On the Roots of Fuzzy Sets - Fuzzy Logic: An Introductory Course for Engineering Students - page 14

( ) = −

with A x 10 x , results

0

, if 3 x 10

μ small ( x ) = μ big ( 10 − x ) =

( 3 − x )
3

, if 0 x 3 ,

with graphics

that shows the neutral region ( 3 , 7 ) . This pair is regular.

1.2.2 Negations

Let it P be a predicate in X , and P = notP its negate. The only we can say about

P ↂ − 1

the relation between P and P is that it is , since

P

, ,

If x is less P than y then y is less not P than x

P ↂ − 1 or, equivalently, . We can also easily agree that,

P

• If μ P ( x ) = ʱ , then μ P ( x ) = ˉ

• If μ P ( x ) = ˉ , then μ P ( x ) = ʱ.

Let it N : L ₒ L be a function such that

1. If a b , then N ( b ) N ( a ) ,

2. N (ʱ) = ˉ , and N (ˉ) = ʱ ,

ⓦ μ P an L -degree for P , since with such a function N ,itis μ P = N

x P y ⃒ y P x ⃒ μ P ( y ) μ P ( x )

⃒ N (μ P ( x )) N (μ P ( y )) ⃔ μ P ( x ) μ P ( y ).

Hence, given an L-degree μ P of P in X , with each function N verifying (1)

and (2), we get the L-degree μ P = N ⓦ μ P . Such functions N are called negation

functions .
Provided the negation function does verify