A First Course in FUZZY and NEURAL CONTROL

92 CHAPTER 3. FUZZY LOGIC FOR CONTROL

We say that each of these binary operations isdualto the other, relative to the
complement. You will be asked in the exercises to verify that these laws hold.
An equivalent way to view these laws is

A∨B=(A^0 ∧B^0 )

0

andA∧B=(A^0 ∨B^0 )

0

from which you can see that knowing either of the binary operations and the
complement completely determines the dual binary operation.

3.3.2 Triangularnorms,conorms,andnegations.........

Although minimum, maximum, and complement are the most common oper-
ations used for AND, OR, and NOT, there are many other possibilities with
fuzzy sets. Some of these are described in the next sections.

Triangular norms Some situations call for a different notion of AND, and
in most cases these notions are supplied bytriangular norms, usually referred
to ast-norms. The notion of AND suggested in Equations 3.6 ñA(x)∧B(x)
ñ is an example of a t-norm. At the end of this section, we look at some
nonassociative generalizations of AND and OR.
The general definition of a t-norm is the following.

Definition 3.2At-normis a binary operation◦:[0,1]◊[0,1]→[0,1]satis-
fying for allx,y,z∈[0,1]

1.x◦y=y◦x(◦is commutative)

2.x◦(y◦z)=(x◦y)◦z(◦is associative)

3.x◦1=1◦x=x( 1 is an identity)

4.y≤zimpliesx◦y≤x◦z(◦is increasing in each variable)

The term ìbinary operationî simply means that a t-norm is a function of
two variables. In the literature, you will oftenfind t-norms written as functions
of two variables with the notationx◦y=T(x,y). Using this notation, the
commutative, associative, and identity laws look like

1.T(x,y)=T(y,x)

2.T(x,T(y,z)) =T(T(x,y),z)

3.T(x,1) =T(1,x)=x

We use the notationx◦ybecause it is simpler and because binary notation
is commonly used for intersection (A∩B), minimum (x∧y) and, as in the next
example, product (where the symbol is suppressed entirely except when needed
for clarity).