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A historical account of types of fuzzy sets and their
relationships
Humberto Bustince, Senior Member, IEEE, Edurne Barrenechea, Miguel Pagola, Javier Fernandez,
Zeshui Xu, Senior Member, IEEE,
Benjamin Bedregal, Javier Montero, Hani Hagras, Fellow, IEEE ,
Francisco Herrera, Membe r, IEEE, and Bernard De Baets, Member, IEEE.
Abstract—In this work we review the d efinition and basic
properties of the different types of fuzzy sets that have appeared
up t o now in the li terature. We also analyze the relationships
between them and enu merate some of the applications in which
they have been used.
Index Terms—Type-2 fuzzy set; Set-valued fuzzy set; Hesitant
fuzzy set; Interval-valued fuzzy set; Atanassov intuitionistic
fuzzy set; Interval type-2 fuzzy sets; Interval-valued Atanassov
intuitioni st ic fuzzy set; Neutrosophic set; Bipolar-valued fuzzy
set; Fuzzy multiset; Fuzzy rough set; Fuzzy soft set; Multi-polar-
valued fuzzy set.
I. INTRODUCTION
It h a s been widely accepted, from the definition of Fuzzy
Sets (FSs) in 1965 [163] and its generalization by Goguen in
1967 [61] (L-FSs), that the main obstacle in their application
is the attribution of membership deg rees to the elements, since
these depend on the application and the co ntext. For this
reason, Zadeh [164], [165] elaborated on the fact that in f uzzy
logic everything is allowed to be a matter of d egree (where the
degree could be fuzzy). Henc e, in 1971 [1 64], Zadeh presente d
the concept o f Type-n Fuzzy Sets (TnFSs), which includes
Type-2 Fuzzy Sets (T2FSs).
Since 1971, several different types of FSs have been in-
troduced, some of them aimed at solving the problem of
constructing the membership degrees of the elements to the
FS, and others focused on represen ting the uncertainty linked
to th e considered problem in a way different from the one
proposed by Zadeh.
H. Bustince, E. Barrenechea, M. Pagola, J. Fernandez are with the De-
partamento de Autom´atica y Computaci´on, Universidad P´ublica de N avarra,
Campus Arrosadia s/n, P.O. Box 31006, Pamplona, Spain.
H. Bustince, E. Barrenechea are with the Institute of Smart Cities, Univer-
sidad Publica de Navarra, Campus Arrosadia s/n, 31006 Pamplona, Spain.
Z. S. Xu is with the Business School, Sichuan University, Chengdu 610064,
China.
B. Bedregal is with the Departamento de Inform´atica e Matem´atica Apli-
cada, Universidade Federal do Rio Grande do Norte, Campus Universit´ario
s/n, 59072-970 Natal, Brazil.
J. Montero is with the Facultad de Ciencias Matem´aticas, Universidad
Complutense, Plaza de las Ciencias 3,28040 Madrid, Spain.
H. Hagras is with the The Computational Intelligence Centre, School of
Computer Science and Electronic Engineering, University of Ess ex, Wivenhoe
Park, Colchester, CO43SQ, UK.
F. Herrera is with the Department of Computation and Artificial Intelli-
gence, CITIC-UGR, Universidad de Granada, Spain and King Abdulaziz Univ,
Fac Comp & Informat Technol North Jeddah, Jeddah 21589, Saudi Arabia.
B. De Baets is with the Department of Mathematical Modelling, Statistics
and Bioinformatics, Ghent University, Coupure links 653, B-9000 Gent,
Belgium
In Table I a historical sequence of the appearance of the
different type s of fuzzy sets is displayed. In Table II we present
a list of the acronyms.
TABL E II
NAMES AND ACRONYMS
Name
Acronym
Atanassov Intuitionistic Fuzzy Sets AIFSs
Bipolar-Valued Fuzzy Sets of Lee BVFSLs
Bipolar-Valued Fuzzy Sets of Zhang BVFSZs
Complex Fuzzy Sets
CFSs
Fuzzy Sets FSs
Fuzzy Rough Sets FRSs
Fuzzy Soft Sets FSSs
Grey Sets
GSs
Hesitant Fuzzy S ets HFSs
Interval Type-2 Fuzzy Sets IT2FSs
Interval-Valued Atanassov
Intuitionistic Fuzzy Sets IVAIFSs
Interval-Valued Fuzzy Sets IVFSs
m-Polar-Valued Fuzzy Sets mPVFSs
Neutrosophic Sets
NSs
Pythagorean Fuzzy Sets PFSs
Set-Valued Fuzzy Sets SVFSs
Shadow Sets SSs
Type-2 Fuzzy Sets
T2FFs
Type-n Fuzzy Sets TnFSs
Typical Hesitant Fuzzy Sets THFSs
Vague Sets
VSs
Related to Table I, the goals of this work are the following:
1.- To introduce the definition and basic properties of each
of the types of FSs.
