LOYOLA COLLEGE AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

AB 28

THIRD SEMESTER – NOV 2006

ST 3875 – FUZZY THEORY AND APPLICATIONS

 

 

Date & Time : 06-11-2006/9.00-12.00   Dept. No.                                                       Max. : 100 Marks

 

 

 

SECTION – A

Answer ALL the Questions                                                                    (10 x 2 = 20 marks)

 

1. Define a fuzzy set. Give an example.
2. Define α–cut and strong α –cut of a fuzzy set.
3. Define height of a fuzzy set. What is a normal fuzzy set?
4. What is the axiomatic skeleton for fuzzy complements?
5. Give two examples of fuzzy t-conorm that are frequently used as fuzzy unions.
6. If X = {0, 1, 2, 3, 4} and A is a fuzzy set defined by the membership function

A(x) = x / 4, find the scalar cardinality of A

7. Give an example of fuzzy set operations that constitute a dual triple.
8. Distinguish between direct and indirect methods of constructing membership

functions.

 

9. Define an ‘Artificial Neural Network’.
10. State the formal definition of ‘Knowledge’.

 

 

SECTION – B

Answer any FIVE Questions                                                                    (5 x 8 = 40 marks)

 

11. Prove that a fuzzy set A on R is convex if and only if A(λx₁ + (1 – λ) x₂) ≥

min[A(x₁), A(x₂)]   for all x₁, x₂  R and all where min denotes the

minimum operator.

 

12. Let A_i F(X) for all iI, where I is an index set. Then prove that

and .

13. Explain the extension principle for fuzzy sets.
14. Prove that the standard fuzzy intersection is the only idempotent t-norm.
15. Let X = R and let A be a fuzzy set defined by the membership function

x – 1, 1 ≤ x ≤ 2

A (x) =       3 – x , 2 ≤x ≤ 3

0,    otherwise

Plot the membership function and the ½ -cut and ¼ -cut of A. Also find the support and core and state whether it is a normal fuzzy set.

 

 

 

 

16. Define an increasing generator and decreasing generator and their Pseudo-inverses.

Give an example for both and find their Pseudo-inverses.

 

17. Discuss the indirect method of constructing membership functions with one expert.
18. Describe a multilayer feed forward network with a neat diagram.

 

SECTION -C

Answer any TWO Questions                                                                 (2 x 20 = 40 marks)

 

19. (a) Let A, B F(X). Then prove that the following properties hold good for all

.

 

(b) Give an example to show third decomposition theorem.                              (15 + 5)

 

20. (a) State and prove First decomposition theorem.

(b) Prove that every fuzzy complement has at most one equilibrium.                 (12 + 8)

21. Let X ={x₁, ..,x₄} be a universal set and suppose three experts E1, E2, E3 have

specified the valuations of these four as elements of two fuzzy sets A and B as given

in the following table:

Membership in A                       Membership in B

Element	E1	E2	E3
x1 x2 x3 x4	1 0 1 1	1 1 0 1	0 1 1 1

Element	E1	E2	E3
x1 x2 x3 x4	0 1 0 0	1 0 0 1	0 1 1 0

 

 

 

 

 

 

Assuming that for set A, the evaluations by the three experts have to be given

weights as c₁ = ½, c₂ = ¼, c₃ = ¼ and for set B as equal weights, find the degree of

membership of the four elements in A and in B. Also, evaluate the degree of

membership in A∩B using the Standard intersection and Bounded difference  function and that in AUB  using the Standard union and Drastic Union..

 

22. (a)Describe the basic model of a neuron with a neat diagram, labeling its elements

and explaining the notations.

(b)Discuss the three basic types of activation functions.

 

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