1. Introduction to Fuzzy Logic
Fuzzy Sets
Shadi T. Kalat
Session number
03/18/2016

2. Fuzzy Sets Fuzzy sets Crisp sets:
In mathematics, fuzzy sets are sets whose elements have degrees of membership
Crisp sets:
In a crisp set, an element is either a member of the set or not.

3. Fuzzy Sets Characteristic function Membership function 𝜇(𝑥)
characteristic function of a subset A of some set X, maps elements of X to the range {0,1}, the function that indicates membership in a set
Membership function 𝜇(𝑥)
𝐴= 𝑥, 𝜇 𝐴 𝑥 𝑥∈𝑋}
Membership grade
𝑛∈[0, 1]

4. Fuzzy Sets Discrete Universe Continuous Universe Membership Grade
X=Age
X= Number of children

5. Fuzzy Sets A fuzzy set (A) could be written in two forms:
𝐴= 𝑥 𝑖 ∈𝑋 𝜇 𝐴 ( 𝑥 𝑖 )/ 𝑥 𝑖
Discrete Universe
𝐴= 𝑋 𝜇 𝐴 (𝑥)/𝑥
Continuous Universe

6. Fuzzy Sets Support Height Core Support and Core
Membership Grade
Support and Core
Support
Height
Core
𝑆𝑢𝑝𝑝𝑜𝑟𝑡 𝐴 ={𝑥∈𝑋| 𝜇 𝐴 𝑥 >0}
𝐻𝑒𝑖𝑔ℎ𝑡 𝐴 =𝑀𝑎𝑥 𝜇 𝐴 𝑥 ∀𝑥∈𝑋
𝐶𝑜𝑟𝑒 𝐴 ={𝑥| 𝜇 𝐴 𝑥 =1}

7. Fuzzy Sets 𝛼−cut Level set Crossover point Crossover points
𝐴 𝛼 ={𝑥∈𝑋| 𝜇 𝐴 𝑥 >𝛼}
Height
∧ 𝐴 ={𝛼| 𝜇 𝐴 𝑥 =𝛼}
Core
𝐶𝑟𝑜𝑠𝑠𝑜𝑣𝑒𝑟 𝐴 ={𝑥| 𝜇 𝐴 𝑥 =0.5}
α-cut

8. Fuzzy Sets
Convexity
𝜇 𝐴 𝜆 𝑥 1 + 1−𝜆 𝑥 2 ≥ min 𝜇 𝐴 𝑥 1 , 𝜇 𝐴 𝑥 𝜆∈[0,1]
Convex
Non-convex

9. Fuzzy Sets
Scalar Cardinality
𝐴 = 𝑥 𝑖 ∈𝑋 𝜇 𝐴 ( 𝑥 𝑖 )

10. Membership Functions Triangular membership function
𝑇𝑟𝑖𝑎𝑛𝑔𝑢𝑙𝑎𝑟 𝑥,𝑎,𝑏,𝑐 =max⁡( min 𝑥−𝑎 𝑏−𝑎 , 𝑐−𝑥 𝑐−𝑏 ,0)
Membership Grade

11. Membership Functions Trapezoidal membership function
𝑇𝑟𝑎𝑝𝑒𝑧𝑜𝑖𝑑𝑎𝑙 𝑥,𝑎,𝑏,𝑐,𝑑 =max⁡( min 𝑥−𝑎 𝑏−𝑎 , 𝑑−𝑥 𝑑−𝑐 ,0)
Membership Grade

12. Membership functions Gaussian membership function
Membership Grade

13. Membership functions Generalized bell membership function
𝑔𝑏𝑒𝑙𝑙 𝑥,𝑎,𝑏,𝑐 = 𝑥−𝑐 𝑎 2𝑏
Membership Grade

14. Operations on fuzzy sets
Equivalence
Fuzzy subset
Fuzzy negation
𝜇 𝐴 𝑥 = 𝜇 𝐵 𝑥 𝑥∈𝑋
Membership Grade
𝜇 𝐴 𝑥 ≤ 𝜇 𝐵 𝑥 𝐴⊆𝐵
𝜇 𝐴 𝑥 = 1−𝜇 𝐴 𝑥
Membership Grade

