Rough Real Functions and Intuitionistic L-fuzziness
Zolt
´
an Ern
˝
o Csajb
´
ok
a
Department of Health Informatics, Faculty of Health Sciences, University of Debrecen,
S
´
ost
´
oi
´
ut 2-4, HU-4406 Ny
´
ıregyh
´
aza, Hungary
Keywords:
Rough Set Theory, Intuitionistic Fuzzy Sets, L-fuzzy Sets, Intuitionistic L-fuzzy Sets.
Abstract:
This work has been motivated by developing tools to manage rough real functions. Rough real function is a
real function attached to a special Cartesian coordinate system. Its values are categorized via the x and y axes.
Some papers establish a connection between the rough real functions and the intuitionistic fuzzy sets to achieve
the set goal. Until now, rough real functions could only take values from the unit interval. This paper presents
the possible extension of the previous methods to more realistic rough real functions. However, care must be
taken to ensure that the selected tools are semantically consistent with the nature of the rough real functions.
1 INTRODUCTION
In the mid-1990s, Z. Pawlak, based on the rough set
theory (RST) (Pawlak, 1982; Pawlak, 1991; Pawlak
and Skowron, 2007), initiated studying the rough cal-
culus of real functions (Pawlak, 1994; Pawlak, 1995a;
Pawlak, 1996; Pawlak, 1997).
Relying on the representations of rough real func-
tions, some papers established a connection between
the rough real functions and the intuitionistic fuzzy
sets (Csajb
´
ok, 2020; Csajb
´
ok, 2022). So far, these
methods have only dealt with rough real functions that
take values in the unit interval.
Very early, in 1967, the idea was born that the co-
domain of fuzzy sets should be a lattice, usually a
complete distributive lattice L ((Pradera et al., 2007),
p.17). That was the L-fuzzy set which was proposed
by J. Goguen (Goguen, 1967). Soon, in 1984, K.
Atanassov and S. Stoeva further generalized L-fuzzy
set to intuitionistic L-fuzzy set, see (Atanassov and
Stoeva, 1984) and (Atanassov, 1999).
Many papers combine rough set theory with fuzzy
sets (Dubois and Prade, 1980; Dubois and Prade,
1987; Dubois and Prade, 1990; Dubois and Prade,
1992), and with intuitionistic fuzzy sets (Abdunabi
and Shletiet, 2021; Tripathy, 2006; Zhou and Wu,
2011; Zhou et al., 2009).
Additional articles study dependency relation-
ships between generalized fuzzy set theories. In sev-
eral cases, equivalences between two theories have
been demonstrated (Deschrijver and Kerre, 2003;
a
https://orcid.org/0000-0002-6357-0233
Cornelis et al., 2003; Hatzimichailidis and Pa-
padopoulos, 2007). An exhaustive list of possible
connections, with appropriate literature references,
can be found in ((Pradera et al., 2007), Ch. 3). In
(Deschrijver and Kerre, 2003), Figure 1 graphically
depicts some important relationships.
This paper is not about combining rough set the-
ory and intuitionistic L-fuzzy sets. It starts from a
nonnegative rough real function f and finds one or
more special intuitionistic L-fuzzy sets that closely
relate to f semantically. Since the paper deals with
rough real functions, for the sake of simplicity, let
L be a closed real interval, i.e. a linearly (totally)
ordered complete lattice with the usual ≤ ordering.
Section 2 summarizes some basic notations. Sec-
tion 3 defines the rough real functions. Section 4 out-
lines the representations of rough real functions. Sec-
tion 5 discusses the notions of roughness and fuzzi-
ness, whereas Section 6 shows the roughness and L-
fuzziness. Section 7 presents some basic facts about
the roughly derived intuitionistic L-fuzzy sets.
2 BASIC NOTATIONS
To avoid misunderstandings, we summarize the main
notations used in this paper.
Let U and V be two nonempty sets.
A function f is denoted by f : U → V , u 7→ f (u)
with domain U and codomain V ; u 7→ f (u) is the
assignment or mapping rule of f . V
U
denotes the set
of all functions from U into V .
Csajbók, Z.
Rough Real Functions and Intuitionistic L-fuzziness.
DOI: 10.5220/0011553100003332
In Proceedings of the 14th International Joint Conference on Computational Intelligence (IJCCI 2022), pages 183-190
ISBN: 978-989-758-611-8; ISSN: 2184-3236
Copyright © 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)
183