2.- To study the relationships between th e different types
of fuzzy sets. In particular, to show that Hesitant Fuzzy
Sets (HFSs) are, conceptually, the same a s Set-Valued
Fuzzy Sets (SVFSs), as defined by Grattan-Gu inness.
However, Torra provided an explicit definition for unio n
and intersection of HFSs, wherea s this was not the case
for Grattan-Guinness’ SVFSs.
3.- To analyze the difference between Interval-valued Fuzzy
Sets (IVFSs) and Interval Type-2 Fu zzy Sets (IT2FSs),
and to show that both are related to HFSs.
4.- To analyze the r e la tionships be twe en IVFSs, Atanassov
Intuitionistic Fuzzy sets (AIFSs), Interval- Valued
Atanassov Intuitionistic Fuzzy Sets (IVAIFSs), HFSs,
SVFSs and T2FSs.
5.- To highlight som e applications of each of the types of
FSs.
Figure 1 presents a clear snapshot on the relatio nships
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TABL E I
HISTORY OF THE TYPES OF FUZZY S ETS.
Year Event
1971 Zadeh introduces the idea of Type-n Fuzzy Set (TnFS) and therefore, Type-2 Fuzzy S et (T2FS) [164].
1975 Zadeh presents the definition of Type-n Fuzzy Set [165].
1975 Sambuc proposes the concept of an Interval-Valued Fuzzy Set (IVFS) under the name of Φ-Flou Sets [124]. Zadeh suggests
the same notion of interval-valued fuzzy set in [165] as a particular case of type-2 fuzzy sets. In 1976 Grattan-Guinness speaks
of IVFSs [66] and, in the eighties, Gorzalczany [62], [63], [64], [65] and Turksen [138], [139], [140] finally fix the naming
and first properties of IVFSs.
1976 Mizumoto and Tanaka [104], Dubois and Prade [49] in 1979 and in 1998 Mendel and Karnik [80] propose the mathematical
definition of a T2FS, as well as the first operations on such sets.
1976 Grattan-Guinness presents the notion of Set-Valued Fuzzy Set (SVFS ) [66] as well as some operations based on previous
developments for many-valued algebras [161].
1983 Atanassov presents the definition of Atanassov Intuitionistic Fuzzy Set (AIFS) [4], [5].
1986 Yager gives the idea of Fuzzy Multiset [156].
1989 Atanassov and G argov present the notion of Interval-Valued Atanassov Intuitionistic Fuzzy Set (IVAIFS) [6].
1989 Grey Sets (GSs) are defined by Deng [43].
1990 Dubois and Prade introduce the definition of Fuzzy Rough Set [50].
1993 Gau and Buehrer define the concept of Vague Set (VS) [60].
1996 Zhang presents the definition of Bipolar Valued Fuzzy Set (BVFSZ) [166]. We call them Bipolar Valued Fuzzy Sets in the
sense of Zhang.
1998 Pedrycz introduces the notion of Shadow Set [115].
2000 Liang and Mendel introduce the idea of Interval Type-2 Fuzzy Set (IT2FS) [81].
2000 Lee introduces a new concept with the name of Bipolar-Valued Fuzzy Set [84]. We call them Bipolar Valued Fuzzy Sets in
the sense of Lee (BVFSL ).
2001 Maji, Biswas and Roy introduce the notion of Fuzzy Soft S et [88].
2002 Smaradache introduces the concept of Neutrosophic Set [132].
2002 Kandel introduces the concept of Complex Fuzzy Set [119].
2006 Mendel et al. present their mathematical definition of IT2FS [94].
2010 Torra introduces the notion of Hesitant Fuzzy Set (HFS) [137].
2013 Yager gives the idea of Pythagorean Fuzzy Set (PFS) [157].
2014 Bedregal et al. introduce the notion of Typical Hesitant Fuzzy Set (THFS) [14].
2014 Mesiarova-Zemankova et al. present the concept of m-Polar-Valued Fuzzy Set (mPVFS) [100].
among the extensions. T2FSs encompass SVFSs and hence
also hesitant sets, which include IVAIFSs. The latter contain
IVFSs, which are mathematically identical to AIFSs. Finally,
FSs are in cluded in all of them.
