INT. J. BIOAUTOMATION, 2016, 20(S1), S1-S6

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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: krat@bas.bg

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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