For any S ⊆ U, f (S) = { f (u) | u ∈ S} ⊆ V is the
direct image of S. f (U ) is the range of f .
Let S ⊆ U. S
c
is the complement of S with respect
to U. If f ∈ V
U
, the complement of f (S) with respect
to V is denoted by f
c
(S) instead of ( f (S))
c
.
P (U) is the set of subsets of U, that is the power
set of U.
U ×V is the Cartesian product of U and V .
If U,V ⊆ R and f ,g ∈ V
U
, the operation f  g,
 ∈ {+, −, ·, /}, the relation f  g, ∈ {=, ≤, <
, ≥, >}, the intervals [ f ,g], ] f ,g], [ f , g[, ] f , g[, and
the ordered pair ( f ,g) are understood pointwise.
Let R, R
≥0
, R
+
represent the real numbers, non-
negative real numbers, and positive real numbers,
respectively.
If a,b ∈ R and a ≤ b, [a, b] = {x ∈ R | a ≤ x ≤
b} and ]a,b[ = {x ∈ R | a < x < b} are closed and
open intervals. The degenerate interval [a,a] = {a}
is identified with the real number a ∈ R. It is easy to
interpret the open-closed ]a,b] and closed-open [a,b[
intervals.
By [a,b], ]a,b], [a,b[, and ]a,b[ we mean the
closed interval [a, b].
|[a,b] | is the length of [a,b].
(·,·) denotes an ordered pair.
Let [n] = {0, 1,..., n} ⊆ N be a finite set of
natural numbers. Accordingly, ]n] = {1,.. .,n},
[n[ = {0,1,... , n −1}, and ]n[ = {1, ...,n − 1}.
3 ROUGH REAL FUNCTIONS
Pawlak’s rough real calculus relies on the notion of
Pawlak’s approximation space.
Let U be a nonempty set. Then PAS (U) =
(U,B,D
B
,`,u) is a Pawlak’s approximation space if
• B = Π (U) is a partition of U; its equivalence
classes are called base sets.
• D
B
is defined with inductive definition:
/
0 ∈ D
B
,
B ⊆ D
B
; if D
1
,D
2
∈ D
B
, then D
1
∪ D
2
∈ D
B
.
The members of D
B
are called definable sets.
• Lower and upper approximation operators ` and
u are defined as
– ` : P (U) → D
B
, S 7→ ∪{B ∈ B | B ⊆ S};
– u : P (U) → D
B
, S 7→ ∪{B ∈ B | B ∩S 6=
/
0}.
The main features of Pawlak’s approximation
space are characterized with the following notions:
• For any S ∈ P (U), the boundary of S is bnd(S) =
u(S) \`(S).
It is easy to check that bnd(S) is also definable.
• S ∈ P (U) is crisp (exact) if `(S) = u(S), i.e.,
bnd(S) =
/
0.
• S ∈ P (U) is rough (inexact) if it is not exact, i.e.,
bnd(S) 6=
/
0.
The distinguishing feature of Pawlak’s approx-
imations spaces is that the notions of exactness and
definability coincide. Formally, D ∈ D
B
iff `(D) =
D = u(D).
For each S ∈ P (U), the approximation operators `
and u divide U into three mutual disjoint regions:
• POS(S) = `(S), positive region of S;
• NEG(S) = U \ u(S) = u
c
(S), negative region of S;
• BND(S) = bnd(S), boundary region of S.
Common semantic interpretations of these regions
are the following:
• positive region POS(S) = `(S) consists of all ele-
ments of U which certainly belong to S;
• upper approximation u(S) = POS(S) ∪ BND(S)
consists of all elements of U which possibly be-
long to S;
• negative region NEG(S) = U \ u(S) = u
c
(S) con-
sists of of all elements of U which certainly do not
belong to S.
After these, the starting notions of the rough
calculus are the categorized interval and the rough
coordinate system. Then the rough real function is
a real function attached to a rough coordinate system
(Pawlak, 1996; Pawlak, 1997; Pawlak, 1994; Pawlak,
1995b; Pawlak, 1995a).
Let I denote a closed interval of the form I = [0,a]
(a ∈ R
+
).
Definition 1. A categorization of I is a strictly mono-
tone, finite sequence I
S
= {x
i
}
i∈[n]
⊂ R
≥0
, where
n ≥ 1 and 0 = x
0
< x
1
< · · · < x
n
= a.
Elements of I
S
are called the categorization points
of I; I equipped with a categorization I
S
is called the
I
S
-categorized interval. 
Let E
S
denote the equivalence relation generated
by I
S
. That is for every x,y ∈ I, xE
S
y if x = y = x
i
∈ I
S
for some i ∈ [n], or x,y ∈ ]x
i
,x
i+1
[ for some i ∈ [n[.
Then the partition generated by E
S
is
I/E
S
= {{x
0
},]x
0
,x
1
[,{x
1
},...,]x
n−1
,x
n
[,{x
n
}},
where [x
i
,x
i
] = {x
i
} (i ∈ [n]).
The members of I/E
S
are called the rough num-
bers; within the rough numbers, the singletons formed
by the categorization points are also called rough
integers or roughly isolated points.
The block containing x ∈ I is denoted by JxK
E
S
.
To make the blocks of the partition I/E
S
easier
to handle technically, its members are enumerated as
follows:
FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications
184

