The Operation of Fuzzy Set

LESSON 02

Dewi Yanti Liliana, M.Kom

Learning Objectives

• Introduces various operations of fuzzy sets

• Introduces the concepts of disjunctive sum, distance, difference, conorm and t – conormoperators

Material sources is taken from First Course on Fuzzy Theory and Applications, Lee, K.H.

Standard Operations of Fuzzy Set

• Complement set A , union , and intersection represent the standard operations of fuzzy theory and are arranged as:

• Note that the two characteristics of crisp set operators does not hold in fuzzy set.

• The reason for this occurrence is that the boundary of complement of A is ambiguous.

Fuzzy Complement(Requirement for complement function)

• Complement set of A ( ) carries the sense of negation.

• Complement function C maps the membership function of fuzzy set A to [0,1] (written by ).

• Fuzzy complement function should satisfy two axioms:

• Axioms C1 and C2 are fundamental requisites to be a complement function.

• These two axioms are called “axiomatic skeleton”.

• Additional axioms are:

Example of Complement Function

• The four axioms for complement hold in standard complement operator

Example of Complement Function

• The following is a complement function satisfying only the axiomatic skeleton.

Example of Complement Function• The following complement function is

continuous (C3) but not holds C4

When a = 0.33, C(0.33) = 0.75 in this function. However since C(0.75)= 0.15 ≠ 0.33, C4 does not hold .

Popular Complement Function

• Yager’s Function

• When w = 1, the Yager’s function becomes the standard complement function C(a) = 1 – a.

Yager Complement Function

Fuzzy Partition

Fuzzy Union

Fuzzy Union

the union and OR logic are identical.

Examples of Union Function

Yager Union Function

Other Union Operations

Fuzzy Intersection

Axioms for Intersection Function

Examples of Intersection

intersection and AND logic are equivalent.

Examples of Intersection

Yager Intersection Function

Other Intersection Operations

Other Operations in Fuzzy Set1. Disjunctive Sum

• Disjunctive sum is the name of operation corresponding “exclusive OR” logic.

• Definition (Simple disjunctive sum)

By means of fuzzy union and fuzzy intersection, definition of the disjunctive sum in fuzzy set is allowed just like in crisp set.

Example of Disjunctive Sum

Example of simple disjunctive sum

Definition (Disjoint sum)

• The key idea of “exclusive OR” is elimination of common area from the union of A and B. With this idea, we can define an operator ∆ for the exclusive OR disjoint sum as follows.

Disjoint Sum

Difference in Crisp Set

Difference in Fuzzy Set

Distance in Fuzzy Set

• The concept ‘distance’ is designated to describe the difference.

• Measures for distance in fuzzy setare defined in:

1. Hamming Distance

2. Euclidean Distance

3. Minkowski distance

1. Hamming Distance

2. Euclidean Distance

Euclidean Distance

3. Minkowski distance

Cartesian Product of Fuzzy Set

t-norms and t-conorms

• There are two types of operators in fuzzy sets:

– t-norms (triangular-norm)

– t-conorms (triangular-conorm) or also called as s-norms

• All tnorm and t-conorm functions follow these relations.

How about these operators?