Wireless Sensor and Actuator Networks for Lighting Energy ...

Convex combination:
the relation
The convex combination of fuzzy sets A, B and is denoted by (A, B; ) with
(A, B;) = A + B (3.12)
where ´ is the complement of . The resulting membership function of this operation
is
Cartesian product and fuzzy relation:
f (A,B;)
(x) = f
(x) f A
(x) + ( 1 f
(x)) f B
(x), x X (3.13)
The Cartesian product of two fuzzy sets A in space X and B in space Y with
membership functions f A (x) and f B (y) respectively is a fuzzy set in the product space
XY with membership function
f A B
(x, y) = min{ f A
(x), f B
(y)}, x X, y Y (3.14)
A fuzzy relation in space X is a fuzzy set A in product space XX with membership
function f A (x 1 , x 2 ) for all x 1 , x 2 in X.
Fuzzy sets induced by mappings:
Let B be a fuzzy set in space Y with membership function f B (y), and T be a
mapping from X to Y. The fuzzy set A in space X induced by the inverse mapping T -1
has the membership function f A (x) defined by
for all x in X mapped into Y by T.
f A
(x) = f B
(y), y Y (3.15)
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