Fuzzy Uncertainty

A classical or crisp set A can be defined as a collection of objects or elements of a universal set X. The elements of the set (say A) may be defined by using the characteristic function XA, which is

XA : X ^ {0, 1}, where X is the universal set.

XA indicates the membership of the element x e X if xA(x) = 1 and its non-membership if XA(x) = °.

Definitions

In the following, some important definitions related to fuzzy sets are introduced and explained.

Fuzzy Set

If X is a collection of objects or elements (denoted by x), then a fuzzy set A in X is a set of ordered pairs:

where m A (x) is the membership function of x.

Example 3.1

Let us consider a fuzzy set A = real numbers larger than 15 (Zimmermann 1991) as A = {x, ma(x))|x G x|,where

Support of a Fuzzy Set

The support of a fuzzy set A is the crisp set of elements x e X that has non-zero membership grades in A.

The support of a fuzzy set A may be written as

a-Level Set of a Fuzzy Set

The a-levd set Aa of a fuzzy set A is the crisp set of all elements x e X that belongs to the fuzzy set A at least to the degree a e [0, 1].

The a-level set Aa+ with

is called strong a-level set of the fuzzy set A.

Convexity of a Fuzzy Set

A fuzzy set A is convex if

In other words, a fuzzy set is convex if all a-level sets are convex.

Height of a Fuzzy Set

The height h (A) of a fuzzy set A is the largest membership grade obtained by any element in that set.

A fuzzy set A is called normal when h (A) = 1 and subnormal if h (A) < 1.

 
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