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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LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

BA 27

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – November 2008

    ST 3875 – FUZZY THEORY AND APPLICATIONS

 

 

Date : 10-11-08                 Dept. No.                                        Max. : 100 Marks

Time : 9:00 – 12:00

SECTION – A

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

1. Define Archimedean t-conorm.
2. Write the axiomatic skeleton of fuzzy t-norm.
3. Define Drastic fuzzy union.
4. Explain increasing generator.
5. Define arithmetic operations on intervals.
6. Define membership function and give an example.
7. Find the core of the fuzzy set whose membership function is given by                                         f(x) = exp [– (x – 3)²]
8. Give an example of a trapezoidal shaped membership function.
9. Explain the sigmoid function used for activation.
10. Present the motivation for fuzzy clustering.

 

SECTION -B

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

1. Define  &  and prove that , .

 

1. Prove that ,.

 

1. Define dual triple and show that given a t-norm i and an involutive fuzzy complement c , the binary operation u on [0,1] defined by  for all  is a t-co-norm such that  is a dual triple.

 

1. Prove that if  for every ,  then . What will happen when .

 

1. Prove that A is a fuzzy subset of B if and only if    ^αA  ^αB α [0,1].

 

1. Prove that (i) = ^α( ) and  (ii)   ^α ()
2. Explain the Lagrange interpolation method for constructing membership function from sample data.

 

1. Describe the architecture of a multi-layer feed-forward network.

 

 

 

 

)

SECTION -C

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

 

1. (a) Prove that A is a fuzzy number if and only if there exists a closed interval

such that  where

is monotonically increasing continuous from the right such

that  and r is monotonically

decreasing continuous from the left such that r.

(b) Write a short note on Linguistic variables.                                                (15+5)

 

1. Let  and B

Find the four basic operations for the fuzzy numbers A and B and also find the

corresponding fuzzy numbers.

 

1. (a) Define equilibrium of a fuzzy complement and show that every fuzzy complement has atmost one equilibrium. Also show that a continuous fuzzy complement has a unique equilibrium.

(b) If a fuzzy complement c has an equilibrium e_c, then a ≤ c(a) iff a ≤ e_c and

a ≥ c(a) iff a ≥ e_c.                                                                                           (14 + 6)

 

1. (a) Explain the direct method with multiple experts for constructing membership  function.

(b) Let X ={x₁, ..,x₅} be a universal set and suppose four experts E1, E2, E3, E4

have specified the valuations of the 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	E4
x1 x2 x3 x4 x5	0 1 1 0 1	1 0 0 1 0	1 0 1 0 1	0 1 1 0 0

Element	E1	E2	E3	E4
x1 x2 x3 x4 x5	1 0 1 0 1	1 0 1 0 0	1 0 0 1 0	0 1 0 1 1

 

 

 

 

 

 

For the set A, the four experts are to be given weights c₁ = 1/3, c₂ = 1/4, c₃ = 1/4,

c₄ = 1/6 and for set B, the weights are all equal for the four experts. With these

weights find the degrees of membership of the five elements in A and in B.

Also, evaluate the degrees of membership in A ∩ B using the standard

            intersection and in A U B using the algebraic sum operators.                  (8 + 12)

 

 

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