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模糊理論
學號：
姓名：
101/12
一、 是非題(28%)
(
) 1. Every fuzzy number is a normal and convex fuzzy set.
(
) 2. Distributive lattices are lattices for which the operations of join and meet distribute
over each other. Fuzzy numbers can be ordered partially in a natural way and that this
partial ordering forms a distributive lattices.
(
) 3. If a function f is continuous from the right at point a then
(
) 4. Norm operations is one kind of binary aggregation operations that satisfy the properties
of monotonicity, commutativity, and associativity of t-conorms, but replace the
boundary conditions of t-conorms with weaker boundary condition h(0, 0) = 0 and h(1,
1) = 1.
(
) 5. Let A  [a1 , a2 ] , 0 = [0,0] and 1 = [1,1]. Arithmetic operations on closed intervals
.
satisfy the following useful properties: A  A  0 and A/ A  1.
(
) 6. Given two closed intervals  A = [a1, a2] and  B = [b1, b2], and define min(  A, B)
 [min( a1 , b1 ), min( a2 , b2 )] , then two fuzzy numbers A and B can be ordered as AB
iff min(  A,  B)   A .
(
) 7. A fuzzy relation is symmetric iff R(x,y) = R(y,x) for all x, y  X . When R(x,y) > 0 and
R(y,x) > 0 implies that x = y for all x, y  X , the relation R is called asymmetric.
(
) 8. A crisp relation R(X, X) is antireflexive if
(
) 9. A fuzzy relation R(X, X) is  -reflexive if R( x, x)   for all x  X , where 0    1 .
(
) 10. A fuzzy relation R(X, X) is max-min transitive if R( x, z)  max min[ R( x, y), R( y, z)]
x, x  R for some x  X .
yY
is satisfied for each pair
x, z  X .
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(
) 11. Let R be a crisp relation defined on X  X , where X is the set of all university
courses and R represents the relation “is a prerequisite of.” Then the relation R is
antireflexive, asymmetric, and transitive.
(
) 12. A fuzzy equivalence relation (similarity relation) is reflexive, symmetric, and
transitive.
(
) 13. A partial ordering on a set X that contains a greatest lower bound and a least upper
bound for every subset of two elements of X is called a linear ordering.
(
) 14. Quasi-ordering relations are symmetric and reflexive.
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二、 複選題(12%)
(
)
1. Which ones are the required properties of a fuzzy number A?
(A) A must be a normal fuzzy set.
(B) A must be a continuous function.
(C)

A must be a closed interval for every   (0,1] .
(D) The support of A must be bounded.
(
)
2. Which one does not belong to the averaging operations?
(A) Archimedean t-conorms
(B) norm operations
(C) ordered weighted averaging (OWA) operations (D) generalized means
(
) 3. Which one is incorrect?
(A). u max (a, b)  a  b  ab  min( 1, a  b)  max( a, b)
(B) imin (a, b)  max(0, a  b  1)  ab  min(a, b)
(C)
ab, a  b  ab, cs
is a dual triple, where cs indicates the standard complement.
(D) Let c indicates any fuzzy complement, u (c(a), c(b))  c(i(a, b)) .
(
) 4. Which ones are correct? Let
(A) MIN(A, B)(x) is
(B) MAX(A, B)(x) is
(C) MAX(A, B) = MAX(B, A)
(D) MIN[A, MAX(A,C)] = MAX[A, MIN(A,C)].
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(
) 5. Which one is not the axiomatic requirement of aggregation operations h?
(A) h(0, 0,..., 0)  0 and h(1,1,...,1)  1 .
(B) h is monotonic increasing in all its arguments.
(C) h is a continuous function.
(D) h is an idempotent function; that is h(a, a,..., a)  a for all a  [0,1] .
(
)
6. Which ones are the properties of partial ordering?
(A) There exists at most one first member and at most one last member.
(B) There may exist several maximal members and several minimal members.
(C) If a first member exists, then only one minimal member exists, and it is identical
with the first members.
(D) The first and last members of a partial ordering relation correspond to the last and
first members of the inverse partial ordering, respectively.
三、 簡答題
1. (4%) Consider a t-norm i (a, b)  ab and the Sugeno class of fuzzy complements
c (a) 
1 a
1  a
(   1)
By applying the equation u (a, b)  c (i(c (a), c (b))) , please show the class of t-conorms.
2.
(4%) For any   (0,1] , let  A  [  a1 ,  a2 ] ,  B  [  b1 ,  b2 ] , and  X  [  x1 ,  x2 ] denote,
respectively, the  -cuts of A, B, and X in our equation. Then the equation A+X = B has a
solution iff:
(1)
.
(2)
.
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3. (6%) Given a increasing generator as follows:
g w (a )  a w for any a  [0,1] (w > 0)
For any a, b  [0,1] , we have u(a, b)  g ( 1) ( g (a)  g (b)) .
Please show
(1) (2%) g w( 1) ( z ) , and
(2) (2%) its union function.
(3) (2%) Given two tables shown as follows:
Can you tell me which one contains larger value of w?
4.
(6%) Define generalized means as follows:
Please answer the following questions:
(1) (2%) The upper bound of h = ?
(2) (2%) If  = -1, then h =?
(3) (2%) If h is equal to the arithmetic mean, then  = ?
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5. (6%) Define an ordered weighted averaging (OWA) operations as follows:
Please answer the following questions:
(1) (2%) Let w =〈0.3,0.1,0.2,0.4〉, then hw(0.6,0.9,0.2,0.7) = ?
(2) (2%) Please show the upper bound of hw.
(3) (2%) If hw is equal to the arithmetic mean, w = ?
ANS:
6. (6%)Consider two triangular-shape fuzzy numbers A and B defined as follows:
 0

A( x)   ( x  1) / 2
(3  x) / 2

for x  1 and x  3
for  1  x  1
for 1  x  3 ,
0

B( x)   ( x  1) / 2
(5  x) / 2

Please answer the following questions:
(1)(3%) Please calculate  ( A  B) .
(2)(3%) Calculate the fuzzy number A  B .
5
for x  1 and x  5
for 1  x  3
for 3  x  5 ,
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7. (6%) Given two fuzzy number A and B,
for x  3 and x  5
0

A( x)   x  3 for 3  x  4
 5  x for 4  x  5 ,

If A • X = B, please calculate

0

B( x)  ( x  12) / 8
(32  x) / 12

for x  12 and x  32
for 12  x  20
for 20  x  32 ,
X and find X.
8. (6%) A fuzzy relation R(X,X) represented by the following membership matrix is a similarity
relation.
Please draw the partition tree for the similarity relation.
ANS:
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9. (6%) Let MIN and MAX be binary operations on R defined by (4.17) and (4.18), respectively.
Then for any A, B, C  R , the followring property hold:
MIN[MIN(A,B),C]=MIN[A, MIN(B,C)]
10. (4%) Given a fuzzy relation and its projections shown as Table 5.1. Please calculate its cylindric
closure cyl(R12, R13, R23).
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11. (6%) The transitive closure of a crisp relation R ( X , X ) is defined as the relaiton that is
transtive, contains R ( X , X ) , and has the fewest possible members. For fuzzy relations, this last
requirement is generalized such that the elements of the transitive closure have the smallest
possible membership grades that still allow the first two requirements to be met. Please
calculate the transitive max-min closure RT ( X , X ) for a fuzzy relation R ( X , X ) defined by
the membership matrix
0.7 0.5 0
0
0
0
R
 0 0.4 0

0 0.8
0
0
1
0

0
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