Remarks on the intuitionistic fuzzy sets

Fuzzy Sets and Systems 51 (1992) 117-118 North-Holland

117

Short Communication

Remarks on the intuitionistic fuzzy sets Krassimir T. Atanassov

i.e., x ~ F × [0, 1] × [0, 1],

Institute for Microsystems, Sofia, Bulgaria

A = {((y, #E(Y), YE(Y)), #A((Y, #e(Y), 7e(Y))),

7A((Y, #E(Y), 7e(Y)))) l (Y, I~e(Y), Ye(Y)) e E}

Received May 1991 Revised August 1991

Abstract: The question of the relation between some Intuitionistic Fuzzy Set (IFS) A and the universe F which is a universe of the IFS E where the latter is a universe of A, is discussed.

Keywords: Fuzzy set; intuitionistic fuzzy set; universe.

Let E be a fixed universe and let A be an Intuitionistic Fuzzy Set (IFS) [1] over E with the form

A = {(x, #A(x), ~A(X)) IX ~ E}, where the functions

#A(X):E--~[O, 1~] and

yA(x):E-*[O, 1]

define the degree of membership and the degree of non-membership of the element x • E to the set A, which is a subset of E, respectively, and for every x e E, 0 ~ IAA(X) "~ )fA(X ) ~

1.

Let F be another universe and let the set E (obviously, every IFS is an ordinary set, also) be an IFS over F with the form E = {(y, #e(Y), Ye(Y)) l Y e F}. Therefore the element x • E has the form

x = (y, #E(Y), Ye(Y)),

and there exists bijection between the E- and F-elements from x- and y-types, respectively. Thus, for y- and for x-elements, we can use everywhere the notation 'y'. By A / E we denote the fact that A be an IFS over E. If the degrees of membership and nonmembership of the element y to set A in frames of universe E are #A(Y) and 7A(Y) and the element lY,#A(Y),YA(Y)) has degrees of membership and non-membership to set E in frames of universe F given by #E(Y) and YE(Y), then we define A = {(y, #e(Y) • IZA(Y)), 7e(Y) " YA(Y))

Obviously, from A / E and B/E follows that sets A and B have equal y-elements. All intuitionistic fuzzy operations, relations and operators (see [1]) will be transformed directly over the new objects. For example, the most general case, when the relations A/E, B/F, E/G, F/G are valid, the IFS A N B over the universe G has the form A A B = { (y, min(#e(y) • #a(Y), #v(Y) " #B(Y)), max(yE(y)" YA(Y), Ye(Y)" YB(Y))) I

y.eG}. From the above follows directly

definition the

following

Theorem. IrA~E, E / F and F / G , then:

(a)

A = {(y, #F(Y)" #e(Y)" IdA(Y),

(b)

YF(Y)" Ye(Y)" 7A(Y)) l Y • G}, A/(E/F) = (A/E)/F.

Correspondence to: K.T. Atanassov, Institute for Microsystems, Lenin Boul. 7 km., Sofia-l184, Bulgaria.

l y ~ F).

0165-0114/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved

118

K.T. Atanassov / lntuitionistic fuzzy sets

All the results above can be transformed for the case of the ordinary fuzzy sets, as follows: if A is a fuzzy set over universe E and E is a fuzzy set over universe F, then A is a fuzzy set over universe F in the form Z = {(y,

#e(Y)" I-tA(Y)) l Y e F),

where #A and /z~ are degrees of membership with the above sense.

Remark. I must note the fact that in March 1991 I understood for the concept IFS that of Gaisi Takeuti and Satako Titani [4]. We have used this concept in a different sense and obviously independently: my first definition is in June 1983 in Bulgarian [2] and in August 1983 in English

[3]; at this time Takeuti and Titani's paper was in press.

References [1] K. Atanssov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96. [2] K. Atanassov, Intuitionistic fuzzy sets, VII ITKR's Session, Sofia (June (1983) (Deposed in Central Scientific-Technical Library of Bulgarian Academy of Sciences, 1985) (in Bulgarian). [3] K. Atanassov and S. Stoeva, Intuitionistic fuzzy sets, Polish Syrup. on Interval & Fuzzy Mathematics, Poznan (Aug. 1983) 23-26. [4] G. Takeuti and S. Titani, Intuitionistic fuzzy logic and intuitionistic fuzzy set theory, J. of Symbolic Logic 49 (1984) 851-866.