LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

M.Sc. DEGREE EXAMINATION – STATISTICS

NO 45

THIRD SEMESTER – APRIL 2008

ST 3875 – FUZZY THEORY AND APPLICATIONS

 

 

 

Date : 30-04-08                  Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

 

 

SECTION – A

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

 

1. Define a fuzzy t-conorm.
2. Give an example of a discontinuous t-norm. Justify?
3. Define pseudo inverse of a increasing continuous function on [0, 1].
4. What are the two methods for defining fuzzy arithmetic?
5. State the law of excluded middle and the law of contradiction.
6. Define α-cut and strong α-cut of a fuzzy set.
7. Give a rough graphical depiction of the membership function of a convex fuzzy set.
8. State the Axiomatic Skeleton for fuzzy complements.
9. Give an example for a parametric class of membership functions.
10. Define an ‘Artificial Neural Network’.

 

 

SECTION – B

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

 

11. (a) Prove that the standard fuzzy union is the only idempotent t-conorm.

(b). Prove that , for all a,b [0, 1].                   (4+4)

12. Given

Determine

 

13. Prove that u(a,b)=c(i(c(a), c(b))) is a t-conorm for all a,b[0, 1], where c is the involutive fuzzy complement.
14. Under what conditions distributive law hold good for fuzzy numbers? Justify your answer with suitable examples.
15. If c: [0,1] → [0,1] is involutive and monotonically decreasing, show that it is continuous and that c(0) = 1 and c(1) = 0.
16. Explain the ‘indirect method with one expert’ for constructing membership functions.
17. What is an activation function? Explain the three basic types of activation functions.
18. Discuss the problem of fuzzy clustering with an example.

 

 

 

 

 

 

 

SECTION -C

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

19. (a) Let A and B be two fuzzy numbers. If

 

and

Determine the product fuzzy number (A . B) and the division (A/B).

 

(b) Explain the basic arithmetic operations on the intervals.                            (16+4)

 

20. (a) Prove the characterization theorem for fuzzy numbers.

(b) Let A and B be fuzzy numbers.  Prove that   is

also a fuzzy number, where * is one of the basic arithmetic operations.      (10+10)

 

21. (a) Explain the ‘direct method with multiple experts’ for constructing membership

functions.

(b) Let X ={x₁, ..,x₅} be a universal set and suppose three experts E1, E2, E3 have

specified the valuations of these five 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 x5	0 1 1 0 1	1 0 0 1 1	1 0 1 0 1

Element	E1	E2	E3
x1 x2 x3 x4 x5	1 0 1 0 0	1 0 1 0 1	1 0 0 1 0

 

 

 

 

 

 

 

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

weights as c₁ = 1/4 , c₂ = 1/2, c₃ =1/4 and for set B as equal weights, find the

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

degree of membership in A∩B using the Algebraic prroduct operator and in

AUB using the Drastic union operator.                                                         (6 +14)

 

22. (a) Briefly explain the three fundamental problems of ‘Pattern Recognition’.

(b) Describe the single-layer and multi-layer feed forward and recurrent neural

network architectures.                                                                                    (6 +14)

 

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