1. Jan Jantzen www.inference.dk 2013
A set is a collection of objects
A special kind of set
Fuzzy Sets
Jan Jantzen
2013

2. Summary: A set ... This object is not a member of the set
This object is a member of the set
A classical set has a sharp boundary

3. ... and a fuzzy set This object is not a member of the set
This object is a member of the set to a degree, for instance 0.8. The membership is between 0 and 1.
A fuzzy set has a graded boundary

4. Example: High and low pressures

5. Example: "Find books from around 1980"
This could include 1978, 1979, 1980, 1981, and 1982

6. Maybe even 41 or 42 could be all right?

7. Example: A fuzzy washing machine
If you fill it with only a few clothes, it will use shorter time and thus save electricity and water.
There is a computer inside that makes decisions depending on how full the machine is and other information from sensors.
Samsung J1045AV capacity 7 kg

8. A rule (implication) IF the machine is full THEN wash long time
Condition
Action.
The internal computer is able to execute an if—then rule even when the condition is only partially fulfilled.

9. IF the machine is full … Example: Load = 3.5 kg clothes Classical set
true
false
Example: Load = 3.5 kg clothes
Classical set
Linear fuzzy set
Nonlinear fuzzy set
Three examples of functions that define ‘full’. The horizontal axis is the weight of the clothes, and the vertical axis is the degree of truth of the statement ‘the machine is full’.

10. … THEN wash long time long time could be t = 120 minutes
The duration depends on the washing program that the user selects.

11. Decision Making (inference)
Rule. IF the machine is full THEN wash long time
Measurement. Load = 3.5 kg
Conclusion. Full(Load) × t = 0.5 × 120 = 60 mins
The machine is only half full, so it washes half the time.
Analogy
The dots mean 'therefore'

12. A rule base with four rules
IF machine is full AND clothes are dirty THEN wash long time
IF machine is full AND clothes are not dirty THEN wash medium time
IF machine is not full AND clothes are dirty THEN wash medium time
IF machine is not full AND clothes are not dirty THEN wash short time
There are two inputs that are combined with a logical 'and'.

13. Theory of Fuzzy Sets

14. Lotfi Zadeh’s Challenge
Clearly, the “class of all real numbers which are much greater than 1,” or “the class of beautiful women,” or “the class of tall men,” do not constitute classes or sets in the usual mathematical sense of these terms (Zadeh 1965).

15. {Live dinosaurs in British Museum} = 
Sets
The set of positive integers
The set of
belonging to
for which
{Live dinosaurs in British Museum} = 
The empty set

16. Fuzzy Sets
{nice days}
{adults}
Membership function
Much greater than

17. Tall Persons Degree of membership 150 160 170 180 190 200 0.2 0.4 0.6
0.2
0.4
0.6
0.8 1
Height [cm]
Membership
fuzzy
crisp
Membership function
Universe

18. Fuzzy (http://www.m-w.com)
adjective
Synonyms: faint, bleary, dim, ill-defined, indistinct, obscure, shadowy, unclear, undefined, vague
Unfortunately, they all carry a negative connotation.

19. 'Around noon'
Triangular
Trapezoidal
Smooth versions of the same sets.

20. The 4 Seasons 1 Membership 0.5 Time of the year Spring Summer Autumn
Seasons have overlap; the transition is fuzzy.
0.5 1
Time of the year
Membership
Spring
Summer
Autumn
Winter
we are here

21. Summary
A set
A fuzzy set

22. Operations on Fuzzy Sets

23. Set Operations
Union
Intersection
Negation
Classical
Fuzzy

24. Fuzzy Set Operations A B
Union
Intersection
Negation

25. Example: Age The negation of 'very young' Primary term Primary term
The square root of Old
The square of Young

26. Operations Here is a whole vocabulary of seven words.
Each operates on a membership function and returns a membership function. They can be combined serially, one after the other, and the result will be a membership function.

27. Cartesian Product
The AND composition of all possible combinations of memberships from A and B
The curves correspond to a cut by a horizontal plane at different levels

28. Example: Donald Duck's family
Suppose,
nephew Huey resembles nephew Dewey
nephew Huey resembles nephew Louie
nephew Dewey resembles uncle Donald
nephew Louie resembles uncle Donald
Question: How much does Huey resemble Donald?

29. Solution: Fuzzy Composition of Relations
Dewey Louie
Huey =
Donald
0.8
0.9
0.5
0.6
Relation ? ?
0.8
0.5
Huey Dewey Donald
Huey Louie Donald
0.9
0.6 ?

30. If—Then Rules If x is Neg then y is Neg If x is Pos then y is Pos
approximately equal

31. Key Concepts Universe Membership function Fuzzy variables
Set operations
Fuzzy relations
All of the above are parallels to classical set theory

32. Application examples Database and WWW searches
Matching of buyers and sellers
Rule bases in expert systems