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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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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      LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034

YB 44

M.Sc. DEGREE EXAMINATION – STATISTICS

THIRD SEMESTER – April 2009

ST 3875 – FUZZY THEORY AND APPLICATIONS

 

 

 

Date & Time: 27/04/2009 / 9:00 – 12:00     Dept. No.                                                       Max. : 100 Marks

 

 

SECTION – A

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

1. Write the axiomatic skeleton of fuzzy t-conorm.
2. Define Archimedean t-norm.
3. Define Drastic fuzzy intersection.
4. Write a short note on fuzzy number.
5. If , find
6. Define a fuzzy variable and give an example.
7. Define scalar cardinality of a fuzzy set.
8. State the role of a ‘knowledge engineer’ in constructing fuzzy sets.
9. Distinguish between direct methods and indirect methods of constructing membership functions.
10. Define an artificial neural network.

 

SECTION – B

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

11. Define &  and prove that ,.

 

12. Prove that , where  and  denote drastic and yager class of t-norm.

 

13. Let the triple be a dual generated by an increasing generator. Prove that fuzzy operations  satisfy the law of excluded middle and the law of contradiction. Also prove that  does not satisfy distributive law.

 

14. Let and B. Find the 4 basic operations for the fuzzy numbers A and B.

 

15. Prove under usual notations: (i) ^α(A^c) = ( ^{(1 – α) +}A)^c (ii)  = ^α+ ()
16. State the axiomatic skeleton and desirable requirements for fuzzy complements. Prove that if the monotonic and involutive axioms are satisfied, then the boundary and continuity conditions are satisfied.

 

17. Let X ={x₁, x₂ ,x₃} be a universal set and suppose two experts E₁ and E₂ have specified the valuations of these three as elements of two fuzzy sets A and B as

given in the following table:

Membership in A                   Membership in B

Element	E1	E2
x1	0.6	0.5
x2	0.2	0.3
x3	0.8	0.6

Element	E1	E2
x1	0.2	0.4
x2	0.9	0.7
x3	0.6	0.3

 

 

 

 

 

Assuming that for set A, the experts have to be given weights as c₁ = 0.7 and c₂ = 0.3 and that for set B, the weights are c₁ = 0.2, c₂ = 0.8, find the degree of membership of the three elements in A and in B. Also, find the degree of membership in AUB by bounded sum operator.

 

18. State the three different classes of network architectures and briefly describe any one of them with a diagram.

 

SECTION -C

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

19. (a) Let i be a t-norm and strictly increasing and continuous function in (0,1) such that g(0)=0, g(1)=1. Prove that the function ,  where denotes pseudo inverse of g is a t-norm.

(b) Prove that the triples  and  are dual with

respect to any fuzzy complement.                                                                    (15+5)

 

20. Let MIN and MAX be binary operations on the set of all fuzzy numbers. Prove that for any fuzzy numbers A, B, C the following properties hold:

(a) MIN(A,MIN(B,C))=MIN(MIN(A,B),C)

(b) MAX(A,MAX(B,C))=MAX(MAX(A,B),C)

(c) MIN(A,MAX(A,B))=A

(d) MAX(A,MIN(A,B))=A

(e) MIN(A,MAX(B,C))=MAX(MIN(A,B), MIN(A,C))

(f) MAX(A,MIN(B,C))=MIN(MAX(A,B), MAX(A,C))

 

21. (a)Explain the indirect method of constructing a membership function with one expert.

(b) State the role of ‘activation function’ in neural networks. Describe the three basic types of activation functions.                                                               (10 + 10)

 

22. (a) Briefly explain the three practical issues in ‘Pattern Recognition’.

(b) State the problem of ‘Fuzzy Clustering’ and present the Fuzzy c-means

algorithm.                                                                                                 (6 + 14)

 

 

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