B
0
= {x
0
},B
1
= ]x
0
,x
1
[,
B
2
= {x
1
},B
3
= ]x
1
,x
2
[,. . .,B
2n
= {x
n
}.
The blocks formed by the categorization points
have even indices, i.e., i ≡ 0 (mod 2) (i ∈ [2n]),
while the open intervals have odd indices, i.e.,
i ≡ 1 (mod 2) (i ∈ [2n]).
Blocks of I/E
S
and B
i
’s are used interchangeably:
• If x
i
∈ I
S
for some i ∈ [n], Jx
i
K
E
S
= B
2i
= {x
i
};
furthermore Jx
i
K
E
S
= B
i
= [x
i
,x
i
] = [x
i
,x
i
] = {x
i
}.
• If x ∈ I but x /∈ I
S
, x ∈ JxK
E
S
= B
2i+1
= ]x
i
,x
i+1
[
for some i ∈ [n[; furthermore JxK
E
S
= B
2i+1
=
]x
i
,x
i+1
[ = [x
i
,x
i+1
].
In terms of rough set theory, E
S
is an indiscerni-
bility relation on I. The members of I/E
S
are called
the base sets, and any union of them are the definable
sets. By definition,
/
0 is definable. Their collection is
denoted by D
I/E
S
. In rough calculus, intervals of the
form [0,x] (x ∈ I) are approximated. Their Pawlak’s
lower and upper approximation operators `
S
and u
S
on I are defined as
• `
S
([0,x]) = ∪{Jx
0
K
E
S
∈ I/E
S
| Jx
0
K
E
S
⊆ [0, x]},
• u
S
([0,x]) = ∪{Jx
0
K
E
S
∈ I/E
S
| Jx
0
K
E
S
∩ [0, x] 6=
/
0}.
In sum, PAS (I)
S
=(I,I/E
S
,D
I/E
S
,`
S
,u
S
) is a Pawlak’s
approximation space on I.
Let PAS(I) and PAS(J) be two Pawlak’s approx-
imation spaces defined on the intervals I = [0,a] and
J = [0, b] on the x-axis and y-axis, respectively. Their
categorizations are I
S
= {x
0
= 0, x
1
,.. . ,x
n
= a} and
J
P
={y
0
=0,y
1
,.. . ,y
m
=b}. The equivalence classes
of J/E
P
can be enumerated in the same way as those
of I/E
S
. Latter classes are referred to as C
j
’s ( j ∈
[2m]).
A Cartesian coordinate system equipped its x and
y axes with Pawlak’s approximation spaces PAS (I)
S
and PAS (J)
P
is called the (I
S
,J
P
)–rough coordinate
system.
A function f ∈ J
I
attached to a rough coordinate
system is called the (I
S
,J
P
)–rough real function.
Throughout the paper, if there is no confusion, the
ordered pair (I
S
,J
P
) will be omitted.
4 REPRESENTATIONS OF
ROUGH REAL FUNCTIONS
To manage rough real functions, they must be repre-
sented. There are two types of representations, dis-
crete and non-discrete ones. For our purposes, the
non-discrete representations are the right ones. These
representations may happen either pointwise or
blockwise. The former comes from Pawlak.
In pointwise representation, lower and upper
approximations are assigned to all function values
point by point on the interval I.
Definition 2 (Pawlak, 1994). (Pawlak, 1994) Let
f ∈ J
I
be an (I
S
,J
P
)–rough real function.
The pointwise (I
S
,J
P
)–lower and (I
S
,J
P
)–upper
approximations of f are the functions
f : I → J
P
, x 7→ max{y∈J
P
| y ≤ f (x)},
f : I → J
P
, x 7→ min{y∈J
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
0
⊆ I if f (x) = f (x) for
all x ∈ I
0
, otherwise f is pointwise inexact (rough) on
I
0
. 
Definition 3. Let f ∈ J
I
be an (I
S
,J
P
)–rough real
function.
The blockwise lower and upper approximations of
f ∈ J
I
are the functions
f
←→
: I → J
P
, x 7→ max{y ∈J
P
| y ≤ inf f (JxK
E
S
)},
←→
f : I → J
P
, x 7→ min{y ∈J
P
| y ≥ sup f (JxK
E
S
}.
f is blockwise exact on B
i
if f
←→
(B
i
) =
←→
f (B
i
); other-
wise f is blockwise inexact (rough) on B
i
.
f is blockwise exact on a definable set D ∈ D
I/E
S
if f is blockwise exact on its all constituents B
i
⊆ D;
otherwise f is blockwise inexact (rough) on D.
In particular, f is blockwise exact on I if f is
blockwise exact on all base sets. 
Remark 1. Definition 2 and Definition 3 do not use
the I
S
categorization directly. However, these defini-
tions deal with rough real functions, which are treated
in rough coordinate systems by definition, so I
S
is not
ignorable. 
Remark 2. Although the approximation functions f
←→
and
←→
f are formally defined point by point, they are
constant on every block because inf f (JxK
E
S
) and
sup f (JxK
E
S
) are constant on every B
i
. Thus, the use
of the term ”blockwise” is justifiable. 
Proposition 1. Let f ∈ J
I
be an (I
S
,J
P
)-real function.
(a) f is pointwise exact at x ∈ I if and only if f (x) =
y
j
∈ J
P
for some j ∈ [m].
(b) f is pointwise inexact at x ∈ I if and only if
f (x) ∈

