4.2 Intuitionistic fuzzy sets
4.2.2 Some properties of intuitionistic fuzzy sets
The presentation of these properties is similar to the presentation in previ-ous section 4.1 in classical and fuzzy setting.
Definition 4.13. LetA = {< x, µA(x), νA(x) > |x ∈ X}be an intuitionistic fuzzy set in the universeX. Now,Ais a empty set,
A=∅⇐⇒ ∀x∈X :µA(x) = 0∧νA(x) = 1.
Intuitively an empty set does not contain any elements and that should be a fact where is no room for uncertainty. If a set is propably empty it does not qualify as an empty set. In this case both the membership functionµA equals0and the non-membership function equals1for all elements of the universe X, which means that the intuitionistic fuzzy set A in definition 4.13 does not contain any elements. Furthermore, the hesitation margin πA(x) equals 0 for all elements of the universe X, so it sure that in the whole universe X there do not exist a single element which could be so
4.2. INTUITIONISTIC FUZZY SETS 53 much as partly a member in the empty set. This definition of the intu-itionstic fuzzy empty set is actually exactly same as the fuzzy empty set [Zadeh, 1965, p. 340].
Definition 4.14. [Atanassov, 2012, p. 17] Let A and B be intuitionistic fuzzy sets in the universeX,µAandµBbe their corresponding membership functions, andνAandνB be their corresponding non-membership functions,
A={< x, µA(x), νA(x)>|x∈X}andB ={< x, µB(x), νB(x)>|x∈X}.
AequalsB,
A=B ⇐⇒ ∀x∈X :µA(x) =µB(x)∧νA(x) = νB(x).
Now, intuitionistic fuzzy sets are defined by their membership and non-membership functions and if these two functions are equal, so must be also the case with both intuitionistic fuzzy sets and vice versa.
Definition 4.15. [Atanassov, 2012, p. 17] Let A and B be intuitionistic fuzzy sets in the universeX,µAandµBbe their corresponding membership functions, andνAandνB be their corresponding non-membership functions,
A={< x, µA(x), νA(x)>|x∈X}andB ={< x, µB(x), νB(x)>|x∈X}.
Ais a subset ofB,
A ⊂B ⇐⇒ ∀x∈X :µA(x)≤µB(x)∧νA(x)≥νB(x).
Furthermore, if in the previous definition 4.15 the intuitionistic fuzzy set B is replaced with A, it is easy to see that the partial inequalities still hold and, therefore, any intuitionistic fuzzy set is its own subset. More-over, if an intuitionistic fuzzy setAis a subset of an intuitionistic fuzzy set B, it means that every elementx is included ”more” in the intuitionistic fuzzy set Bthan in the intuitionistic fuzzy setA. This notion of ”more” is considered in a such way, that an element is both as much or more member and as much or less non-member.
Theorem 4.3. Let A and B be intuitionistic fuzzy sets in the universe X, µA and µB be their corresponding membership functions, and νA and νB be their corresponding non-membership functions,
A={< x, µA(x), νA(x)>|x∈X}andB ={< x, µB(x), νB(x)>|x∈X}.
Intuitionistic fuzzy setsAandBare equal,
A=B ⇐⇒A⊂B∧B ⊂A.
First, letA =B. Now, from the definition 4.15 and from fact that each set is its own subset follows that A ⊂ B ∧B ⊂ A is true. Next, letA ⊂ B ∧B ⊂ A hold. By the definition 4.15 both ∀x ∈ X : µA(x) ≤ µB(x)∧ νA(x) ≥νB(x)and∀x ∈X :µB(x)≤ µA(x)∧νB(x)≥ νA(x)and therefore
∀x ∈ X : µA(x) = µB(x) ∧νA(x) = νB(x), which by the definition 4.14 mean thatA=B and the theorem 4.3 holds.
The basic operations between intuitionistic fuzzy sets are defined in similar manner than operations over fuzzy sets. However, the difference is that also the non-membership function plays a role in the definition.
Definition 4.16. [Atanassov, 2012, p. 17],[Li, 2014, pp. 9-10],[Szmidt, 2014, pp. 17-18] Let A and B be intuitionistic fuzzy sets in the universe X, µA and µB be their corresponding membership functions, andνA andνB be their corre-sponding non-membership functions,
A ={< x, µA(x), νA(x)>|x∈X}andB ={< x, µB(x), νB(x)>|x∈X}.
