Lfuzzy Sets and Intuitionistic Fuzzy Sets
ABSTRACT Summary. In this article we firstly summarize some notions on Lfuzzy sets, where L denotes a complete lattice. We then study a special case of Lfuzzy sets, namely the “intuitionistic fuzzy sets”. The importance of these sets comes from the fact that the negation is
being defined independently from the fuzzy membership function. The latter implies both flexibility and e.ectiveness in fuzzy
inference applications. We additionally show several practical applications on intuitionistic fuzzy sets, in the context of
computational intelligence.

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ABSTRACT: research of fuzzy set theory, which is applied in fuzzy topology [23], fuzzy algebras [4 5], fuzzy measure and analysis [610], fuzzy optimization and decision [111 2], fuzzy reasoning [1314], fuzzy logic [1 5]and related domains. The cut01/2011;  [Show abstract] [Hide abstract]
ABSTRACT: In this paper, we study a generalization of group, hypergroup and nary group. Firstly, we define intervalvalued fuzzy (anti fuzzy) nary subhypergroup with respect to a tnorm T (tconorm S). We give a necessary and sufficient condition for, an intervalvalued fuzzy subset to be an intervalvalued fuzzy (anti fuzzy) nary subhypergroup with respect to a tnorm T (tconorm S). Secondly, using the notion of image (anti image) and inverse image of a homomorphism, some new properties of intervalvalued fuzzy (anti fuzzy) nary subhypergroup are obtained with respect to infinitely ∨distributive tnorms T (∧distributive tconorms S). Also, we obtain some results of Tproduct (Sproduct) of the intervalvalued fuzzy subsets for infinitely ∨distributive tnorms T (∧distributive tconorms S). Lastly, we investigate some properties of intervalvalued fuzzy subsets of the fundamental nary group with infinitely ∨distributive tnorms T (∧distributive tconorms S).Information Sciences 10/2008; 178(20):3957–3972. · 3.89 Impact Factor  SourceAvailable from: Sifeng Liu
Article: Operations of grey sets
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Page 1
Lfuzzy Sets and Intuitionistic Fuzzy Sets
Anestis G. Hatzimichailidis and Basil K. Papadopoulos
Democritus University of Thrace, Dept. of Civil Engineering
Xanthi 67100, Greece
hatz ane@hol.gr, papadob@civil.duth.gr
Summary. In this article we firstly summarize some notions on L−fuzzy sets, where
L denotes a complete lattice. We then study a special case of L−fuzzy sets, namely
the “intuitionistic fuzzy sets”. The importance of these sets comes from the fact that
the negation is being defined independently from the fuzzy membership function.
The latter implies both flexibility and effectiveness in fuzzy inference applications.
We additionally show several practical applications on intuitionistic fuzzy sets, in
the context of computational intelligence.
1 Introduction
This work presents a novel fuzzy implication. Then, its extends it to intuition
istic fuzzy sets [14, 16]. Finally, it proposes useful geometric interpretations
regarding intuitionistic fuzzy sets [15].
This chapter is organized as follows. Sections 2 and 3 summarize basic
notations and definitions including useful geometrical interpretations. Section
4 presents a fuzzy implication. Section 5 presents an extension of the aforeme
nioned implication to intuitionistic fuzzy sets. Finally, section 6 summarizes
the contribution of this work.
2 L−Fuzzy Sets
The definition and notations can be found in [12, 19].
Definition 1. Let X be a nonempty, ordinary set and let L be a complete
lattice. An L− fuzzy subset, on X, is a mapping:
A : X → L
That is, the family of all L−fuzzy subsets, of X, is just LXconsisting of
all mappings from X to L. Here, LXis called an L−fuzzy space, X is called
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2Hatzimichailidis and Papadopoulos
the carrier domain, of each L−fuzzy subset of it, and L is called the value
domain, of each L−fuzzy subset of X (see [12],[19]).
Many authors study fuzziness in the case L = [0,1], and use the word
“fuzzy”, and not “L−fuzzy”, to describe it. So, the word “fuzzy” possesses
two levels of meaning. In the first case it means “[0,1]−fuzzy” and in the other
describes all fuzzy L = [0,1] cases including both L−fuzzy and [0,1] cases.
In the special case, in which L = [0,1] × [0,1] is equipped with the order
≤, where (x1,y1) ≤ (x2,y2), if and only if x1≤ x2and y1≥ y2, we consider
the intuitionistic fuzzy set.