f
(x), f (x)

= ]y
j
,y
j+1
[ with y
j
,y
j+1
∈ J
P
for some j ∈ [m[.
Proof. (a) It is straightforward. (Csajb
´
ok, 2022),
Proposition 1.
(b) It is the contrapositive of (a). 
Rough Real Functions and Intuitionistic L-fuzziness
185

Proposition 1 (a) geometrically means that f is
pointwise exact at a point in I iff at this point f
touches or intersects a horizontal line segment y = y
j
for some y
j
∈ J
P
.
Proposition 2 ((Csajb
´
ok, 2022), Proposition 2).
Let f ∈ J
I
be an (I
S
,J
P
)-real function.
f is blockwise exact on B
i
∈ I/E
S
if and only if
f (x) = y
j
∈ J
P
on B
i
for some j ∈ [m].
Proposition 2 geometrically means that f is block-
wise exact on B
i
iff f coincides on B
i
with a horizontal
line segment y = y
j
for some y
j
∈ J
P
.
It is easy to check that the blockwise exactness on
a block implies the pointwise exactness on the same
block. The reverse statement, however, is not true.
Indeed, if f is pointwise exact on an open B
i
, it may
occur that f (x)=y
j
on B
0
( B
i
and f (x)=y
j
0
on B
i
\B
0
for some y
j
6= y
j
0
∈ J
P
. Then, on B
i
, f is pointwise
exact but not blockwise.
It is easy to check the following simple but impor-
tant statement.
Lemma 1. Let f ∈ J
I
be an (I
S
,J
P
)–rough real func-
tion. Then
f
←→
≤ f ≤ f ≤ f ≤
←→
f .
Let f ∈ J
J
. Then, for every f (x) ∈ J, positive,
negative, and boundary regions can be defined based
on the pointwise approximation of f as follows:
POS
−
( f (x)) = `
S
([0, f (x)])
= ∪{Jx
0
K
E
S
∈ I/E
S
| Jx
0
K
E
S
⊆ [0, f (x)]}
=

0, f (x)

NEG
−
( f (x)) = J \ u
S
([0, f (x)]) = [0,b]
\ ∪ {Jx
0
K
E
S
∈ I/E
S
| Jx
0
K
E
S
∩ [0, f (x)] 6=
/
0}
=

[0,b] \

0, f (x)

=

f (x),b

, if f (x) ∈ J
P
[0,b] \

0, f (x)

=

f (x),b

, if f (x) 6∈ J
P
BND
−
( f (x)) = u
S
([0, f (x)]) \ `
S
([0, f (x)])
=




0, f (x)

\

0, f (x)

=
/
0, if f (x) ∈ J
P

0, f (x)

\

0, f (x)

=

f (x), f (x)

,
if f (x) 6∈ J
P
Similar formulas can be derived based on block-
wise approximations of f :
POS
↔
( f (x)) = `
S

0, f (x)

=

0, f
←→
(x)

NEG
↔
( f (x)) =
=
(

0,b

\

0,
←→
f (x)

=

←→
f (x), b

, if f (x) ∈ J
P

0,b

\

0,
←→
f (x)

=

←→
f (x), b

, if f (x) 6∈ J
P
BND
↔
( f (x)) =
=






0, f (x)

\

0, f (x)

=
/
0, if f (x) ∈ J
P

0,
←→
f (x)

\

0, f
←→
(x)

