# 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 *X _{A},* which is

*X _{A}* :

*X*^ {0, 1}, where

*X*is the universal set.

*X _{A}* indicates the membership of the element

*x e X*if x

_{A}(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 A_{a} 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 A_{a}+ 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.