# ( )1. a Crisp Set Whose Members Can Be Labeled by the Positive Integers Is Called a Countable

( )1. A crisp set whose members can be labeled by the positive integers is called a countable set.Every countable set is finite.

( )2. A crisp set A in Rn is convex iff, for every pair of points r and s in A, all points located on the straight-line segment connecting r and s are also inA. For example, A= [0,2]U[3,5] is not convex.

( )3. The height of a fuzzy set A is the largest membership grade obtained by any element in that set..

( )4. The dual point of any membership grade is equal to its complemented value whenever the complement is continuous. If the complement is not continuous, either the dual point does not exist or it does not coincide with the complement point.

( )5. The cardinality of a crisp set A (|A|) is the number of members of a finite set A. For example, , where is the power set of A.

( )6. Let be a partition on a crisp set A, it can be define as follows: , where , is a partition on A iff for each pair ,, and .

( )7. Given two partitions and , is a refinement of iff each block of is included in some block of .

( )8.The -cut of a fuzzy set A is the crisp set that contains all the elements of the universal set X whose membership grades in A are greater than the specified value of .

( )9. Membership functions of convex fuzzy sets are concave functions

( )10.-cuts of a convex fuzzy set should be convex for all .

( )11. The stand fuzzy intersection is the strongest fuzzy intersection. The standard fuzzy intersection produces the smallest fuzzy set among the fuzzy sets produced by all possible fuzzy intersection.

( )12.Normality and convexity are cutworthy properties when we operate on fuzzy sets by the standard operations of intersection and complement.

( )13. Let A, B be fuzzy sets,

( )14.-cuts and strong -cuts are always monotonic decreasing with respect to .

( )15. The standard fuzzy intersection and fuzzy union are both cutworthy when applied to two fuzzy sets.

( )16.Given a t-norm defined as , if p converges to 0, then its value is converge to ab.

( )1. Let R denote a set of real number, which one isincorrect?

(A)If there is a real number r such that for every , then r is called an upper bound of R.

(B)If a real number r is called the supremum of Rthan r is an upper bound.

(C)If there is a real number s such that for every , then A is bounded below by s.

(D)If s is a lower bound of Rthen s is called the infimum of R.

( )2. Which one defines the domains of fuzzy set of type 2?

(A) .

(B) , where indicates a fuzzy power set of [0,1].

(C) , where indicates a fuzzy power set of X.

(D) , where indicates a power set of [0,1].

( )3. Let A, B be fuzzy sets defined on a finite universal set X, which formula can define the degree of subsethood of A in B?

(A) (B)

(C).(D)

( )4. Let A, B be fuzzy sets, which one is incorrect?

(A) .(B).

(C).(D) implies .

( )5. Which one isincorrect?For all

(A)(B)

(C)(D)

( )6. Which one is incorrect?

(A) If c is a fuzzy complement, then c has a unique equilibrium.

(B) The standard fuzzy intersection is the only idempotent t-norms.

(C) Archimedean t-norm is a continuous t-norm that satisfies subidempotency.

(D) There is at most one dual point for each particular fuzzy complement c and membership grade of value a.

( )7. Which axiom is not satisfied by the following fuzzy complement function?

(A) and .

(B).

(C) c is a continuous function.

(D) For all , if , then .

1. (4%) Given membership functions as follows:

Let and

Please draw the following sets.

Where

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1. (6%) Consider a membership function is defined as follows:

Please find and .

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1. (4%) Given the Sugeno class of fuzzy complement as follows:

Please show the equilibrium of the fuzzy complement function.

ANS:

1. (4%) Let be an arbitrary crisp function. Then for any and all , when f is fuzzified by the extension principle, please give an example show that .
1. (6%) Given a increasing generator as follows:

, where

For any , we have . Please show (1) and (2) its complement function.

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1. (3%) Given a decreasing generator as follows:

,

For any , please show .

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1. (4%) Given a decreasing generator as follows:

for any (w > 0)

For any , we have .

Please show its intersection function.

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1. (6%) Please finish the following graph.
1. (6%) Let

(1)(2%) Please draw the function g.

(2)(2%) Please show the pseudo-inverse of g.

(3)(2%) Given now , please find , where .

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1. (10%) Please show that the standard fuzzy intersection is the only idempotent t-norm.

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