Fuzzy Arithmetic - Fuzzy Logic: An Introductory Course for Engineering Students - page 140

6.2 Fuzzy Numbers

In the usual scientific computation is rather unusual to consider exact numbers, as in
the case of taking

1416, or √ 2

ˀ ≈ 3 . ≈ 1 . 4142. In some cases, an interval containing

the number is considered, for example, √ 2 . Fuzzy numbers are just

a “fuzzification” of this last idea. For example, the fuzzy number “approximately 2”
can be taken either as the fuzzy set,

∈[ 1 . 412 , 1 . 413 ]

1

0 1 1.9 2.1 2

or as the fuzzy set.

1

0 1 1 1.9 2.1 2 2.5 3

That kind of fuzzy sets are known as fuzzy numbers . Their definition is the fol-
lowing.

] R is a fuzzy number, provided there is a Definition 6.2.1 A fuzzy set μ ∈[ 0 , 1

closed interval [ a , b ]∈R , such that

1. μ( x ) = 1, for all x ∈[ a , b ]

2. It exists a function L : ( −∞ , a ) ₒ[ 0 , 1 ] , that is continuous and non-decreasing,

such that μ( x ) = L ( x ) , for all x ∈ ( −∞ , a ) .

3. It exists a function R : ( b , +∞ ) ₒ[ 0 , 1 ] , that is continuous and decreasing, such

that μ( x ) = R ( x ) , for all x ∈ ( b , +∞ ) .