Chapter 2 The Operation of Fuzzy Set

FUZZY MATHEMATICAL ANALYSIS

ÁNGEL GARRIDO DEPARTAMENTO DE MATEMÁTICAS FUNDAMENTALES FACULTAD DE CIENCIAS UNED

OUTLINE • • • • •

Standard operations on fuzzy sets: Fuzzy complement Fuzzy union Fuzzy intersection Other operations in fuzzy set  Disjunctive sum  Difference  Distance  Cartesian product

• t-norms and t-conorms

STANDARD OPERATIONS ON FUZZY SETS • Complement

 A ( x)  1   A ( x), x  X

• Union

 AB ( x)  max(  A ( x), B ( x)), x  X

• Intersection

 AB ( x)  min(  A ( x), B ( x)), x  X

FUZZY COMPLEMENT • C:[0,1][0,1]

FUZZY COMPLEMENT

FUZZY COMPLEMENT • Axioms C1 and C2 called “axiomatic skeleton” are fundamental requisites to be a complement function, i.e., for any function C:[0,1][0,1] that satisfies axioms C1 and C2 is called a fuzzy complement. • Additional requirements

FUZZY COMPLEMENT • Example 1 : Standard function

Axiom C1 Axiom C2 Axiom C3 Axiom C4

FUZZY COMPLEMENT • Example 2 :

Axiom C1 Axiom C2 X Axiom C3 X Axiom C4

FUZZY COMPLEMENT • Example 3:

Axiom C1 Axiom C2 Axiom C3 X Axiom C4

FUZZY COMPLEMENT • Example 4: Yager’s function

Axiom C1 Axiom C2 Axiom C3 Axiom C4

FUZZY COMPLEMENT • Fuzzy partition If m subsets are defined in X, m-tuple (A1, A2,…,Am) holding the following conditions is called a fuzzy partition.

FUZZY UNION

FUZZY UNION • Axioms U1 ,U2,U3 and U4 (called “axiomatic skeleton”) are fundamental requisites to be a union function, i.e., for any function U:[0,1]X[0,1][0,1] that satisfies axioms U1,U2,U3 and U4 is called a fuzzy union. • Additional requirements

• Example 1 : Standard union

Axiom U1 Axiom U2 Axiom U3 Axiom U4 Axiom U5 Axiom U6

• Example 2: Yager’s union function

Axiom U1 Axiom U2 Axiom U3 Axiom U4 Axiom U5 X Axiom U6

FUZZY UNION

• Some frequently used fuzzy unions – Probabilistic sum (Algebraic Sum):

U as ( x, y)  x  y  x  y – Bounded Sum (Bold union):

U bs ( x, y)  min{1, x  y} – Drastic Sum:

max{ x, y}, if min{ x, y}  0 U ds ( x, y )   1, x, y  0 

– Hamacher’s Sum

x  y  (2   ) x  y U hs ( x, y )  ,  0 1  (1   ) x  y

FUZZY UNION

FUZZY INTERSECTION

FUZZY INTERSECTION • Axioms I1 ,I2,I3 and I4 (called their “axiomatic skeleton”) are fundamental requisites to be a intersection function, i.e., for any function I:[0,1]X[0,1][0,1] that satisfies axioms I1,I2,I3 and I4 is called a fuzzy intersection. • Additional requirements

• Example 1 : Standard intersection function

Axiom I1 Axiom I2 Axiom I3 Axiom I4 Axiom I5 Axiom I6

• Example 2: Yager’s intersection function

Axiom I1 Axiom I2 Axiom I3 Axiom I4 Axiom I5 X Axiom I6

FUZZY INTERSECTION

• Some frequently used fuzzy intersections - Probabilistic product (Algebraic product):

I ap ( x, y)  x  y - Bounded product (Bold intersection):

I bd ( x, y)  max{ 0, x  y  1} - Drastic product:

min{ x, y}, if max{ x, y}  1 I dp ( x, y )   0, x, y  1 

- Hamacher’s product x y I hp ( x, y )  ,  0   (1   )( x  y  x  y )

FUZZY INTERSECTION

OTHER OPERATIONS • Disjunctive sum (exclusive OR)

OTHER OPERATIONS

OTHER OPERATIONS

MORE OPERATIONS • Disjoint sum (elimination of common area)

• Difference  Crisp set

 Fuzzy set : Bounded difference

OTHER OPERATIONS • Example  Simple difference

• Example  Bounded difference

• Distance and difference

• Distance  Hamming distance

OTHER OPERATIONS  Euclidean distance

 Relative Euclidean distance  Minkowski distance (w=1-> Hamming and w=2-> Euclidean)

• Cartesian product  Power

 Cartesian product

• Example: • A = { (x1, 0.2), (x2, 0.5), (x3, 1) } • B = { (y1, 0.3), (y2, 0.9) }

T-NORMS AND T-CONORMS (OR SNORMS)

T-NORMS AND T-CONORMS (S-NORMS)

T-NORMS AND T-CONORMS (S-NORMS) • Duality of t-norms and t-conorms

 Applying complements

 ( x, y )  1  T (1  x,1  y)  1  T ( x, y)  T ( x, y), : t - conorms T : t - norms  De Morgan’s laws