Fuzzy Relations - Fuzzy Logic: An Introductory Course for Engineering Students - page 120

For what concerns the antisymmetric property, t ij > 0 and t ji >

=

0, implies i j .

For example, the matrix

⊛ ⊞

100 . 70

⊝ ⊠

0 . 6100 . 7

00 . 510 . 8

0 . 7001

is antisymmetric.

Let's define the binary relation, [ μ ] [ ˃ ] , between n × n matrices if t ij s ij

for all 1 i , j n , provided [ μ ]= ( t ij ) , [ ˃ ]= ( s ij ) . With such definition,

•[ μ ] reflects a T -transitive fuzzy relation μ , if and only if, [ μ ]↗ T [ μ ] [ μ ] .

The proof is as follows.
a. If [ μ ] is T -transitive, from

T (μ( x i , x j ), μ( x j , x k )) μ( x i , x k ),

or T ( t ij , t jk ) t ik ,itis Max

1 j n T

( t ij , t jk ) t ik . That is, [ μ ]↗ T [ μ ] [ μ ] .

b. If [ μ ]↗ T [ μ ] [ μ ] ,or Max

1

T ( t ij , t jk ) t ik , follows T ( t ij , t jk ) t ik , for all

j n

1 i , j n . That is, μ is T -transitive.

• If μ is reflexive and T -transitive, it is [ μ ]↗ T [ μ ]=[ μ ] . since

t ik = T ( 1 , t ik ) = T ( t ii , t ik ) Max

1 j n

T ( t ij , t jk ) t ik ,

implies

t ik =

(

t ij , t jk )

[ μ ]=[ μ ]↗ T [ μ ] .

Max
1

j n T

,or

Remark 4.3.1 The definitions given in this section contain the case of the corre-
sponding classical crisp definitions,
• If R ↂ X × X is a classical reflexive relation in X , its membership function μ R

reflects ( x , x ) ∈ R for all x ∈ X ,by μ R ( x , x ) = 1.

• If R ↂ X × X is a classical symmetric relation in X,

( x , y ) ∈ R ⃔ ( y , x ) ∈ R is reflected by μ( x , y ) = μ( y , x ).

• If R ↂ X × X is antisymmetric,

( x , y ) ∈ R & ( y , x ) ∈ R ⃔ x = y

μ R ( , ) = μ R ( , ) = (> ) ⃒ =

is reflected by x y y x 1 0 x y .