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INT. J. BIOAUTOMATION, 2016, 20(S1), S1-S6
S1
Intuitionistic Fuzzy Sets Krassimir T. Atanassov*
Copyright © 1983, 2016 Krassimir T. Atanassov Copyright © 2016 Int. J. Bioautomation. Reprinted with permission
How to cite:
Atanassov K. T. Intuitionistic Fuzzy Sets, VII ITKR Session, Sofia, 20-23 June 1983
(Deposed in Centr. Sci.-Techn. Library of the Bulg. Acad. of Sci., 1697/84)
(in Bulgarian). Reprinted: Int. J. Bioautomation, 2016, 20(S1), S1-S6.
I.
Let E be an arbitrary fixed set and A be its subset (denote A ⊂ E). In the theory, the fact that
element x of set E belongs to A is denoted by x ∈ A. .1
There, it is introduced the characteristic function : [0,1]A
Eµ → , defined by
1, if ( )
0, if A
x Ax
x Aµ
∈=
∈/.
Fuzzy subset [1] of E is each set with the form * *{ , ( ) | }( )A
x x x E Aµ⟨ ⟩ ∈ = , where * : { | 0 1A
E a aµ → ≤ ≤ & }a M∈ is a function, determining the degree of membership of the
element x to some set A , about ,M that is some numerical set. If {0,1}M = , then this
function coincides with function .A
µ By *µ , in [1], different relations and operations over fuzzy sets are introduced:
{ }
{ }
* * * *
* * * *
* * * *
* * *
* * * * * *
* *
(1) iff ( )( ( ) ( ));
(2) iff ( )( ( ) ( ));
(3) iff ( )( ( ) ( ));
(4) , ( ) , and ( ) 1 ( );
(5) , ( ) , and ( ) min( ( ), ( ));
(6) ,
µ µ
µ µ
µ µ
µ µ µ
µ µ µ µ
A B
A B
A B
AA A
A B A B A B
A B x E x x
A B x E x x
A B x E x x
A x x x E x x
A B x x x E x x x
A B x
∩ ∩
⊂ ∀ ∈ ≤
= ∀ ∈ =
≠ ∃ ∈ ≠
= ∈
= =
∪
= −
∈
=
∩
{ }
{ }{ }{ }
* * * *
* * * * * *
* * * *
* *
* * * *
. .
* * * * *
( ) , and ( ) max( ( ), ( ));
(7) ( ) ( );
(8) ;
(9) ( ) , ( ) ( ) & ;
(10) . , , and ( ). ( ) ;
(11) , , and ( ) ( ) (
µ µ µ µ
µ µ
µ µ µ µ
µ µ µ µ µ
A B A B A B
M A A
A B A B A B
A B A B A B A
x x E x x x
A B A B A B
A B A B
B x x x M x E M M
A B x x x
A B x x x+ +
∪ ∪
∩
∈ =
⊗ =
− =
′ ′= ∈ ∈ ⊂
= =
+ = = + −
∪ ∩
∩
P
{ }*). ( ) .µB
x x
* Current affiliation: Bioinformatics and Mathematical Modelling Department
Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences
105 Acad. G. Bonchev Str., Sofia 1113, Bulgaria, E-mail: [email protected]
INT. J. BIOAUTOMATION, 2016, 20(S1), S1-S6
Behind all these definitions, the Axiom for Excluded Middle is seen, i.e., it is supposed that iffor the element x is not valid that x ∈ A, then x 6∈ A and opposite.
Our next research will be directed t introducing of definitions similar to the above ones, butwith lack of this Axiom, i.e., we will obtain a theory from intuitionistic type. Let us supposethat there is a situation, in which besides the cases x ∈ A and x 6∈ A to be possible a third case –when we cannot determine which of the two cases is valid. Therefore,
µ∗A(x)+ µ
∗A(x) ≤ 1.
When there is “=”, we obtain the ordinary fuzzy sets, in the opposite case, the new objects mustbe used – the intuitionistic fuzzy sets.
Let us discuss one example. Let E = {0,1,2, ...,10}, M = {0,1} ⊂R,
A = {n|n ∈ E& the equation x3− y2 = n has more than 2 solutions}.
For example, µ∗A(0) = 1, µ∗A(1) = 0, but µ∗A(7) cannot be determined, because it is not knownwhether the equation x3− y2 = n has other solutions, except (2,1) and (32,181).
Let us put ν∗A(x) = µ∗A(x). Therefore,
0≤ µ∗A(x),ν
∗A(x) ≤ 1, 0≤ µ
∗A(x)+ν
∗A(x) ≤ 1.
The intuitionistic fuzzy set has the form:
{〈x, µ∗A(x),ν
∗A(x)〉|x ∈ E}.
