8 Fuzzy Logic and Fuzzy Based Multi Criteria Decision Analysis
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collected in two main processes with different components ranging from 0 or 1 and
0,1 sequently (Kiral and Uzun 2017, 2019). These processes express the classical
and Fuzzy sets.
8.2.1 The Mathematical Expression of Fuzzy Sets
Fuzziness is defined mathematically as a derived mathematics of logic and set theories
that enable evaluation as derived and proposed by Zadeh Lofti in 1965 (Zadeh 1965).
The mathematical expression is detailed as follows:
A fuzzy set ˜
A ∈ I R is a set of pairs as:
A =
x, μ ˜
A (x
)|x ∈ I R
(1)
Therefore, μ ˜
A : I R → [0, 1] and μ ˜
A (x) denotes the membership function of the
A (Kiral 2018).
The fuzzy set is represented by the mathematical expression in various ways;
these processes are illustrated below.
Where “U ” is discrete and finite:
˜
A =
μ ˜
A (x 1 )
x 1
+
μ ˜
A (x 2 )
x 2
+
μ ˜
A (x 3 )
x 3
+ . . .
=
n
i=1
μ ˜
A (x i )
x i
(2)
where “U ” is continuous and infinite:
˜
A =
∫
μ ˜
A (x)
x
(3)
The expression above show, each element of set is defined by the symbol
summation, where U represented the universe of information.
8.2.2 Logical Operations of the Fuzzy Sets
The Fuzzy set operation is connected by union, complement operation and intersection on fuzzy sets. These outlined fuzzy set complements are characterized by
different mathematical expressions. The following is a relationship expressed to
describe the fuzzy sets, union and intersection.
Union:
μ ˜
A∪ ˜
B (x) = μ ˜
A ∨ μ ˜
B , ∀x ∈ U
(4)
∨ represents the ‘max’ operation.