3 Intuitionistic Fuzzy Sets
The definitions and notations, that we are going to use in this Section, can
be found in [1, 2, 3, 4, 5, 6, 13, 15].
3.1 Some basic notions
Atanassov (see [1]) suggested, in 1983, a generalization of a classical fuzzy set,
that was named intuitionistic fuzzy set1. We define such a set as follows.
Definition 2. An intuitionistic fuzzy set (IFS) A, in X, is an object of the
following form
A = {?x, µA(x), νA(x)? : x ∈ X},
where the functions
µA: X → [0,1]
and
νA: X → [0,1]
define the degree of membership and the degree of nonmembership, of an
element x ∈ X and, evenmore, for each x ∈ X
0 ≤ µA(x) + νA(x) ≤ 1.
Now, if πA(x) = 1−µA(x)−νA(x), then πA(x) is called the degree of non
determinance, of an element x ∈ X, to the set A, where πA(x) ∈ [0,1], ∀x ∈ X.
It can be easily verified that each fuzzy set is a particular case of the
intuitionistic fuzzy set. Moreover, if A is a fuzzy set, then πA(x) = 0, ∀x ∈ X.
We are particularly interested in the study of fuzzy sets for the case that
L = [0,1], because many practical problems can be solved within this frame
work. In the same manner, a great interest is growing for the intuitionistic
fuzzy sets, with L = [0,1] × [0,1].
1The term “ intuitionistic” is controversial in the area of Fuzzy Logic [7, 10].
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Lfuzzy Sets and Intuitionistic Fuzzy Sets3
3.2 Geometric interpretations of IFSs
In the following we give two geometrical interpretations of intuitionistic fuzzy
sets, which introduced in the literature (see [1, 13]).
(i) Let X denote the universe set. Let us also consider the euclidean plane,
with the Cartesion coordinates (see Figure 1), and let F be defined as
follows:
F = {P/(p = ?a,b?) : a,b ≥ 0, a + b ≤ 1}
Let A ⊆ X be a fixed set. Then, we can construct a function fA: X → F,
such that if x ∈ X, then
fA(x) = p = ?a,b? ∈ F,
where
0 ≤ a + b ≤ 1.
Note that the coordinates have been fixed such that a = µA(x) and b =
νA(x).
< 0, 1>
fA x
X
P?
< 0, 0 > <1, 0 >
Fig. 1. Atanassov’s geometrical interpretation of intuitionistic fuzzy sets.
(ii) The geometrical interpretation given below ([15]) could also be regarded
as a generalization of the corresponding one given in [1].
Let T be an arbitrary triangle. It is known, from the euclidean geometry,
that for an arbitrary internal point P, of T, the following relation holds:
dµ
vµ
+dν
vν
+dπ
vπ
= 1,
where dµ, dν, dπdenote the distances between P and the sides µ, ν and
π, of the respective hights (see Figure 2).
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4Hatzimichailidis and Papadopoulos
We now assume that X is the universe set, and A ⊆ X is a fixed set. We
then define a function fA: X → T, as follows:
fA(x) = p = ?a,b? ∈ T, ∀x ∈ X,
where
a,b ∈ [0,1]
and
0 ≤ a + b ≤ 1.
The coordinates a and b are defined by the relations:
a = µA(x) =dµ
vµ,
b = νA(x) =dν
vν
and
1 − πA(x) =dπ
vπ
.
<0,0>
<1,0>
<0,1>
fA(x)
?
?
?
??
??
??
d?
d?
d?
fA
X x
Fig. 2. Geometrical interpretation of some concepts in the intuitionistic fuzzy logics.
3.3 Operations on IFS
For any two IFSs, A and B, several relations and operations have been in
troduced (see references [1][6]). Here we shall introduce only those which are
closely related to this article, namely the properties:
1. A ⊆ B ⇔ µA(x) ≤ µB(x) and νA(x) ≥ νB(x), ∀x ∈ X
2. A ⊇ B ⇔ B ⊆ A
3. A = B ⇔ µA(x) = µB(x) and νA(x) = νB(x), ∀x ∈ X
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Lfuzzy Sets and Intuitionistic Fuzzy Sets5
4.¯A = {?x, νA(x), µA(x)? : x ∈ X}
5. A ∩ B = {?x, min(µA(x), µB(x)), max(νA(x), νB(x))? : x ∈ X}
6. A ∪ B = {?x, max(µA(x), µB(x)), min(νA(x), νB(x))? : x ∈ X}
7. A+B = {?x,(µA(x) + µB(x) − µA(x) · µB(x)),(νA(x) · νB(x))? : x ∈ X}
8. A · B = {?x,(µA(x) · µB(x)),(νA(x) + νB(x) − νA(x) · νB(x))? : x ∈ X}
9. A@B =
x,(µA(x)+µB(x))
22
10. A$B =
??