For it, (1)-(3), (7) and (8) are not changed and the rest equalities obtain the form:
(4) A∗ = {〈x,ν∗A(x), µ∗
A(x)〉|x ∈ E},
and µ∗A(x) = νA(x),ν∗A(x) = µA(x);
(5) A∗∩B∗ = {〈x, µ∗A∩B(x),ν∗A∩B(x)〉|x ∈ E},
and µ∗A∩B(x) = min(µ∗A(x), µ∗B(x)),ν∗A∩B(x) = max(µ∗A(x), µ∗B(x));
(6) A∗∪B∗ = {〈x, µ∗A∪B(x),ν∗A∪B(x)〉|x ∈ E},
and µ∗A∪B(x) = max(µ∗A(x), µ∗B(x)),ν∗A∪B(x) = min(µ∗A(x), µ∗B(x));
(9) PM(E) = {{〈x, µ∗A(x),ν∗A(x)〉|µ∗A(x) ∈M′,ν∗A(x) ∈M′′}|x ∈ E, M′ ⊂M,M′′ ⊂M};
(10) A.B = {〈x, µ∗A.B,ν∗A.B〉|x ∈ E}, ,and µ∗A.B = µ∗A(x).µ
∗B(x),ν
∗A.B = ν∗A+B = ν∗A(x)+ν∗B(x)−ν∗A(x).ν
∗B(x);
(11) A+B = {〈x, µ∗A+B,ν∗A+B〉|x ∈ E}, ,and µ∗A+B = µ∗A(x)+ µ∗B(x)−µ∗A(x).µ
∗B(x),ν
∗A+B = ν∗A(x).ν
∗B(x).
Similarly to [1], the following assertions are valid.
Theorem 1: Operations ∩ and ∪ are commutative, associative, distributive between them fromleft and right sides, idempotent and satisfying De Morgan’s Laws.
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INT. J. BIOAUTOMATION, 2016, 20(S1), S1-S6
Theorem 2: Operations . and + are commutative, associative and satisfying De Morgan’s Laws.
Theorem 3: Operations . and + are distributive from left and right sides about operations∩ and ∪.
II.
In fuzzy sets theory, the Hemming’s distance between two sets is defined by
D(A,B) = ∑x∈E|µ∗A(x)−µ∗B(x)|
=∫R|µ∗A(x)−µ∗B(x)|dx, for M = R,
and Euclid’s distance is defined by
E (A,B) =√
∑x∈E
(µ∗A(x)−µ∗B(x))2
=√∫
R∑
x∈E(µ∗A(x)−µ∗B(x))2dx, for M = R.
Analogues of these definitions, here have the forms:
D ′(A,B) = 12 ∑
x∈E(|µ∗A(x)−µ∗B(x)|+ |ν∗A(x)−ν∗B(x)|)
= 12∫R(|µ∗A(x)−µ∗B(x)|+ |ν∗A(x)−ν∗B(x)|)dx, for M = R,
and Euclid’s distance is defined by
E ′(A,B) = 1√2
√∑
x∈E((µ∗A(x)−µ∗B(x))2 +(ν∗A(x)−ν∗B(x))2)
= 1√2
√∫R((µ∗A(x)−µ∗B(x))2 +(ν∗A(x)−ν∗B(x))2)dx, for M = R.
III.
We will call that set Aα ,β ⊂ A is a set of level (α ,β ) if
Aα ,β = {x|x ∈ E&µ∗A(x) ≥ α&ν
∗A(x) ≤ β}.
Theorem 4:a) ∀β (0≤ β ≤ 1) : α2 ≥ α1→ Aα2,β ⊆ Aα1,β ,
b) ∀α(0≤ α ≤ 1) : β1 ≥ β2→ Aα ,β1 ⊆ Aα ,β2 .
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INT. J. BIOAUTOMATION, 2016, 20(S1), S1-S6
IV.
Now, we define two operators, that transform each intuitionistic fuzzy set to a fuzzy set. Theyare analogous of operators “necessity” and “possibility”, that are defined in the modal logic.For each set A⊂ E:
A∗ = {〈x, µ∗A(x),ν
∗A(x)〉|x ∈ E}= {〈x, µ
∗A(x),1−µ
∗A(x)〉|x ∈ E},
♦A∗ = ♦{〈x, µ∗A(x),ν
∗A(x)〉|x ∈ E}= {〈x,1−ν
∗A(x),ν
∗A(x)〉|x ∈ E}.
The connection between these two operators is given by the following:
Theorem 5: For each intuitionistic fuzzy set A∗:
a) A∗ = ♦A∗,
b) ♦A∗ = A∗.
Really,
A∗ = {〈x,ν∗A(x), µ∗A(x)〉|x ∈ E}= {〈x,ν∗A(x),1−ν∗A(x)〉|x ∈ E}
= {〈x,1−ν∗A(x),ν
∗A(x)〉|x ∈ E}= ♦A∗.
♦A∗ = ♦{〈x,ν∗A(x), µ∗A(x)〉|x ∈ E}= {〈x,1−µ∗A(x), µ∗A(x)〉|x ∈ E}
= {〈x, µ∗A(x),1−µ
∗A(x)〉|x ∈ E}= A∗.
Therefore, the two operators are defined correctly. The connection between them is given by:
Theorem 6: For each intuitionistic fuzzy set A∗:
A∗ ⊆ A⊆♦A∗.
References1. Kaufmann A. (1977). Introduction a la Theorie des Sour-ensembles Flous, Paris, Masson.
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INT. J. BIOAUTOMATION, 2015, 20(S1), S1-S6
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