15. Operations on fuzzy sets
Sugeno’s complement
Yager’s complement
𝑁 𝑠 𝑎 = 1−𝑎 1+𝑠𝑎 𝑠>−1
𝑁 𝑤 𝑎 = 1− 𝑎 𝑤 1 𝑤 𝑤>0

16. Operations on fuzzy sets
Fuzzy sets A and B
Fuzzy set NOT A
Fuzzy sets A OR B
Fuzzy sets A AND B

17. 2D fuzzy set
𝑅= 𝑋×𝑌 𝜇 𝑅 (𝑥,𝑦) (𝑥,𝑦)
Membership Grade

18. T-Norm (Triangle norm)
𝑇:𝐼×𝐼⟶𝐼
𝜇 𝐴∩𝐵 𝑥 =𝑇 𝜇 𝐴 𝑥 , 𝜇 𝐵 𝑥 = 𝜇 𝐴 (𝑥)× 𝜇 𝐵 (𝑥)
𝑇 0,0 =0
𝑇 𝑎,1 =𝑇 1,𝑎 =𝑎
𝑇 𝑎,𝑏 =𝑇 𝑏,𝑎
𝑇(𝑇 𝑎,𝑏 ,𝑐)=𝑇(𝑎,𝑇 𝑏,𝑐 )
𝐼𝑓 𝑎≤𝑏 𝑇 𝑎,𝑐 ≤𝑇 𝑏,𝑐

19. T-norms Min Algebraic product Bounded product Drastic product
𝑇 𝑎𝑝 𝑎,𝑏 =𝑎𝑏
𝑇 𝑏𝑝 𝑎,𝑏 = max 𝑎+𝑏−1,0 =0∧(𝑎+𝑏−1)
𝑇 𝑑𝑝 𝑎,𝑏 = 𝑎 𝑖𝑓 𝑏=1 𝑏 𝑖𝑓 𝑎= 𝑖𝑓 𝑎,𝑏<1

20. T-norms
Min A&B
Algebraic product

21. T-norms
Bounded product
Drastic product

22. T-norms 𝐻𝑎𝑚𝑎𝑐ℎ𝑒𝑟: 𝑎𝑏 𝛾+(1−𝛾)(𝑎−𝑏−𝑎𝑏) 𝛾≥0
𝐻𝑎𝑚𝑎𝑐ℎ𝑒𝑟: 𝑎𝑏 𝛾+(1−𝛾)(𝑎−𝑏−𝑎𝑏) 𝛾≥0
𝐷𝑜𝑚𝑏𝑖:1− min −𝛼 𝜓 + 1−𝑏 𝜓 1 𝜓 , 𝜓>0

23. S-norms 𝑆:𝐼×𝐼→𝐼 S 1,1 =1 S 𝑎,0 =𝑆 0,𝑎 =𝑎 S 𝑎,𝑏 =𝑆 𝑏,𝑎
𝐼𝑓 𝑎≤𝑏 𝑇 𝑎,𝑐 ≤𝑇 𝑏,𝑐

24. S-norms Max Algebraic sum Bounded sum Drastic sum
𝑇 𝑎𝑠 𝑎,𝑏 =𝑎+𝑏−𝑎𝑏
𝑇 𝑏𝑠 𝑎,𝑏 = min 𝑎+𝑏,1
𝑇 𝑑𝑠 𝑎,𝑏 = 𝑎 𝑖𝑓 𝑏=0 𝑏 𝑖𝑓 𝑎=0 1 𝑖𝑓 𝑎,𝑏<0