T2FSs
Fig. 1. Representation of the inclusion relationships between different types
of fuzzy sets.
The paper starts by presen ting the concep ts of FSs and
L-FSs (in Sections 2 and 3, respectively). In Section 4, we
analyze T2FSs and their relationship with other types of FSs.
Section 5 is devoted to Set-Valued Fuzzy Sets and Hesitant
Fuzzy Sets. In Section 6, we study IVFSs and in Section 7
we analyze the case of IT2FSs. We then review in Section 8
AIFSs and the specific cases of Neutrosophic sets, BVFSZs
and BVFSLs. We discuss in Sections 9, 10 and 11 IVAIFSs,
Fuzzy Multisets an d n-Dimensional Fuzzy Sets. In Section 12
we recall the definitions of Fuzzy Rough Sets, Fuzzy Soft Sets
and Multi-Valued Fuzzy Sets. We finish with some c onclusions
and references.
II. FUZZY SETS
Łukasiewicz, together with Lesniewski, founded in the
twenties of the XXth century a school of logic in War-
saw that became one of the most important mathe matical
teams in the world, and among whose members was Tarsk i.
Łukasiewicz introduced the idea of distributing the truth values
unifor mly on the interval [0, 1]: if n values are considered,
then 0,
1
n−1
,
2
n−1
, ··· ,
n−2
n−1
, 1 are the possible truth values; if
there is an infinite number of truth values, one should take
Q ∩ [0, 1]. The negation is defined as n(x) = 1 − x, while
x ⊕y = min(1, x + y) is also fixed.
In the line of Łukasiewicz’s studies, Zadeh ([163]) intro-
duced fuzzy sets in his 1965 work, Fuzzy Sets. His ideas on
FSs were soon applied to different areas such as artificial in-
telligence, natural langua ge, decision making, exp e rt systems,
neural networks, control theory, etc.
From now on, we denote by X a non-empty universe, either
finite or infinite.
Definition 2. 1: A fuzzy set (or type-1 fuzzy set) A on X
is a mapping A : X → [0, 1].
The value A(x) is referred to as the membership degree of
the element x to the fuzzy set A.
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An equivalent definition is given by A = {(x, µ
A
(x)) | x ∈
X} with µ
A
: X → [0, 1], explicitly providing the membership
degree of each x ∈ X. Note tha t this definition attempts to
identify the FS with the graph of the mapping A given in Defi-
nition 2.1. Many other notations have been used for fuzzy sets
in the literature. For example, in the early years of FS theo ry,
a common way to describe a FS A (see [104], for instance) on
a finite universe X was A =
P
n
i=1
µ
A
(x
i
)/x
i
, whereas the
notation A =
R
X
µ
A
(x
i
)/x
i
was used for an infinite universe
X. These definitions lead to impor ta nt notational pro blems
and misconceptions, since in fact no sum mation or integration
is taking place.
We denote by FS(X) the class of FSs on the universe X
(note that, in fact, FS(X) = [0, 1]
X
). A partial order relation
≤
F
on FS(X) can be defined as follows. Given A, B ∈
FS(X), A ≤
F
B if the inequality A(x) ≤ B(x) holds for
every x ∈ X. Equivalently, we have the following important
result.
Proposition 2.1: [163] (FS(X), ∪
F
, ∩
F
) is a complete lat-
tice, where, for every A, B ∈ FS(X), union and intersection
are defined, respectively, by
A ∪
F
B(x) = max (A(x), B(x)) , and (1)
A ∩
F
B(x) = min(A(x), B(x)) . (2)
It is important to re call that a lattice L is a partially ordered
set where, for each pair of eleme nts, there exist a supremum
and an infimum. If there exist a supremum and an infimum
for every subset of L, then the lattice is called c omplete.
The first criticism to FS theory concerns the order rela-
tion ≤
F
. Despite Zadeh p resented FSs in order to represent
uncertainty, ≤
F
happens to be a crisp relationship. This
fact h as led Willmott [146], Bandler and Kohout [10] and
others to consider the concept of inclusion measur e. These
measures have been widely used in field s such as mathematical
morphology [42], or image processing [76].