=

←→
f (x), f
←→
(x

,
if f (x) 6∈ J
P
5 ROUGHNESS AND FUZZINESS
A fuzzy set (FS) on U is a function µ ∈ [0,1]
U
(Zadeh,
1965). It is called the membership function. F S (U )
denotes the family of all fuzzy sets on U.
Let I = {[a,b] | 0 ≤ a ≤ b ≤ 1}.
If µ
A
,ν
A
∈ F S (U) with µ
A
≤ ν
A
, the function
µ
IV FS
A
: U → I, u 7→ [µ
A
(u),ν
A
(u)] forms an interval-
valued fuzzy set (IVFS) on U (Gorzałczany, 1987).
An IVFS is also denoted by µ
IV FS
A
= [µ
A
,ν
A
].
Let µ
A
,ν
A
∈ F S(U) with 0 ≤ µ
A
+ ν
A
≤ 1.
Intuitionistic fuzzy set (IFS) on U is the function pair
µ
IFS
A
= (µ
A
,ν
A
) (Atanassov, 1986; Atanassov, 1999;
Atanassov, 2012). µ
A
is the membership, ν
A
is the
nonmembership, and π
A
= 1 − µ
A
− ν
A
∈ F S (U) is
the hesitancy or indeterminacy function.
The family of all intuitionistic fuzzy sets on U is
denoted by I F S (U).
Any µ ∈ F S (U) may be viewed as a special IFS
µ
IFS
A
with the membership function µ
A
= µ and the
derived nonmembership function ν
A
= 1 − µ, i.e.,
µ
IFS
A
= (µ,1 − µ). It is clear that an µ
IFS
A
∈ I F S (U) is
a fuzzy set iff π
A
= 0 or, what is the same, µ
A
+ν
A
= 1.
It is well known that every IVFS [µ
A
,ν
A
]
corresponds to an IFS (µ
A
,1 − ν
A
). On the contrary,
every IFS (µ
A
,ν
A
) corresponds to an IVFS [µ
A
,1−ν
A
]
(Atanassov and Gargov, 1989; Bustince and Burillo,
1996).
In order to establish a connection between the
rough real functions and the intuitionistic fuzzy sets,
let us change the PAS(J) Pawlak’s approximation
space on the y-axis for PAS([0,1]). That is, on the
y-axis, let us take the closed interval [0,1] with
the categorizations P
[0,1]
= {y
0
= 0,y
1
,.. . ,y
m
= 1}.
PAS(J) remains unchanged. Last, let f ∈ [0,1]
I
be a rough real function attached to the (S
I
,P
[0,1]
)–
coordinate system.
The pointwise lower and upper approximations
f
, f ∈ [0,1]
I
of f are fuzzy sets, and f ≤ f holds.
Hence, f
IV FS
pw
=

f , f

forms an interval-valued fuzzy
set. Then f
IFS
pw
=

f ,1 − f

forms a pointwise intu-
itionistic fuzzy set on I.
Similarly, the blockwise lower and upper approx-
imations f
←→
,
←→
f ∈ [0,1]
I
of f are fuzzy sets, and
f
←→
≤
←→
f holds. Thus f
IV FS
bw
=

f
←→
,
←→
f

is an interval-
valued fuzzy set, and f
IFS
bw
=

f
←→
,1 −
←→
f

is a block-
wise intuitionistic fuzzy set on I.
In terms of intuitionistic fuzzy set theory, f , f
←→
are IFS membership, 1 − f , 1 − f
←→
are nonmember-
ship, and
• π
−
f
= 1 − f −

1 − f

= f − f ,
FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications
186

• π
←→
f
= 1 − f
←→
−

1 −
←→
f

=
←→
f − f
←→
are IFS hesitancy functions.
If IFS

f ,1 − f

is regarded as IVFS

f , f

, it can
be observed that for every x ∈ I where f (x) is inexact,
|BND
−
( f (x))| = |

f (x), f (x)

| = |

f (x), f (x)