12. A ? ?B =
x,2
νA(x)+νB(x)
??
,(νA(x)+νB(x))
?
: x ∈ X
?
?
?
??
x,?µA(x) · µB(x),?νA(x) · νB(x)
x,
2·(µA(x)·µB(x)+1),
??
where if µA(x) = µB(x) = 0, then
0, then
: x ∈ X
?
: x ∈ X
?
11. A ∗ B =
µA(x)+µB(x)νA(x)+νB(x)
2·(νA(x)·νB(x)+1)
νA(x)·νB(x)
: x ∈ X
?
?
µA(x)·µB(x)
µA(x)+µB(x),2
,
µA(x)·µB(x)
µA(x)+µB(x)= 0 and if νA(x) = νA(x) =
νA(x)·νB(x)
νA(x)+νB(x)= 0.
We introduce four operators, over IFS, with the following definitions.
Definition 3. The necessity operator is defined as:
?A = {< x,µA(x),1 − µA(x) >: x ∈ X}
and the possibility operator is defined as:
♦A = {< x,νA(x),1 − νA(x) >: x ∈ X}
The above operators are similar to the operators of necessity and possi
bility that are defined in some modal logic, and they have no counterparts in
the ordinary fuzzy set theory.
The operator below is an extension of the operators ? and ♦, but it can
be extended even further.
Definition 4. Let a ∈ [0,1] be a fixed number. Given an IFS A, the operator
Dais defined as follows:
Da(A) = {< x,µA(x) + a · πA(x),νA(x) + (1 − a) · πA(x) >: x ∈ X}.
Definition 5. Let a,b ∈ [0,1] and a + b ≤ 1. The operator Fa,b, for the IFS,
A is defined as:
Fa,b(A) = {< x,µA(x) + a · πA(x),νA(x) + b · πA(x) >: x ∈ X}.
In the figures below, we give the geometric interpretations for the operators
Daand Fa,brespectively.
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6Hatzimichailidis and Papadopoulos
<1,0>
f??(x)
<0,0>
<0,1>
?
f?(x)
f? ?(x)
?
?
?1
?2
fDa (?)(x)
Fig. 3. The point fD α(x) belongs to the segment defined by the points f? A(x) and
f♦ A(x), and its exact position depends on the value of the arguments α ∈ [0,1].
?1
?2
?
?
?
<1,0>
<0,1><0,0>
f??(x)
fF?,?(?)(x)
( )
f??(x)
f?(x)
Fig. 4. The point fFα,β(A)(x) is an internal point of the triangle with vertices
fA(x), f? A(x) and f♦ A(x). The point depends on the value of the argumets a,b ∈
[0,1], for which a + b ≤ 1.
4 Implications on Fuzzy Sets
The definitions and notations can be found in [9],[11],[18] and [20].
Let X denote a universe of discourse. Then, a fuzzy set A, in X, is defined
as a set of ordered pairs A = {< x,µ(x) >: x ∈ X}, where the function
µA: X → [0,1] defines the degree of membership, of the element x ∈ X [22].
Definition 6. A binary operation i, on the unit interval, (i.e. a [0,1]×[0,1] →
[0,1] mapping) is called a fuzzy intersection, if it is an extension of the
classical Boolean intersection i(a,b) ∈ [0,1], for every a,b ∈ [0,1] where
i(0,0) = i(0,1) = i(1,0) = 0 and i(1,1) = 1.
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Lfuzzy Sets and Intuitionistic Fuzzy Sets7
A canonical model of fuzzy intersections is the family of triangular norms
(briefly the t−norms).
Definition 7. A t−norm is a function of the form:
i : [0,1] × [0,1] → [0,1],
which is commutative, associative, nondecreasing, and i(a,1) = a, a ∈ [0,1].
A t−norm is called Archimedean, if it is continuous, and for a ∈ (0,1),
i(a,a) < a it is nilpotent, if it is continuous and ∀a ∈ (0,1), there exists a
ν ∈ N, such that i
forms, in the nilpotent and in the nonnilpotent ones. Moreover, those which
are not nilpotent are called strict.
ν
??? ?
