A First Course in FUZZY and NEURAL CONTROL

3.3. COMBINING FUZZY SETS 101

both hold. These are called theDe Morgan laws.They may or may not hold
for other pairs of t-norms and t-conorms. When they do, we say that◦isdual
to∗via the negation^0 , and we call the triple◦,∗,^0 aDe Morgan system
on the unit interval. For example,∧,∨, 1 −xis a De Morgan system. Tables
3.1 and 3.2 provide strict and nilpotent De Morgan systems if you pair t-norms
and t-conorms with common names and parameters. The duality between the
t-norm/t-conorm pairs in Tables 3.1 and 3.2 is via the negation 1 −x, except
for the AczÈl-Alsina case where the duality is via the negatione
ln^1 x
.Inthe
nilpotent case, it is also natural to use the duality given by the t-norm or t-
conorm residual, but that is not done here.
The Frank t-norms are the solutions to the functional equation

x 4 y+x 5 y=x+y

wherex 5 y=1−((1−x) 4 (1−y))is the t-conorm dual to 4 relative to the
negation 1 −x. The FrankóDe Morgan systems are the triples for which the
Frank t-norm and Frank t-conorm use the same parameter and the negation is
1 −x. See page 95 for a complete list of Frank t-norms.

3.3.3 Averagingoperators .....................

In the most general sense, averaging operations are aggregation operations that
produce a result that lies between the minimum and maximum. Averaging op-
erators represent some kind of compromise, and are of fundamental importance
in decision-making. Averaging operators are, in general, not associative.

Definition 3.4Anaveraging operator (ormean) is a continuous binary
operation
⊕:[0,1]◊[0,1]→[0,1]

satisfying

1.x⊕x=x(idempotent)

2.x⊕y=y⊕x(commutative)

3.x≤yandu≤vimpliesx⊕u≤y⊕v(monotonic)

For an averaging operator⊕on[0,1],itisalwaystruethat

x∧y≤x⊕y≤x∨y

In other words, the average ofxandyliesbetween xandy.Toseethis,just
observe that

x∧y=(x∧y)⊕(x∧y)≤x⊕y≤(x∨y)⊕(x∨y)=(x∨y)

which follows from the idempotent and monotonic properties of averaging oper-
ators. Continuous t-norms and t-conorms are averaging operators. The arith-
metic mean(x+y)/ 2 is, of course, an averaging operator, as are geometric