|
= f (x) − f (x) = π
−
f
(x). (1)
That is, the length of the boundary interval of an
inexact value f (x) is exactly equal to the hesitancy
degree π
−
f
(x). Thus the hesitancy function establishes
a relationship between the rough real functions and
their derived intuitionistic fuzzy sets.
6 ROUGHNESS AND
L–FUZZINESS
6.1 L-fuzziness
The essence of L-fuzziness is that the membership
and many other fuzziness functions take their values
from a lattice. This lattice most commonly is a com-
plete one. In the literature, many proposals expand
or narrow the features of the complete lattice; how-
ever, we use the classical approaches. For the details
of lattice theory, see (Gr
¨
atzer, 2011).
(L,≤
L
) is a lattice if
• L is a nonempty set;
• ≤
L
⊆ L × L is a partially ordered relation on L;
• and any two elements a,b ∈ L have supremum
sup{a,b} and infimum inf {a,b}.
If L is a lattice, any finite subset of L has supre-
mum and infimum.
A partially ordered set in which every subset has
supremum and infimum is called a complete lattice.
Definition 4. Let L be a complete lattice and N :L→L
be an unary operator on L with the following proper-
ties:
• a ≤ b if N(b) ≤ N(a) (a,b ∈ L) (order-reversing);
• N(N(a)) = a (a ∈ L) (involutive).
N is called the involutive negation.
Such a lattice is denoted by (L,≤
L
,N). If no
confusion arises, (L,≤
L
,N) is simplified as L. 
Definition 5 (Goguen, 1967). Let (L,≤
L
,N) be a
complete lattice provided with an involutive negation
N. An L-fuzzy set (LFS) on U is a lattice valued
function µ
L
∈ L
U
.
µ
L
is called the membership function. The sub-
script L is dropped if no confusion arises. 
F S
L
(U) denotes the family of all L-fuzzy sets on U.
Definition 6 ((Atanassov, 1999), Definition 3.1).
Let (L, ≤
L
,N) be a complete lattice provided with an
involutive negation N.
Let µ
A
,ν
A
∈ F S
L
(U) be two L-fuzzy sets on U
satisfying the following condition:
µ
A
(u) ≤
L
N(ν
A
(u)) (u ∈ U). (2)
Then intuitionistic L-fuzzy set (ILFS) on U is a func-
tion pair µ
ILFS
A
= (µ
A
,ν
A
). µ
A
is the L-membership,
and ν
A
is the L-nonmembership function. 
The family of all intuitionistic L-fuzzy sets on U
is denoted by I F S
L
(U).
Definition 7. Let µ
ILFS
A
,µ
ILFS
B
∈ I F S
L
(U). Then
µ
ILFS
A
= µ
ILFS
B
if µ
A
= µ
B
and ν
A
= ν
B
; (3)
µ
ILFS
A
⊆ µ
ILFS
B
if µ
A
≤
L
µ
B
and ν
A
≥
L
ν
B
; (4)
µ
ILFS
µ
ILFS
A
∩µ
ILFS
B
= µ
ILFS
A
∩ µ
ILFS
B
(5)
=

inf

µ
A
,µ
B

,sup

ν
A
,ν
B

;
µ
ILFS
µ
ILFS
A
∪µ
ILFS
B
= µ
ILFS
A
∪ µ
ILFS
B
(6)
=

sup

µ
A
,µ
B

,inf

ν
A
,ν
B

.
6.2 Rough Real Functions and
L–fuzziness
Let f ∈ J
I
be an (I
S
,J
P
)–rough real function with its
pointwise and blockwise representations as defined in
Section 4.
J = [0, b] is a linearly (totally) ordered complete
lattice with the usual ≤ ordering. On (J,≤), let us
define the standard involutive negation as
N
S
: J → J, x 7→ b − x.
Proposition 3. N
S
is a unary operation, and the
involution and order-reserving conditions hold for it.
Proof. It is easy to see that
• N
S
is a unary operation on J because if x ∈ J,
b − x ∈ J;
• involution holds because
N
S
(N
S
(x)) = N
S
(b − x) = b − (b − x) = x (x ∈ J);
• order-reserving holds because if x ≤ y,
N
S
(x) = b − x ≥ N
S
(y) = b − y (x,y ∈ J).

Rough Real Functions and Intuitionistic L-fuzziness
187

In the following the co-domain of the J-fuzzy and
intuitionistic J-fuzzy sets will be the lattice (J,≤,N
S
).
Proposition 4. Let f ∈ J
I
be an (I
S
,J
P
)–rough real
function.
(i) By the pointwise lower and upper approxima-
tions f , f of f , the function pair

f ,b − f

forms
an intuitionistic J-fuzzy set on I.
(ii) By the blockwise lower and upper approximati-
ons f
←→
,
←→
f of f , the function pair

f
←→
,b −
←→
f

forms an intuitionistic J-fuzzy set on I.
Proof. (i) f , f ∈ J
I
are J-fuzzy sets, and b − f ∈ J
I
is
also a J-fuzzy set. Furthermore, for every x ∈ J,
f (x) ≤ f (x) = b − (b − f (x)) = N
S
(b − f (x)).
That is Equation (2) holds. Hence,

f ,b − f

forms
an intuitionistic J-fuzzy set on I.
(ii) With the same argument as in (i), it can be
proved that

