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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