(a,...,a) = 0. So, the Archimedean norms appear in two
Definition 8. A function n : [0,1] → [0,1] is called a negation, if it is non
increasing, i.e. n(a) ≤ n(b), if a ≥ b, and n(0) = 1, n(1) = 0.
A negation, n, is called strict, if and only if n is continuous and strictly
decreasing (n(a) < n(b), if a > b, for all a,b ∈ [0,1]). A strict negation, n,
is called strong, iff it is selfinverse, i.e. n(n(a)) = a, for all a ∈ [0,1]. The
most important, and widely used strong negation, is the standard negation
ns: [0,1] → [0,1], given by nS= 1 − a.
A function u : [0,1] × [0,1] → [0,1], satisfying the properties:
1. u(a,0) = a, for all a ∈ [0,1],
2. u(a,b) ≤ u(c,d), if a ≤ c and b ≤ d,
3. u(a,b) = u(b,a), for all a,b ∈ [0,1],
4. u(u(a,b),c) = u(a,u(b,c)), for all a,b,c ∈ [0,1]
is called a triangular conorm (or t−conorm).
A fuzzy implication, g, is a function of the form:
g : [0,1] × [0,1] → [0,1],
which defines (for any possible truth values a, b, of some given fuzzy proposi
tions p, q respectively) the truth value I(a,b), of the conditional proposition
“ if p then q”.
The function g should be an extension of the classical implication, from
the domain {0,1} to the domain {0,1}, of truth values in fuzzy logic.
Definition 9. The implication operator, of the classical logic, it a mapping:
m : {0,1} × {0,1} → {0,},
which satisfies the conditions:
m(0,0) = m(0,1) = m(1,1) = 1 and m(1,0) = 0
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8Hatzimichailidis and Papadopoulos
These conditions are the least ones that we can demand, from a fuzzy
implication operator. In other words, the fuzzy implications collapse to the
classical implication, when the truth values are restricted to 0 and 1:
g(0,0) = g(0,1) = g(1,1) = 1 and g(1,0) = 0
One way of defining g in classical logic is to use the logicformula:
g(a,b) = ¯ a ∨ b,
for all a,b ∈ {0,1}.
Another way is to employ the formula:
g(a,b) = max{x ∈ {0,1} : a ∧ x ≤ b},
for all a,b ∈ {0,1}.
The extensions of these equations in fuzzy logic are, respectively:
g(a,b) = u(n(a),b) (1) and
g(a,b) = sup{x ∈ [0,1] : i(a,x) ≤ b} (2),
for all a,b ∈ [a,b], where u, i and n denote a fuzzy union, a continuous fuzzy
intersection and a fuzzy negation, respectively. Furthermore, if u and i are
dual with respect to n, we say that a t−conorm i and a t−conorm u are
dual with respect to a fuzzy negation n, iff n(i(a,b)) = u(n(a),n(b)) and
n(u(a,b)) = i(n(a),n(b)), for all a,b ∈ [0,1].
The fuzzy implications, that are obtained from (1), are usually referred to
the literature as S−implications (S is often used for denoting t−conorms), and
the fuzzy implications that are obtained from (2) are called R−implications.
Moreover, the formula g(a,b) = ¯ a ∨ b may also be rewritten (due to the
law of absorption of negation in classical logic), as either:
g(a,b) = ¯ a ∨ (a ∧ b)
or
g(a,b) = (¯ a ∧¯b) ∨ b
The extensions of these equations, in fuzzy logic, are, respectively:
g(a,b) = u(n(a),i(a,b)) (3) and
g(a,b) = u(i(n(a),n(b)),b) (4),
where u, i and n are required to satisfy the De Morgan laws.
The fuzzy implications that are obtained from (3) are called QL−implications,
since they were originally employed in quantum logic.
Identifying various properties of the classical implication, and generalizing
them appropriately, leads to the following properties, which may be viewed
as reasonable axioms of fuzzy implications (see [18]):
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Lfuzzy Sets and Intuitionistic Fuzzy Sets9
(A1) a ≤ b implies that g(a,x) ≥ g(b,x) (monotonicity in first argument). This
means that the truth value of fuzzy implications increases, as the truth
value of the antecedent increases.
(A2) a ≤ b implies that g(x,a) ≤ g(x,b) (monotonicity in second argument).
This means that the truth value of of fuzzy implications increases, as the
truth value of the consequent increases.
