1. Intelligent System Course
Fuzzy Logic (1)
Intelligent System Course

2. Apples, oranges or in between?
Group of apple
Group of orange O A O A O A O A A O O A O A

3. Lotfi Zadeh

4. Thinking in Fuzzy O A O O A O A O A O A A O A Fuzzy group of apples
Fuzzy group of oranges

5. Crisp vs Fuzzy Crisp sets handle only 0s and 1s
Fuzzy sets handle all values between 0 and 1
Crisp
No Yes
Fuzzy
No Slightly Mostly Yes

6. Fuzzy Set Notation
Continuous
Discrete

7. Continuous Fuzzy Membership Function
The set, B, of numbers near to two is x
B(x)

8. Discrete Fuzzy Membership Function
The set, B, of numbers near to two is x
B(x)

9. Example Example Fuzzy Sets to Aggregate...
A = { x | x is near an integer}
B = { x | x is close to 2} x
A(x) x
B(x) 1

10. Fuzzy Union Meets crisp boundary conditions Commutative Associative
Fuzzy Union (logic “or”)
Meets crisp boundary conditions
Commutative
Associative
Idempotent

11. Fuzzy Union - example A+B (x)
A OR B = A+B ={ x | (x is near an integer) OR (x is close to 2)}
= MAX [A(x), B(x)] x
A+B (x)

12. Fuzzy Intersection Meets crisp boundary conditions Commutative
Fuzzy Intersection (logic “and”)
Meets crisp boundary conditions
Commutative
Associative
Idempotent

13. Fuzzy Intersection - example
A AND B = A·B ={ x | (x is near an integer) AND (x is close to 2)}
= MIN [A(x), B(x)] x
AB(x)

14. Fuzzy Complement Meets crisp boundary conditions
The complement of a fuzzy set has a membership function...
Meets crisp boundary conditions
Two Negatives Should Make a Positive
Law of Excluded Middle: NO!
Law of Contradiction: NO!

15. Fuzzy Complement - example
complement of A ={ x | x is not near an integer} x 1

16. Fuzzy Associativity (1)
Min-Max fuzzy logic has intersection distributive over union...
since
min[ A,max(B,C) ]=min[ max(A,B), max(A,C) ]

17. Fuzzy Associativity (2)
Min-Max fuzzy logic has union distributive over intersection...
since
max[ A,min(B,C) ]= max[ min(A,B), min(A,C) ]

18. Fuzzy and DeMorgan’s Law (1)
Min-Max fuzzy logic obeys DeMorgan’s Law #1...
since
1 - min(B,C)= max[ (1-B), (1-C)]

19. Fuzzy and DeMorgan’s Law (2)
Min-Max fuzzy logic obeys DeMorgans Law #2...
since
1 - max(B,C)= min[(1-B), (1-C)]

20. Fuzzy and Excluded Middle
Min-Max fuzzy logic fails The Law of Excluded Middle.
since
min( A,1- A)  0
Thus, (the set of numbers close to 2) AND (the set of numbers not close to 2)  null set

21. Fuzzy and the Law of Contradiction
Min-Max fuzzy logic fails the The Law of Contradiction.
since
max( A,1- A)  1
Thus, (the set of numbers close to 2) OR (the set of numbers not close to 2)  universal set

22. Cartesian Product
The intersection and union operations can also be used to assign memberships on the Cartesian product of two sets.
Consider, as an example, the fuzzy membership of a set, G, of liquids that taste good and the set, LA, of cities close to Los Angeles

23. = liquids that taste good AND cities that
Cartesian Product
We form the set...
E = G ·LA
= liquids that taste good AND cities that
are close to LA
The following table results...
LosAngeles(0.0) Chicago (0.5) New York (0.8) London(0.9)
Swamp Water (0.0)
Radish Juice (0.5)
Grape Juice (0.9)