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)

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

SECTION – B

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

Define & and prove that ,.

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

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.

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

Prove under usual notations: (i) ^α(A^c) = ( ^{(1 – α) +}A)^c (ii) = ^α+ ()
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.

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.

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)

(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)

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)

(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))

(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)

(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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