When using the operations defined in Eqs. (1) and (2)
together with the standard negation, n(x) = 1 − x for all
x ∈ [0, 1], neithe r the law of contradiction nor the law of the
excluded middle hold. Nowadays, the operations in Eqs. (1)
and (2) are expressed in terms of t-norms and t-conorms [28],
[34], [58], [82].
Note that we can d efine a fuzzy set over the set of all fuzzy
sets on a given universe X, leading to level 2 fuzzy sets [83].
Of course this ca n be generalized to level k fuzzy sets [164].
III. A GENERALIZATION: L-FUZZY SETS
Goguen [61] realized that, other than its lattice structure ,
there was no relevant reason to use the interval [0, 1] in the
definition of FSs. This observation led him to the introduction
of the concept of an L-fuzzy set.
Definition 3. 1: Let L be a complete lattice. An L-fuzzy set
A on X is a mapping A : X → L.
Given a com plete lattice L, the class of L-fuzzy sets on
the universe X is denoted by L-FS(X). Note that, with this
notation, if L = [0, 1] (and considering the max and min
operations), then FS(X) is the same as L-FS(X). Again,
L-FS(X) can be endowed with a partial order relation, which
is induced by the lattice structure of L as follows. Given
A, B ∈ L-FS(X), A ≤
L
B if the inequality A(x) ≤
L
B(x)
holds for eve ry x ∈ X, where ≤
L
denotes the order relation
of the lattice L. Equivalently, we have the following result.
Proposition 3.1: [61] (L-FS(X), ∪
LF
, ∩
LF
) is a complete
lattice, where, for every A, B ∈ L-FS(X), union and inter-
section are defined, respectively, by:
A ∪
LF
B(x) = A(x) ∨B(x) , and
A ∪
LF
B(x) = A(x) ∧B(x) ,
where ∨ is the greatest lower bound or meet operation and ∧
is the least upper b ound or join operation.
From Proposition 3.1 it is clear that FSs are a special case
of L-fuzzy sets fo r which L = [0, 1] and the maximum and
minimum take the role of the join and meet, respective ly. The
notion of L-FS allows some types of FSs to be emcompassed
within a single theoretical f ramework.
IV. TYPE-2 FUZZY SETS
A. Origin of the Concept
In 1971, and using the ideas given in [18], Zadeh settled in
his work [164] that the problem of estimating the membership
degrees of the elements to the fuzzy set is related to abstraction
-a problem that plays a central role in pattern recognition.
Therefore, the determination of the membership d egre e of
each element to the set is the biggest p roblem for applying FS
theory. Taking these considerations into account, the concept
of type-2 fuzzy set was given as follows: A T2FS is a FS for
which the membership degrees are expressed as FSs on [0, 1].
Later, on December 11, 2008, Zadeh proposed the follow-
ing definitions in the bisc-group m ailing list:
Definition 4. 1: Fuzzy logic is a precise system of reasonin g,
deduction and computation in which the objects of disco urse
and analysis are associated with information which is, or is
allowed to be, im perfect.
Definition 4. 2: Imperf e ct information is defin e d as infor-
mation wh ic h in o ne or more respects is impr ecise, uncertain,
vague, incomplete, par tially tr ue or partially possible.
On the same date and place, Zadeh made the following
remarks:
1.- In fuzzy logic, everything is or is allowed to be a matter
of degree. Degrees are allowed to be fuzzy.
2.- Fuzzy logic is not a replacement for b ivalent logic
or bivalent-logic- based probability theory. Fuzzy logic
adds to bivalent logic and bivale nt-logic-based probabil-
ity theory a wide range of concepts and techniques for
dealing with imperfect information.
3.- Fuzzy logic is designed to ad dress problems in rea -
soning, deduction and computa tion with imperfect in -
formation which are beyond the reach of traditional
methods based on bivalent logic and bivalent logic-b ased
probability theory.
4.- In fuzzy logic the writing instrument is a spray pen
with precisely known adjustable spray pattern. In bi-
valent logic , the writing instrument is a ballp oint pen
(see Fig. 2, which also appeared in the same place).
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Fig. 2. Representation of the uncertainty in fuzzy logic, which according to
Zadeh can be seen as a spray pen.
5.- The importance of fuzzy logic derives from the fact that
in much of the real world, imperfect information is the
norm rath e r than exception.