Letλ >0be an arbitrary real number. The following basic operations are defined:
1. the complement
AC ={< x, νA(x), µA(x)>|x∈X}, 2. the intersection
A∩B ={< x, min(µA(x), µB(x)), max(νA(x), νB(x))>|x∈X}, 3. the union
A∪B ={< x, max(µA(x), µB(x)), min(νA(x), νB(x))>|x∈X}, 4. the differcence
A−B ={< x, min(µA(x), νB(x)), max(νA(x), µB(x))>|x∈X}, 5. the addition
A+B ={< x, µA(x) +µB(x)−µA(x)µB(x), νA(x)νB(x)>|x∈X}, 6. the multiplication
AB={< x, µA(x)µB(x), νA(x) +νB(x)−νA(x)νB(x)>|x∈X}, 7. the product of an intuitionistic fuzzy set and a real number
λA={< x,1−(1−µA(x))λ,(νA(x))λ >|x∈X}, and 8. the power
Aλ ={< x,(µA(x))λ,1−(1−νA(x))λ >|x∈X}.
Here it is important to notice that the complement reverses the values of membership and non-membership. Furthermore, in the classical setting an element is included either in the set or its complement. Now, in the in-tuitionistic fuzzy setting it feels reasonable, that if an element belongs to
4.2. INTUITIONISTIC FUZZY SETS 55 an intuitionistic fuzzy set with some value, it does not belong to the com-plement of the original set with the same exact value and also vice versa.
Therefore, the reversal of the membership and non-membership functions make sense. The complement is also used to define the difference between intuitionistic two fuzzy sets and the definition is based on the fact that A−B =A∩(BC).
The definition of both the intersection and the union follow the fuzzy definition, only the non-membership function is the difference. Again, it makes sense when considering the subset relation(⊂)given in the defini-tion 4.15. Namely, the intersecdefini-tion(A∩B)of two intuitionistic fuzzy sets, A and B, is always a subset of both original sets and in the intuitionistic fuzzy setting this means that∀x∈X :µA∩B(x)≤µA(x)∧νA∩B(x)≥νA(x) and∀x∈X :µA∩B(x)≤µB(x)∧νA∩B(x)≥νB(x). This, in turn, means that µA∩B(x) =min(µA(x), µB(x))andνA∩B(x) =max(νA(x), νB(x)). Further, if the elementxfrom the intersectionA∩Bwould have smaller membership thanmin(µA(x), µB(x)), it would mean that there would exist an intuition-istic fuzzy setCsuch thatC ⊂A, C ⊂B, A∩B ⊂CandA∩B 6=C, which is a contradiction.
Example 4.6. LetX ={x1, x2, x3, x4}be the finite universe,λ= 2, andAand B be intuitionistic fuzzy sets overX defined as follows:
A= {< x1,0.6,0.2>, < x2,0.5,0.3>, < x3,0.0,0.0>, < x4,0.0,1.0>}, B = {< x1,0.3,0.4>, < x2,1.0,0.0>, < x3,0.2,0.5>, < x4,0.3,0.4>}.
Now, the operations from the definition 4.16 give following results:
1.the complement
AC ={< x1,0.2,0.6>, < x2,0.3,0.5>, < x3,0.0,0.0>, < x4,1.0,0.0>}, 2.the intersection
A∩B ={< x1,0.3,0.4>, < x2,0.5,0.3>, < x3,0.0,0.5>, < x4,0.0,1.0>}, 3.the union
A∪B ={< x1,0.6,0.2>, < x2,1.0,0.0>, < x3,0.2,0.0>, < x4,0.3,0.4>},
4.the differcence
A−B ={< x1,0.4,0.3>, < x2,0.0,1.0>, < x3,0.0,0.2>, < x4,0.0,1.0>}, 5.the addition
A+B ={< x1,0.72,0.08>, < x2,1.0,0.0>, < x3,0.2,0.0>, < x4,0.3,0.4>}, 6.the multiplication
AB ={< x1,0.18,0.52>, < x2,0.5,0.3>, < x3,0.0,0.5>, < x4,0.0,1.0>}, 7.the product of an intuitionistic fuzzy set and a real number
2A={< x1,0.84,0.04>, < x2,0.75,0.09>, < x3,0.0,0.0>, < x4,0.0,1.0>}, 8.the power
A2 ={< x1,0.36,0.36>, < x2,0.25,0.51>, < x3,0.0,0.0>, < x4,0.0,1.0>}.
It should be noted, that all operations presented here involve only membership and non-membership functions, so therefore the two-term presentation was used in the previous example 4.6.