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    MA526

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    Deep Singha
    The document is an exam for a course on fuzzy logic and set theory. It contains 10 questions assessing students' knowledge of key concepts like fuzzy set operations, fuzzy relations, fuzzy numbers, fuzzy control systems, and inference methods. Students are asked to calculate fuzzy set intersections and unions, compare properties of crisp and fuzzy sets, perform approximate reasoning using given fuzzy rules, analyze a relation for equivalence, and operate on and differentiate fuzzy numbers.

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    MA526

    Uploaded by

    Deep Singha
    142 views2 pages
    The document is an exam for a course on fuzzy logic and set theory. It contains 10 questions assessing students' knowledge of key concepts like fuzzy set operations, fuzzy relations, fuzzy numbers, fuzzy control systems, and inference methods. Students are asked to calculate fuzzy set intersections and unions, compare properties of crisp and fuzzy sets, perform approximate reasoning using given fuzzy rules, analyze a relation for equivalence, and operate on and differentiate fuzzy numbers.

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    MA526 (1)

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    The document is an exam for a course on fuzzy logic and set theory. It contains 10 questions assessing students' knowledge of key concepts like fuzzy set operations, fuzzy relations, fuzzy numbers, fuzzy control systems, and inference methods. Students are asked to calculate fuzzy set intersections and unions, compare properties of crisp and fuzzy sets, perform approximate reasoning using given fuzzy rules, analyze a relation for equivalence, and operate on and differentiate fuzzy numbers.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    142 views2 pages

    MA526

    Uploaded by

    Deep Singha
    The document is an exam for a course on fuzzy logic and set theory. It contains 10 questions assessing students' knowledge of key concepts like fuzzy set operations, fuzzy relations, fuzzy numbers, fuzzy control systems, and inference methods. Students are asked to calculate fuzzy set intersections and unions, compare properties of crisp and fuzzy sets, perform approximate reasoning using given fuzzy rules, analyze a relation for equivalence, and operate on and differentiate fuzzy numbers.

    Copyright:

    © All Rights Reserved
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 2

    NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA

    End-Semester (Spring) Examination, 2019


    M. Sc. Mathematics (2 Years and 5 Years Integrated)
    Subject : Mathematics (Fuzzy Logic and Set Theory) DEPT. CODE : MA-526
    FULL MARKS : 50 Duration of Examination : 03 Hours
    Answer all questions; Figures at the right hand margin indicate marks.
    All parts of a question should be answered at one place.
    (This question paper contains 02 pages)
    Q.No. Marks
    1. Given two fuzzy sets 05
    A = { (2,0.4), (3,0.6), (4,0.8), (5,1), (6,0.9), (7,0.5), (8,0.4)},
    B = { (2,0.4), (4,0.8), (5,1), (7,0.6)}
    Determine the intersection and union of A and B by using Hamacher
    operator with =0.25 and 0.5.

    2. i) Compare laws of contradiction and excluded middle between crisp and 05


    fuzzy sets.
    ii)State and prove De Morgan’s laws of t and s norms.

    3. Given Fuzzy propositions 05


     0.6 1 0.2   0.4 1 0.8 0.3 
    A    X , B     Y
     2 3 4   2 3 4 5 
    where X={1,2,3,4}, Y={1,2,3,4,5,6} and RULE 1 is given as
    RULE 1 : IF x is A , THEN y is B .
    Then find the consequent B  for a new antecedent A  as per RULE 2
    given by
    RULE 2 : IF x is A  , THEN y is B 
     0.4 1 0.2 
    where A      
     1 2 3 
    (Using Approximate Reasoning method and use operation as per your
    choice).

    4. Let us consider a fuzzy relation expressed in the following table. Show 05


    whether this is an equivalence relation. (Plot the graph also)
    a b c d
    a 1.0 0.8 0.7 1.0
    b 0.8 1.0 0.7 0.8
    c 0.7 0.7 1.0 0.7
    d 1.0 0.8 0.7 1.0
    5. Write the detail steps for the operations of addition and multiplication of two 05
    trapezoidal fuzzy numbers in term of the operations using their membership
    functions (i.e. not by -cut).

    6. i) If a  {(2,0.9), (3,1), (4,0.2)} , b  {(5,0.7), (6,1), (7,0.3)} , f(x)= 3, x  [2,7] , 05


    b
    Find  f (x )dx
    a
    (ii) There is a fuzzifying function f  ( f1,0.3), ( f2 ,0.8), ( f3 ,0.3),
    f1 ( x )  x, f2 ( x )  x 2 , f3 ( x )  x 3  1. Differentiate the function f at
    x0  0.5 .

    7. Compute the following operations for the Triangular fuzzy Numbers A 05


    =[1,4,8], B =[2,5,6] using -cut operation:
    i) A  B , ii) A  B , iii) , A  B iv) A / B .

    8. Solve [1,2][ x, x ]  [4,8] , by the various possible ways (that you can) and 05
    then comment in each of the solution.

    9. Discuss Mamdani and Sugeno control models and show those by graph for 05
    four rules having three antecedents and single consequent in each.

    10. i) Prove “Modus Tollens” tautology 05


    ii) Define and explain Extension Principle
    iii) Define Fuzzy number
    iv) Define and explain centroid method of defuzzification
    v) Define Fuzzy tolerance.

    -----------x----------

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