All of these considerations justify the use of FS theory
whenever ob je cts are linked to soft concepts, i.e. do not show
clear boundaries. Nevertheless, the way to d esign membership
functions might be non-evident, and faces a wide variety of
difficulties. In suc h circumstances, it seems reasona ble to make
use of the so-c alled g eneralizations (types) of FSs, which
might better accommodate the kn owledge available in the
context of the application. In fact, the introduction of many
of su c h generalizations is directly associate d to the need of
building FSs that allow us to represent objects described
through imperfect information, as well as to represent the lack
of knowledge or uncertainty of the considered experts.
B. Basic Definitions
From the notion of T2FS given by Zadeh in [164], and the
study made in [165], we have the followin g definition.
Definition 4. 3: A T2FS A on X is a mapping A : X →
FS([0, 1]).
From Definition 4.3 it can be seen that, mathematically,
a T2FS is a mapping A : X → [0, 1]
[0,1]
. We denote by
T2FS(X) the class of T2FSs on the universe X.
Note that any A ∈ FS(X) can a lso be seen as a T2FS for
which the membership degree is given by a singleton on [0, 1],
that is:
S(t) =
(
1 if t = A(x)
0 otherwise.
Elaboratin g on Zadeh’s definitions for union and intersec-
tion of FSs, Mizumoto and Tanaka [104] in 1976 and Dubois
and Prade [49] in 1979, pr oposed the following definition of
union and intersection for T2FSs.
Definition 4. 4: For every A, B ∈ T2FS(X),
A ∪
T2F
B(x) = A(x) ∪
F
B(x) , and
A ∩
T2F
B(x) = A(x) ∩
F
B(x) .
Proposition 4.1: (T2FS(X), ∪
T2F
, ∩
T2F
) is a complete lat-
tice.
With the union and intersection given in Definition 4.4 the
classical definitions of union and intersection ∪
F
and ∩
F
given
by Zadeh for FSs [4 8] are not recovered. Consider a finite
universe X = {x
1
, x
2
, x
3
}, and consider the following FSs
on X (see [ 22]):
A = {(x
1
,
1
2
), (x
2
,
1
3
), (x
3
, 1)}, and (3)
B = {(x
1
,
1
4
), (x
2
,
1
2
), (x
3
,
1
7
)}. (4)
We have that, for instance, A ∪
F
B =
{(x
1
,
1
2
), (x
2
,
1
2
), (x
3
, 1)}. Alternatively, let A
T 2
and B
T 2
be
analogo us T2FSs, i.e.:
A
T 2
(x
1
)(t) =
(
1 if t =
1
2
0 otherwise
A
T 2
(x
2
)(t) =
(
1 if t =
1
3
0 otherwise
A
T 2
(x
3
)(t) =
(
1 if t = 1
0 otherwise
and
B
T 2
(x
1
)(t) =
(
1 if t =
1
4
0 otherwise
B
T 2
(x
2
)(t) =
(
1 if t =
1
2
0 otherwise
B
T 2
(x
3
)(t) =
(
1 if t =
1
7
0 otherwise.
Then we have that
A
T 2
∪
T2F
B
T 2
(x
1
)(t) =
(
1 if t =
1
4
or t =
1
2
0 otherwise,
A
T 2
∪
T2F
B
T 2
(x
2
)(t) =
(
1 if t =
1
2
or t =
1
3
0 otherwise
A
T 2
∪
T2F
B
T 2
(x
3
)(t) =
(
1 if t =
1
7
or t = 1
0 otherwise
which does not c oincide with our previous result. Mo reove r,
observe that a T2FS is recovered, instead of a FS.
Alternative definitions of union and intersection have been
provided for T2FSs extending Zadeh’s union and intersec-
tion [7 0], [73]. For example, the operators
(A ⊔
T2F
B)(x) = sup{min(A(y), B(z)) | max(y, z) = x}
(A ⊓
T2F
B)(x) = sup{min(A(y), B(z)) | min(y, z) = x}
are based on Zadeh’s extension principle.
Remark 1: Note that (T2FS(X), ⊔
T2F
, ⊓
T2F
) becomes a
basic algebra, but not a lattice, since the absorption law does
not hold (see [71], [72], [73]).
From the ideas given by Karnik and Mendel [ 80] in 1998,
Mendel and John [93] provide in 2002 the following definition:
Definition 4. 5: A T2FS A is characterized by a ty pe-2
membersh ip function µ
A
(x, u), where x ∈ X and u ∈ J
x
⊆
[0, 1], i.e.,
A = {(x, µ
A
(x, u)) | x ∈ X, u ∈ J
x
⊆ [0, 1]} , (5)
