Fuzzy
Rules
2
Extension Principle & Fuzzy Relations
Extension principle
A is a fuzzy set on X :
A = A ( x1 ) / x1 + A ( x2 ) / x2 ++ A ( xn ) / xn
The image of A under f(.) is a fuzzy set B:
B = B ( x1 ) / y1 + B ( x2 ) / y2 ++ B ( xn ) / yn
where yi = f(xi), i = 1 to n
If f(.) is a many-to-one mapping, then
B ( y ) = max A ( x )
−1
x= f ( y)
Extension Principle &
Fuzzy Relations
– Example:
Application of the extension principle to fuzzy
sets with discrete universes
Let A = 0.1 / -2+0.4 / -1+0.8 / 0+0.9 / 1+0.3 / 2
and f(x) = x2 – 3
Applying the extension principle, we obtain:
B = 0.1 / 1+0.4 / -2+0.8 / -3+0.9 / -2+0.3 /1
= 0.8 / -3+(0.4V0.9) / -2+(0.1V0.3) / 1
= 0.8 / -3+0.9 / -2+0.3 / 1
where “V” represents the “max” operator
Same reasoning for continuous universes
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4
Extension Principle & Fuzzy Relations
Fuzzy relations
– A fuzzy relation R is a 2D MF:
R = {(( x, y ), R ( x, y ))|( x , y ) X Y}
– Examples:
Let X = Y = IR+
and R(x,y) = “y is much greater than x”
The MF of this fuzzy relation can be subjectively defined as:
y−x
, if y x
R ( x, y ) = x + y + 2
0 , if y x
if X = {3,4,5} & Y = {3,4,5,6,7}
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch. 3: Fuzzy rules & fuzzy reasoning
5
Extension Principle & Fuzzy Relations
• Then R can be Written as a matrix:
0 0.111 0.200 0.273 0.333
R = 0 0 0.091 0.167 0.231
0 0 0 0.077 0.143
where R{i,j} = [xi, yj]
– x is close to y (x and y are numbers)
– x depends on y (x and y are events)
– x and y look alike (x and y are persons or objects)
– If x is large, then y is small (x is an observed reading and Y is
a corresponding action)
Dr. Djamel Bouchaffra CSE 513 Soft Computing, Ch. 3: Fuzzy rules & fuzzy reasoning
6
Extension Principle & Fuzzy Relations
– Max-Min Composition
• The max-min composition of two fuzzy relations R1 (defined
on X and Y) and R2 (defined on Y and Z) is
R R ( x , z ) = [ R ( x , y ) R ( y , z )]
1 2 1 2
y
• Properties:
– Associativity:
R (S T ) = ( R S ) T
– Distributivity over union:
R ( S T ) = ( R S ) ( R T )
– Week distributivity over intersection:
R ( S T ) ( R S ) ( R T )
– Monotonicity:
S T (R S) (RT)
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Extension Principle & Fuzzy Relations
• Max-min composition is not mathematically tractable,
therefore other compositions such as max-product
composition have been suggested
– Max-product composition
R R ( x , z ) = [ R ( x , y ) R ( y , z )]
1 2 1 2
y
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Extension Principle & Fuzzy Relations
– Example of max-min & max-product composition
• Let R1 = “x is relevant to y”
R2 = “y is relevant to z”
be two fuzzy relations defined on X*Y and Y*Z respectively
X = {1,2,3}, Y = {,,,} and Z = {a,b}.
Assume that:
0.9 0.1
0.1 0.3 0.5 0.7 0.2 0.3
R 1 = 0.4 0.2 0.8 0.9 R2 =
0.5 0.6
0.6 0.8 0.3 0.2
0.7 0.2
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Extension Principle & Fuzzy Relations
The derived fuzzy relation “x is relevant to z” based on R1
& R2
Let’s assume that we want to compute the degree of
relevance between 2 X & a Z
Using max-min, we obtain:
R1 R 2 ( 2, a) = max0.4 0.9,0.2 0.2,0.8 0.5,0.9 0.7
= max0.4,0.2,0.5,0.7
= 0.7
Using max-product composition, we obtain:
R1 R 2 ( 2, a) = max0.4 * 0.9,0.2 * 0.2,0.8 * 0.5,0.9 * 0.7
= max0.36,0.04,0.40,0.63
= 0.63
Dr. Djamel Bouchaffra
Terimakasih
CSE 513 Soft Computing, Ch. 3: Fuzzy rules & fuzzy
Dr. Djamel Bouchaffra reasoning 10
