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模糊理論
學號：
姓名：
103/12
一、 是非題(15%)
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) 1. All aggregation operations between the standard fuzzy intersection and the standard
fuzzy union are not idempotent.
) 2. The triples (min, max, c) and (imin, umax, c) are dual with respect to any fuzzy
complement c.
) 3. Let < i, u, c> be a dual triple that satisfies the law of excluded middle ( A  A   ) and
the law of contradiction ( A  A  X ). Then < i, u, c> satisfies the distributive laws
( A  ( B  C )  ( A  B)  ( A  C ) ).
) 4. The aggregation operations which are idempotent are usually called averaging
operations, such as generalized means and ordered weighted averaging (OWA)
operations.
) 5. When a norm operation also has the property h(a, 0) = a, it becomes a t-norm. When it
has also the property h(a, 1) = a, it becomes a t-conorm.
) 6. Fuzzy numbers can be ordered partially in a natural way and that this partial ordering
forms a distributive lattices.
) 7. If a function f is continuous from the right at point a then
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) 8. Given a fuzzy relation R(X,X), its domain is a fuzzy set on X, dom R, whose
membership function is defined by dom R(X) = max R( x, y) for each x X .
yY
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) 9. Let R be the fuzzy relation defined on the set of cities and representing the concept
‘very near’, then the relation is reflexive, symmetric, and transitive.
) 10. 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 .
) 11. A fuzzy similarity relation is reflexive, symmetric and transitive.
) 12. A fuzzy relation R(X, X) is  -reflexive if R( x, x)   or every x  X , where
0   1.
) 13. Fuzzy equivalence is a cutworthy property of binary relations R(X, X) since it is
preserved in the classical sense in each  -cut of R.
) 14. A fuzzy relation R(X,X) is max-min transitive if R( x, z)  max min[ R( x, y), R( y, z)]
yY
is satisfied for each pair x, z  X .
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) 15. 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
antisymmetric.
二、 選擇題(4%)
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) 1. Which one is not the required property 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.
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) 2. Which one is not the property 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 minimal member exists, then the first member does not exist.
(D) The first and last members of a partial ordering relation correspond to the last and
first members of the inverse partial ordering, respectively.
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) 3. Let A = [a1,a2], B = [b1,b2], C = [c1,c2],0 = [0,0], and 1 = [1,1], which one of the
following properties is incorrect?
(A) A  0  A  A  0
(B) A  ( B  C )  A  B  A  C .
(C) 0  A  A .
(D) If A  E and B  F , then A  B  E  F .
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) 4. Which one is incorrect?
(A) MIN[MAX(A, C),C] = MIN(A, C).
(B) MIN[C, MAX(A, C)] = C.
(C) MIN[A, MAX(A,C)] = MAX[A, MIN(A,C)]. (D) MAX(A, A) = A.
三、 簡答題
1. (6%) Given a increasing generator as follows:
( p  0) .
g p (a)  1  (1  a) p
For any a, b  [0,1] , we have u(a, b)  g ( 1) ( g (a)  g (b)) .
Please show (1) g s( 1) ( z ) and (2) its union function.
ANS:
2. (3%) Consider a t-norm i (a, b)  ab and the Sugeno class of fuzzy complements
1 a
c (a) 
(   1)
1  a
By applying the equation u (a, b)  c (i(c (a), c (b))) , please show the class of t-conorms. (請
化簡)
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3. (6%) Given a increasing generator as follows:
For any a, b  [0,1] ,
we have i(a, b)  g ( 1) ( g (a)  g (b)  g (1)) and u(a, b)  g ( 1) ( g (a)  g (b)) .
Please show its intersection and union functions.
4. (6%) Define generalized means as follows:
Please answer the following questions:
(1) (2%) The upper bound of h = ?
(2) (2%) If h is equal to the harmonic mean, then  = ?
(3) (2%) If  = 1, then h =?
ANS:
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5. (6%) Define an ordered weighted averaging (OWA) operations as follows:
Please answer the following questions:
(1) (2%) Let w =〈0.4,0.2,0.2,0.1,0.1〉, then hw(0.5,0.1,0.9,0.2,0.7) = ?
(2) (2%) Please show the lower bound of hw, where w = ?
(3) (2%) If hw is equal to the arithmetic mean, w = ?
ANS:
6. (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. How do we know that the equation A+X
= B has a solution?
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

X and find X.
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for x  12 and x  32
for 12  x  20
for 20  x  32 ,
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8. (14%)Consider two triangular-shape fuzzy numbers A and B defined as follows:
Please answer the following questions:
(1)(4%) Please calculate  ( A  B) and  ( A  B) .
(2)(3%) Calculate the fuzzy number A  B .
(3)(3%) Please find the X if A  X  B .
(4)(4%) Let
, please write
the function A/ B .
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9. (6%) Given two fuzzy number A and B,
 0

A( x)  ( x  2) / 3
(4  x) / 3

for x  2 and x  4
for  2  x  1
for 1  x  4 ,
0

B( x)   x  1
3  x

for x  1 and x  3
for 1  x  2
for 2  x  3 ,
Please (1) show MAX(A, B)(x) and (2) draw the graph of MAX(A, B)(x).
ANS:
10. (6%) Please finish the following projection table and the cylindric closure cyl(R12, R3) table.
11. (4%) A crisp partial ordering Q on the set X  {a, b, c, d , e} is defined by its membership
matrix (crisp). The underlined entries in the matrix indicate the relationship of the immediate
predecessor and successor employed in the corresponding Hasse diagram.
(1) (2%) Please draw its corresponding Hasse diagram.
(2) (2%) Please show the first member and the last member of Q.
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12. (6%) The following membership matric defines a fuzzy partical ordering R on the set
X  {a, b, c, d , e} . Please answer the questions.
(1) (2%) Which element is undominated?
(2) (4%) For subset A  {a, b} , the upper bound is the fuzzy set produced by the intersection of
the domainting classes for a and b. What is the upper bound for the set A? What is the least
upper bound?
13. (9%) A fuzzy relation R(X,X) represented by the following membership matrix is a similarity
relation.
(1) (3%)Please show the simple diagram.
(2) (3%)Please draw the partition tree for the similarity relation.
(3) (3%)Suppose the membership matrix also represents a compatibility relation R(X,X), please
show all complete  -covers for the compatibility relation.
ANS:
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14. (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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