f
←→
,b −
←→
f

also forms an intuitionistic
J-fuzzy set on I. 
f
ILFS
pw
=

f ,b − f

∈ I F S
J
(I) is called the point-
wise intuitionistic L-fuzzy set of f .
f
ILFS
bw
=

f
←→
,b −
←→
f

∈ I F S
J
(I) is called the
blockwise intuitionistic J-fuzzy set of f .
They are together called the roughly derived intu-
itionistic J-fuzzy sets of f .
In intuitionistic fuzzy set theory, the hesitancy
function has an important role. However, in the L-
fuzziness context, its definition and role is an interest-
ing question.
In (Atanassov, 1999), Equation (3.3), p. 180), the
following L-hesitancy function is defined:
π
sup
A
: U → L, u 7→ N(sup(µ
A
(u),ν
A
(u)) (7)
where L is a complete lattice provided with the invo-
lutive negation N.
Let us examine the following L-hesitancy function
too, which is analogous with Equation (7):
π
in f
A
: U → L, u 7→ N(inf(µ
A
(u),ν
A
(u)) (8)
According to Equations (7) and (8), with the co-
domain (J, ≤,N
S
), the pointwise J-hesitancy func-
tions π
sup
f ,pw
and π
in f
f ,pw
of f are the following (since
J is linearly ordered, “max” and “min” is applicable
instead of “sup” and “inf”):
π
sup
f ,pw
= N
S
(sup{ f , b − f }) = N
S
(max{ f , b − f })
=

b − f , if f ≥ b − f
b − (b − f ) = f if f < b − f
.
π
in f
f ,pw
= N
S
(inf{ f , b − f }) = N
S
(min{ f , b − f })
=

b − f , if f ≤ b − f
b − (b − f ) = f , if f > b − f
.
Neither π
sup
f ,pw
nor π
in f
f ,pw
establish a connection
between the rough real functions and their derived
intuitionistic fuzzy sets in the sense of Equation 1.
Therefore, in the next definition, a new hesitancy
function is suggested.
Definition 8. Let µ
ILFS
A
= (µ
A
,ν
A
) be an intuitionistic
(J, ≤,N
S
)-fuzzy set on I.
Then its J-hesitancy function π
+
A
is
π
+
A
: I → J, x 7→ N
S
(µ
A
(x) + ν
A
(x)).

First, it is needed to make sure that N
S
in Defini-
tion 8 is an involutive negation. Since, for every x ∈ I,
0 ≤ µ
A
(x) + ν
A
(x)
≤ N
S
(ν
A
) + ν
A
= b − ν
A
+ ν
A
= b,
so (µ
A
+ ν
A
)(I) = J. Then, by Proposition 3, N
S
is
indeed an involutive negation.
Let f ∈ J
I
be a rough real function. The following
proposition shows that, according to Definition 8, the
J-hesitancy functions π
+
f ,pw
and π
+
f ,bw
of the pointwise
and blockwise intuitionistic fuzzy sets of f establish
the required connections between f and its derived
ILFS f
ILFS
pw
and f
ILFS
bw
.
Proposition 5. Let f ∈ J
I
be an (I
S
,J
P
)–rough real
function. Its pointwise and blockwise intuitionistic
J-fuzzy sets are
f
ILFS
pw
=

f ,b − f

and f
ILFS
bw
=

f
←→
,b −
←→
f

.
Then, for every x ∈ I where f (x) is inexact,
(i) π
+
f ,pw
(x) = |BND
−
( f (x))|,
(ii) π
+
f ,bw
(x) = |BND
↔
( f (x))|.
Proof. For every x ∈ I where f (x) is inexact,
π
+
f ,pw
(x) = N
S

f (x) + b − f (x)

= b −

f (x) + b − f (x)

= f (x) − f (x)
= |

f (x), f (x)

| = |BND
−
( f (x))|.
The case π
+
f ,bw
(x) can be proved similarly. 
FCTA 2022 - 14th International Conference on Fuzzy Computation Theory and Applications
188