(A3) g(a,g(b,x)) = g(b,g(a,x)) (exchange propery). This is a generalization of
the equivalence of a ⇒ (b ⇒ x) and b ⇒ (a ⇒ x), which holds for the
classical implication.
(A4) g(a,b) = g(n(b),n(a)), for a fuzzy complement n (contraposition)
(A5) g(1,b) = b (neutrality of truth)
(A6) g(0,a) = 1 (dominance of falsity)
(A7) g(a,a) = 1 (identity)
(A8) g(a,b) = 1, if and only if a ≤ b (boundary condition)
(A9) g is a continuous function (continuity)
One can easily prove that all the S−implications fulfill axioms A1, A2,
A3, A5, A6 and, when the negation is strong, A4. In addition, all the
R−implications fulfill the axioms A1, A2, A5, A6 and A7.
Let now G be the fuzzy implications that are obtained from (4), when a ≤
b, i.e. G(a,b) = u(i(n(a),n(b)),b), when a ≤ b. We also defined G(a,b) = 0,
when a > b. Thus, we have a class of fuzzy implications, which we define as
follows (see also [16]):
Definition 10. Consider the function G : [0,1]×[0,1] → [0,1], with the type:
G(a,b) = [1 − sg(a − b)] · u(i(n(a),n(b)),b) (5),
for all a,b ∈ [0,1] where u, i and n are required to satisfy De Morgan laws,
n is a strong negation and sg(x) =
?1, if x > 0
0, otherwise. Then G(a,b) is a class
of fuzzy implications, which collapse to the classic implication when the truth
values are restricted to 0 and 1, i.e. G(0,0) = G(0,1) = G(1,1) = 1 and
G(1,0) = 0.
Choosing in (5) as a fuzzy union the standard one, u(a,b) = max(a,b) =
a∨b, as a fuzzy intersection the standard i(a,b) = min(a,b) = min(a,b) = a∧b,
and as a negation the standard fuzzy negation nS= 1 − a, we will have the
following fuzzy implication, which we introduce in the proposition below.
Proposition 1. Let g be a function, which is defined for all a,b ∈ [0,1], and
having the type:
gh(a,b) = [1 − sg(a − b)] · [(1 − b) ∨ b],
where sg(x) is given in the previous definition. Then gh(a,b) is a fuzzy im
plication, which collapses to the classical implication when the truth values
are restricted between 0 and 1, i.e gh(0,0) = gh(0,1) = gh(1,1) = 1 and
gh(1,0) = 0.
Page 10
10Hatzimichailidis and Papadopoulos
A graphical interpretation of fuzzy implication gh(a,b) is shown in Fig. 5.
Fig. 5. The graphical representation of the fuzzy implication gh.
5 Implications on Intuitionistic Fuzzy Sets (IFSs)
Let S be a set of propositions p. Let also a valuation function, V, be defined
over S, in the following way:
V (p) =< µ(p),ν(p) >,
where µ(p) + ν(p) ≤ 1.
The function V : S → [0,1] × [0,1] gives the truth and falsity degres of
all propositions in S. The valuation function V assigns to the logical truth T,
V (T) =< 1,0 > and to the logical falsity F, V (F) =< 0,1 >. Also,
V (? p) =< ν(p),µ(p) >,
V (p ∧ q) =< min(µ(p),µ(q)),max(ν(p),ν(q)) >,
V (p ∨ q) =< max(µ(p),µ(q)),min(ν(p),ν(q)) > .
Now, considering the propositions p and q, the two most acceptable defi
nitions of implications, in IFS, are the following:
1. The (maxmin)implication:
V (p ⇒ q) =< max(ν(p),µ(q)),min(µ(p),ν(q)) >
Page 11
Lfuzzy Sets and Intuitionistic Fuzzy Sets11
2. The sg−implication:
V (p ⇒ q) =< 1−(1−µ(q))·sg(µ(p)−µ(q)),ν(q)·sg(µ(p)−µ(q))·sg(ν(q)−ν(p)) >,
?1, if x > 0
Implications in IFS collapse to the classical implication, when the truth
values are restricted between 0 and 1, i.e.
where sg(x) =
0, otherwise
V (p)
< 0,1 > < 0,1 > < 0,1 >
< 0,1 > < 1,0 > < 1,0 >
< 1,0 > < 0,1 > < 0,1 >
< 1,0 > < 1,0 > < 1,0 >
V (q)
V (p ⇒ q)
The following proposition introduces an implication in IFS, which we call
“→ implication” (see [14]).
Proposition 2. Let A and B be two IFSs, and denote A → B = V (A ⇒ B).
