Define Convex Fuzzy Set


Convex Fuzzy Set

A concept of a convex fuzzy set was introduced by Zadeh in the first paper on fuzzy sets. We rewrite the Zadeh’s definition using a simpler notation.

Let A: R” + [0, 1 ] denote a fuzzy set in R” for a given positive integer n. 

DEFINITION 1. A fuzzy set A is convex if
A(tx + (1 ~ t) y) ≥ mint(A(x), ( A(y))     for x, y ∈ Rn, t ∈ (0, 1).................... (1)

A is strongly convex if
A(tx+(l -t)y) > min(A(x), A(y))    for x ≠ y, x, y ⊏ Rn, t ⊏ (0, 1)..........................(2)

Note that any strongly convex fuzzy set is convex. Different properties of convex fuzzy sets are described by Chang , Katsaras and Liu [S], and Lowen [9]. However, we do not know any deeper considerations of strongly convex fuzzy sets. 
          Our considerations are stimulated by the papers of Dubois and Prade [S] and [6] because of the differences between the two definitions of fuzzy number used there.


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