25. S-norms
Max A&B
Algebraic sum

26. S-norms
Bounded sum
Drastic sum

27. Fuzzy Rules Extension Principle F, y, x belong to a crisp set
𝑦=𝑓(𝑥) 𝑥=𝐴 𝑦=𝑓 𝐴
F, y, x belong to a crisp set

28. Fuzzy Rules Assume A is a fuzzy member of X 𝑦=𝑓(𝑥) 𝑥=𝐴 𝑦=𝑓 𝐴 =𝐵
𝑦=𝑓(𝑥) 𝑥=𝐴 𝑦=𝑓 𝐴 =𝐵
𝜇 𝐵 𝑦 = 𝜇 𝐴 (𝑥)
𝜇 𝐵 𝑦 = 𝜇 𝐴 ( 𝑓 −1 (𝑦))
𝜇 𝐵 𝑦 ≜𝑚𝑎𝑥 𝜇 𝐴 (𝑥)
𝑥= 𝑓 −1 (𝑦)

29. Example
𝐴= 0.1 − − −8
𝑆𝑚𝑎𝑙𝑙=
𝑦=|𝑥|
𝑦= 𝑥 2 𝐵=
𝑆𝑚𝑎𝑙 𝑙 2 =

30. Cylindrical extension
𝐶 𝐴 = 𝑥×𝑦 𝜇 𝐴 𝑥 (𝑥,𝑦)
Cylindrical extension of A
Base fuzzy set A
Membership Grade
Membership Grade

31. Projections of fuzzy sets
𝑅 𝑥 = [ max 𝑦 𝜇 𝑅 (𝑥,𝑦)]/𝑥
𝑅 𝑦 = [ max 𝑥 𝜇 𝑅 (𝑥,𝑦)]/𝑦
2-D Membership Function
Projection onto X
Projection onto Y

32. Fuzzy relations
𝑅= 𝑥×𝑦 𝜇 𝑅 𝑥,𝑦 (𝑥,𝑦)
𝜇 𝑅 𝑥,𝑦 = 𝑦−𝑥 𝑥+𝑦 𝑥<𝑦 0

33. Fuzzy Relations Max-Min Composition Max-Dot Product
𝑣∈𝑉, 𝑥∈𝑋, 𝑦∈𝑌
𝜇 𝑅 𝑥,𝑦 =𝑚𝑎𝑥 𝜇 𝑅1 𝑥,𝑣 , 𝜇 𝑅2 𝑣,𝑦
𝑣∈𝑉, 𝑥∈𝑋, 𝑦∈𝑌

34. Fuzzy Relations 𝑅1:𝑋 𝑖𝑠 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑛 𝑉 𝑅1 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑖𝑛 𝑋×𝑉 𝑎𝑛𝑑 𝑅2 𝑖𝑛 𝑉×𝑌
𝑅2:𝑉 𝑖𝑠 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑜𝑛 𝑌
𝑋={1,2,3}
𝑉={𝛼,𝛽, 𝛾, 𝛿}
𝑌={𝑎,𝑏}
𝑅2=
𝑅1=

35. Fuzzy Relations Max-Min Max-Dot
𝜇 𝑅1.𝑅2 2,𝑎 = max 0.4∧0.9,0.2∧0.2,0.8∧0.5,0.9∧0.7 = max 0.4,0.2,0.5,0.7 =0.7
𝜇 𝑅1.𝑅2 2,𝑎 = max 0.4×0.9,0.2×0.2,0.8×0.5,0.9×0.7 = max 0.36,0.4,0.4,0.63 =0.63

36. Linguistic Variables Primary terms: Young, old, …
Negation: Not young, Not old
Hedge: Very old, Extremely young, more or less old

37. Linguistic Variables Concentration (very) Dilation (more or less) Not
Contrast Intensification
𝐶𝑜𝑛 𝐴 = 𝐴 2
Contrast intensifier effect
𝐷𝑖𝑙 𝐴 = 𝐴 0.5
Membership Grade
𝑁𝑜𝑡 𝐴 =−𝐴
𝐼𝑛𝑡 𝐴 = 2 𝐴 ≤ 𝜇 𝐴 𝑥 ≤ 𝐴 ≤ 𝜇 𝐴 𝑥 ≤1