Further Operations on Fuzzy Sets

1
Further Operations on Fuzzy Sets

• Chapter 3

2
Basic Operations

• Complement
• Intersection
• Union

• Simplest forms for these operations
• Agree with Crisp set theory
• Do any other formulas make sense?
• YES!!!!!!

3
Fuzzy Complement

• What properties must any formula for complement
  possess?
• Let c0,1-gt0,1 be a function that transforms
  a membership function of a set into the
  membership function for the complement of the set.

• Boundary Conditions
• Axion C1
• Nondecreasing condition
• Axiom C2

4
Fuzzy Complement

• Definition. Only functions c0,1-gt0,1 that
  satisfies both axioms c1 and c2 can be called
  Fuzzy complements.

• Two other fuzzy complements will now be defined.
• They are not used very often.

5
Sugeno Class Complement
6
Yager Class Complement
7
Fuzzy Union

• What properties must any formula for union
  possess?
• Let s0,1X0,1-gt0,1 be a function that
  transforms membership functions of fuzzy sets A
  and B into the membership function of their union.

• Boundary conditions
• Axiom s1
• Commutative condition
• Axiom s2
• Nondecreasing condition
• Axiom s3
• Associative condition
• Axiom s4

8
Other s-norms

• Dombi
• Dubois-Prade
• Yager
• Drastic Sum
• Einstein Sum
• Algebraic Sum

9
Example 3.1

• Two S-norms
• Models Union

• Recall p percentage of US made parts in a car
• p measure of domesticity of a car
• D Domestic car F Foreign car

10
Theorem 3.1

• For any s-norm s,

• Proof

If b 0
If a0
For nonzero a and b
11
Lemma 3.1

• (See Eqns. 3.8 3.11, pp. 37-38)

12
Lemma 3.1cont.

• (See Eqns. 3.8 3.11, pp. 37-38)

13
Lemma 3.1cont.

• (See Eqns. 3.8 3.11, pp. 37-38)

• LHopitals Rule

14
Lemma 3.1cont.

• (See Eqns. 3.8 3.11, pp. 37-38)

Recall from Calculus
15
Lemma 3.1cont.

• (See Eqns. 3.8 3.11, pp. 37-38)

16
Lemma 3.1cont.

• (See Eqns. 3.8 3.11, pp. 37-38)

• So we have proven that the Dombi s-norm ranges to
  the smallest s-norm as its parameter goes to
  infinity

17
Lemma 3.1cont.

• (See Eqns. 3.8 3.11, pp. 37-38)

18
Lemma 3.1cont.

• (See Eqns. 3.8 3.11, pp. 37-38)

• We have completed proving that the Dombi s-norm
  ranges from the largest to the smallest s-norm as
  its parameter ranges from infinity to 0.