Then,
The “→ implication” collapses to the classical one, when the truth values
are restricted between 0 and 1, i.e.
A → B =
{< x,max(µB(x),νB(x),min(µB(x),νB(x)) >: x ∈ X},
if µA(x) ≤ µB(x) and νA(x) ≥ νB(x)
{< x,0,1 >: x ∈ X},otherwise
V (A)
< 0,1 > < 0,1 > < 0,1 >
< 0,1 > < 1,0 > < 1,0 >
< 1,0 > < 0,1 > < 0,1 >
< 1,0 > < 1,0 > < 1,0 >
V (B) V (A → B)
Also, the “→ implication” will be an extension, in IFS, of the fuzzy impli
cation gh(see [16]) (where ghis the fuzzy implication, which was introduced
in Proposition 1).
Definition 11. An IFS, A, is called an Intuitionistic Fuzzy Tautological Set
(IFTS), iff µA(x) ≥ νA(x), ∀x ∈ X. (see [1]).
In the following theorems (14) we introduce the relations, which the “→
implication” satisfies. These theorems are easily proved (see [14]).
Theorem 1. Let A, B and C be IFSs. Then the following sets are IFTS:
1. A → A
2. A ∩ B → A
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12Hatzimichailidis and Papadopoulos
3. A ∩ B → B
4.¯A → A
5. A → (A ∪ B)
6. B → (A ∪ B)
7. A → (B → A), when B ⊆ A
8. A → B, if A ⊆ B
9. (¯A →¯B) → [(¯A → B) → A], when max(µB(x),νB(x)) ≤ µA(x) and
min(µB(x),νB(x)) ≥ νA(x)
10. [A ∩ (A → B)] → B
11. [(A → B) ∩¯B] →¯A
12. [A → (B → C)] → [(A → B) → (A → C)] when A ⊆ C and
max(µB(x),νB(x)) ≤ max(µc(x),νC(x)),min(µB(x),νB(x)) ≥ min(µC(x),νC(x))
13. [(A → B) ∩ (B → C)] → (A → C), when A ⊆ C
14. (A → C) → [(B → C) → ((A ∪ B) → C))].
Theorem 2. Let A1,A2,...,Aνbe IFSs. If max(µAν−1(x),νAν−1(x)) ≤ µAν(x)
and min(µAν−1(x),νAν−1(x)) ≥ νAν(x), then (((A1→ A2) → ...) → Aν−1) →
Aνis an IFTS.
Theorem 3. Let A and B be two IFSs. Then,
1. ?(A → B) ⊆ (?A) → (?B) and
2. ♦(A → B) ⊇ (♦A) → (♦B), when A ⊆ ?B.
Theorem 4. Let A and B be two IFSs. Then, the following statements are
true:
1. If A ⊆ B, then Da(A → B) ⊇ Da(B).
2. If A ? B, then Da(A → B) ⊆ Da(B).
3. If A ⊆ B, then Fα,β(A → B) ⊇ Fα,β(B).
4. If A ? B, then Fα,β(A → B) ⊆ Fα,β(B).
The geometric interpretation of Theorem 4 is given in Figure 6 (statements
(i)(iv)) and in Figure 7 (statements (iii)(iv)).
6 Conclusion
This chapter has presented a fuzzy implication as well as its extension in intu
itionistic fuzzy sets (IFSs). We pointed out that there is a strong connection
between intuitionistic fuzzy sets and Lfuzzy sets. Therefore, the aforemen
tioned implication might be expressed equivalently using either intuitionistic
fuzzy sets or lattice fuzzy (Lfuzzy) sets.
Page 13
Lfuzzy Sets and Intuitionistic Fuzzy Sets13
<1,0>
<0,0>
<0,1>
fA(x)
fB(x)
f?B(x)
B( )
fx
( )
x
B
f
?
f?B(x)
fDa(B)(x)
( )
x
Bf
)(
)(
xf
BD?
?
Fig. 6. The geometric interpretation of Theorem 4, statements (i)(ii).
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14Hatzimichailidis and Papadopoulos
? + ? ? 1
<1,0>
<0,0>
<0,1>
fA(x)
fB(x)
f?B(x)
( )x
B
f
?
fDa(B)(x)
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fx
f?B(x)
( )x
Bf
( )( )
Da B
fx
?
?
?
( )x
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f
?
fF?,?(B)(x)
Fig. 7. The geometric interpretation of Theorem 4, statements (iii)(iv).
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