M.Sc. Math IV (Fourth) Semester Examination 2014-15
Course Code:MAT402
Paper ID: 0174112
Fuzzy Sets and its Applications
Time: 3 Hours
Max. Marks: 70
Note: Attempt six questions in all. Q. No. 1 is compulsory.
1.
a)
b)
c)
Answer any five of the following (limit your answer to 50
words):
(4x5=20)
Compute the scalar cardinalities for each of the following
fuzzy sets:
i)
A=
+
ii)
B= +
.
iii)
C (x) =
for
.
iv)
D(x) =1 for
.
Let A be a fuzzy set defined by
A=
+
. List all α-cuts and strong α-cuts
of A.
Find the equilibrium of the fuzzy complements
( > -1,
and
d)
.
Determine whether each of following function is an
increasing generator, if it is, find and plot the fuzzy
complement, t - norm and t – conorm generated by it:
i)
g(a) = sin a
ii)
g(a) = t g(a).
e) Calculate the following:
i)
[-1,2] +[1,3]
ii)
[-2,4] - [3,6]
iii)
[ -3,4] . [-3,4]
iv)
[-4,6] / [1,2]
f) The fuzzy relation R is defined on set X1 = { a, b, c }, X2 =
{s, t}, X3 = {x, y}, X4 = {i, j} as follows:
R (X1, X2, X3, X4) =
+
+
+
+
+
i)
Compute the projections R1,2,4, R1,3, and R4
ii)
Compute the cylindric extensions [R1,2,4 ↑ {X3}, [R1,3
↑ {X2, X4}], [R4 ↑ {X1, X2, X3}]
g) For each of the following binary relations on a single set,
state whether the relation is reflexive, irreflexive or
antireflexive, symmetric, antisymmetric, asymmetric or
strictly antisymmetric, and transitive, nontransitive or
antitransitive
i)
“ is a sibling of ”
ii)
“ is a parent of”
iii)
“ is smarter than”
iv)
“ is the same height as”
h) Let P1, P2
Then show that
i)
P1 P P2 implies that P S(Q,R)
ii)
P 1 P2
S(Q , R) where
denotes the standard
fuzzy union.
2.
3.
Prove that a fuzzy set A on R is convex iff A ( 𝛌x1 + (1𝛌)x2) ≥ min {A(x1), A(x2)} for all x1, x2 R and all 𝛌 [0,1]
where min denotes the minimum operator.
(10)
Let ƒ: X→Y be an arbitrary crisp function. Then prove that
for any A Ƒ(X) and all
[0, 1], the following properties
of f fuzzified by the extension principle hold:
(10)
a)
=f
b)
f
4.
et C be a function from [0,1] to [0,1]. Then prove that C is a
fuzzy complement iff there exists a continuous function f
from [0,1] to R such that f(1) =0, f is strictly decreasing and
C(a )= f -1(f(0)-f(a)) for all a [0 ,1].
(10)
5.
Let MIN and MAX be binary operations on R defined by
a)
b)
6.
for all
z R respectively. Then prove that for any A, B, C R the
following properties hold:
(10)
MIN (A, B) = MIN (B, A)
MAX (A, B) = MAX (B, A)
MIN [ MAX (A, B),C ] = MIN[A,MIN(B, C)]
MAX [ MAX (A, B),C ] = MAX[A,MAX(B, C)]
Let i be a t-norm in the equation
P
=
Determine if the above equation has a solution for i = min,
product and bounded difference respectively.
(10)
7.
Le
be the solvability index of fuzzy relation equation 111
defined by
then
prove that
.
8.
(10)
Given
Q
and r =
Determine all the solutions of p Q = r
Where p =
,Q=
,r=
.
(10)