8 Fuzzy Logic and Fuzzy Based Multi Criteria Decision Analysis
51
(c) Distributive Property:
The distributive property uses three fuzzy sets properties, this involves fuzzy sets
˜
A, ˜
B and ˜
C, and states that:
˜
A ∪
˜
B ∩ ˜
C
=
˜
A ∪ ˜
B
∩
˜
A ∪ ˜
C
(11)
˜
A ∩
˜
B ∪ ˜
C
=
˜
A ∩ ˜
B
∪
˜
A ∩ ˜
C
(12)
(d) Idempotency:
The Idempotency technique can be expressed given fuzzy set ˜
A, it can be stated that:
˜
A ∪ ˜
A = ˜
A
(13)
˜
A = ˜
A ∩ ˜
A
(14)
(e) Identity Property:
The property illustrated mathematically using a given fuzzy set ˜
A and a universal
set U, it can be stated that:
˜
A = ˜
A ∩ U
(15)
U = ˜
A ∪ U
(16)
And also:
˜
A = ˜
A ∪ ∅
(17)
∅ = ˜
A ∩ ∅
(18)
(f) Transitivity:
This can be shown by using fuzzy sets ˜
A, ˜
B and ˜
C, the transitivity feature states:
If ˜
A ⊆ ˜
B and ˜
B ⊆ ˜
C then ˜
A ⊆ ˜
C
(19)
(g) Involution:
The involution property is expressed using the following expression, provided a fuzzy
set ˜
A: