Fuzzy Relations - Fuzzy Logic: An Introductory Course for Engineering Students - page 115

Chapter 4
Fuzzy Relations

4.1 What Is a Fuzzy Relation?

A predicate R on a cartesian product X 1 ×···× X n is called a relational (n-ary)

predicate. For example, if X 1 = X 2 =[ 0 , 10 ] , R = close to, ' ( x , y ) is R ', or ' x is

close to y ', is a relational binary predicate.
Analogously, if X 1 = lives in the same borough, or ' x lives

in the same borough than y ', is a relational binary predicate.
A fuzzy relation in X 1 ×···×

X 2 = London ,R =

X n is any function μ : X 1 ×···× X n ₒ[ 0 , 1 ] .

If interpreting μ R ( x 1 ,..., x n ) = 'degree up to which ( x 1 ,..., x n ) is in R ', it is said

that μ R represents the n-ary relational relation R '.

Anyrule'If x is P , then y is Q ' defines the binary predicate Q / P in X × Y given by

( x , y ) is Q / P ⃔ If x is P , then y is Q ,

whose representation, or membership function of the corresponding fuzzy set Q / P

is given by

μ Q / P ( x , y ) = (μ P ₒ μ Q )( x , y ) = J (μ P ( x ), μ Q ( y )),

once a T-conditional J adapted to the meaning of Q / P is selected. A fuzzy relation

μ R is nothing else than a fuzzy set R in X 1 ×···× X n .

For example, if R = 'close to' , is represented by

μ R ( x , y ) = max ( 0 , k | x − y | ), for all x , y ∈[ 0 , 1 ] ,