8.6.8. Algebraic Operations on Fuzzy Sets
In addition to the operations of union and intersection, one can define a number of other ways of forming combinations of fuzzy sets and relating them to one another.
Algebraic product: Given and the algebraic product of and denoted by is defined in terms of the membership functions of and ,
f = f · f
This indicate that 
« .
The algebraic sum of and denoted by + is defined by
f + = f + f
provided that 0 ² f + f ² 1
Convex combination of and
with an arbitrary fuzzy set denoted by
( , ; ) is defined by
( , ; ) = +
written out in terms of membership functions
f( , ; ) (w) = f · f + (1 - f ) · f
A basic property of the convex combination of , and is expressed by
A « B 
(A, B ; L) 
A » B
Given any fuzzy set
satisfying « 
» , one can always find a
fuzzy set such that
= ( , ; ) .
In fact,
f = for w ΠW .