7 SOME BASIC PROPERTIES OF
ROUGHLY DERIVED
INTUITIONISTIC L-FUZZY
SETS
Let f ∈ J
I
be an (I
S
,J
P
)–rough real function. Its
roughly derived intuitionistic J-fuzzy sets are:
f
ILFS
pw
=

f ,b − f

and f
ILFS
bw
=

f
←→
,b −
←→
f

.
Proposition 6. The inclusion relations f
ILFS
bw
⊆ f
ILFS
pw
and f
ILFS
pw
⊆ f
ILFS
bw
fail in general.
Proof. Due to Lemma 1,
• in the case of f
ILFS
bw
⊆ f
ILFS
pw
, f ≤ f
←→
does not hold
in general;
• in the case of f
ILFS
pw
⊆ f
ILFS
bw
, 1 −
←→
f ≥ 1 − f does
not hold in general. 
With the help of common intuitionistic fuzzy set
operations, intersection and union (cf. (Atanassov,
1999), Definition 1.4), in addition by Lemma 1 , we
obtain
Proposition 7. Intersection and union of f
ILFS
pw
and
f
ILFS
bw
are:
(i) f
ILFS
pw∩bw
= f
ILFS
pw
∩ f
ILFS
bw
=

f
←→
,1 − f

;
(ii) f
ILFS
pw∪bw
= f
ILFS
pw
∪ f
ILFS
bw
= ( f ,1 −
←→
f ).
Proof. According to Lemma (1), and Equations (5)
and (6), respectively, we get
(i) f
ILFS
pw∩bw
= f
ILFS
pw
∩ f
ILFS
bw
=

min

f , f
←→

,max

1 − f ,1 −
←→
f

=

f
←→
,1 − f

;
(ii) f
ILFS
pw∪bw
= f
ILFS
pw
∪ f
IFS
bw
=

max

f , f
←→

,min

1 − f ,1 −
←→
f

= ( f , 1 −
←→
f ).

Proposition 8. The following inclusion relations
hold:
(i) f
ILFS
bw
∩ f
ILFS
pw
⊆ f
ILFS
bw
∪ f
ILFS
pw
.
(ii) f
ILFS
bw
∩ f
ILFS
pw
⊆ ( f ,1 − f ).
It means that the ILFS f
ILFS
bw
∩ f
ILFS
pw
is included
in the LFS f if f is considered as ILFS.
(iii) f
IFS
bw
∪ f
IFS
pw
⊆

←→
f , 1 −
←→
f

.
It means that the ILFS f
ILFS
bw
∪ f
ILFS
pw
is included
in the LFS
←→
f if
←→
f is considered as ILFS.
(iv) ( f ,1 − f ) ⊆ f
IFS
bw
∪ f
IFS
pw
.
It means that the LFS f , if it is considered as ILFS,
is included in the ILFS f
ILFS
bw
∪ f
ILFS
pw
.
(v)

f
←→
,1 − f
←→

⊆ f
IFS
bw
∩ f
IFS
pw
.
It means that the LFS f
←→
, if it is considered as
ILFS, is included in the ILFS f
ILFS
bw
∩ f
ILFS
pw
.
Proof.
(i) f
ILFS
bw
∩ f
ILFS
pw
=

f
←→
,1 − f

⊆ ( f , 1 −
←→
f ) = f
ILFS
pw
∪ f
ILFS
bw
,
because f
←→
≤ f and 1 − f ≥ 1 −
←→
f .
(ii) f
ILFS
bw
∩ f
ILFS
pw
=

f
←→
,1 − f

⊆ ( f ,1 − f ),
because f
←→
≤ f and 1 − f ≥ 1 − f .
(iii) f
ILFS
bw
∪ f
ILFS
pw
= ( f , 1 −
←→
f ) ⊆ (
←→
f , 1 −
←→
f ),
because f ≤ f
←→
and 1 −
←→
f ≥ 1 −
←→
f .
(iv) ( f , 1 − f ) ⊆ ( f , 1 −
←→
f ) = f
IFS
bw
∪ f
IFS
pw
,
because f ≤ f and 1 − f ≥ 1 −
←→
f .
(v)

f
←→
,1 − f
←→

⊆

f
←→
,1 − f

= f
ILFS
bw
∩ f
ILFS
pw
,
because f
←→
≤ f
←→
and 1 − f
←→
≥ (1 − f ).
8 CONCLUSION
The paper has shown how the more realistic real func-
tions can connect with intuitionistic L-fuzzy sets. The
boundary region on the rough real function side, and
the hesitancy function on the intuitionistic L-fuzzy set
side, are extremely important for rough real functions.
These two regions are a connecting link, both seman-
tically and syntactically, between the two otherwise
distant areas.
Several goals can be formulated for the future.
The rich tool set of intuitionistic L-fuzzy sets can be
applied to studying rough real functions. Care must
be taken, however, that the tools chosen are consistent
with the semantics of rough real functions. Deciding
on this is not always easy, so the tools must be
carefully considered. The intuitionistic fuzzy tools
applied in rough set theory may also imply that the
rough calculus can have a fertilizing effect on the
intuitionistic fuzzy set theory.
Rough Real Functions and Intuitionistic L-fuzziness
189

ACKNOWLEDGEMENTS
The author would like to thank the anonymous
referees for their useful comments and suggestions.
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