly

strong.

Secondly, let us assume that f intersects the line

segment y = y

at x. In this case, j ∈]m[, and f (x) =

f (x) = f (x) = y

. Hereinafter, the proof is similar to

the previous case.

Case |J f (x)K

|>1. Then, J f (x)K

i−1

i+1

for some j ∈ [2m[, j ≡ 1 (mod 2). Moreover, let us

recall that f (x

) /∈

i−1

i+1

First, if f (x

) < y

i−1

, let g ∈ G

←→

with the

constraint that g(x) = y

i−1

on JxK

. Such a function

g exists, because f

←→

< y

i−1

≤

←→

f holds on JxK

Then, f

←→

< g

←→

= y

i−1

on JxK

, i.e., f

IFS

6= g

IFS

which contradicts the condition that f

IFS

is roughly

FCTA 2020 - 12th International Conference on Fuzzy Computation Theory and Applications

272

strong.

Secondly, if f (x

) > y

i+1

, let g ∈ G

←→

with the

constraint that g(x) = y

i+1

on JxK

. Such a function

g exists, because f

←→

≤ y

i+1

←→

f holds on JxK

Then, g

←→

= y

i+1

←→

f on JxK

, i.e., f

IFS

6= g

IFS

which contradicts the condition that f

IFS

is roughly

strong. 

Corollary 2. A rough real function f ∈ J

is (S

)–

continuous on I if and only if Π

−

= Π

←→

Proof. f is roughly continuous

⇔ f

IFS

is roughly strong by Proposition 8

⇔ f

IFS

= f

IFS

by Proposition 5

⇔ Π

−

= Π

←→

by Propositions 3 2. 

Corollary 3. A rough real function f ∈ J

is (S

)–

continuous on I if and only if G

−

= G

←→

Proof. f is roughly continuous

⇔ f

IFS

is roughly strong by Proposition 8

⇔ f

IFS

= f

IFS

by Proposition 5

⇔ G

−

= G

←→

by Corollary 1 2. 

5 CONCLUSION AND FUTURE

WORK

Rough continuity is a central notion in rough calculus.

This paper has characterized the rough continuity in

three different ways in terms of intuitionistic fuzzy

set theory.

This characterization establishes a connection be-

tween the two theories of uncertainty management,

the rough set theory and intuitionistic fuzzy set

theory. It may allow the application of the means of

intuitionistic fuzzy calculus in rough calculus.

In the future, the investigations can be continued

in several directions. This article has addressed only

one important concept of rough calculus, namely, the

rough continuity. First of all, rough continuity has

some additional features, such as rough discontinuity,

rough Darboux property or Intermediate Value

Property (IVP). The question is how they could also

be captured with the help of IFS tools. Moreover,

the relationships between additional notions of rough

calculus and IFS can also be studied.

Classical Pawlak’s rough set theory has many

different generalizations. The question is whether

they can be captured with IFS tools in one way or

another.

ACKNOWLEDGEMENTS

The author would like to thank the anonymous

referees for their useful comments and suggestions.

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