模糊数学练习题(聊城大学期末考试).doc

模糊数学练习题（聊城大学期末考试）
1.What is a lattice? State the fundamental properties of the lattice.
2. State the Decomposition theorem III on a fuzzy set.
Let the universe, the fuzzy sets A and B as following
.
3.    Calculate the union, intersection, complements ,difference A\B, and level .
4.  State and prove the representation theorem III.
5.    A lattice  is complete if every non-empty subset of L has a supremum and an infimum. Prove the following conclusion that A bounded lattice  is complete  every nonempty subset of L has an infimum.
6.   Let be a fuzzy matrix. Calculate the transitive closure t(A) of A.
Exercise B
1.           What is a Boolean algebra? Give an example of the Boolean algebra.
2.           What is a fuzzy similarity relation? How to get the product of fuzzy relations?
3.           Let  be a Boolean algebra defined by Huntington. Prove  satisfies the idempotency and the absorption laws.
4.           State and prove the Decomposition theorem I on a fuzzy set.
5.           Prove the following proposition. Let  be an algebraic structure consisting of non-void set and two binary operations  and . If  and  satisfy the commutativity, the associativity, the idempotency and the absorption laws, then  is a lattice as a poset
6.           Let f be a X-Y mapping and a fuzzy subset of X, then

1.  What is a fuzzy set? Why to say the fuzzy set is the generalization of the classical set?
2.    Prove the conclusion that ([0,1], max, min, 1-) is completely distributive.
3.  Let  be a Boolean algebra defined by Huntington. Prove  satisfies the idempotency and the absorption laws.
4.     Given the universe , and the fuzzy set  is the following. Calculate the Hamming index of , the Euclidean index of .
5.     Let f be a X-Y mapping and a fuzzy subset of X, then .
6.   If a union operator U is distributive with respect to a intersection operator I and satisfies the condition , then I is idempotent
Exercise D
1.   What is a convex fuzzy set? Let the universe , the fuzzy set  is represented by .  Verify the fuzzy set  is a convex fuzzy set.
2.      Let the universe , and the fuzzy set  is the following .
(1) Calculate the probabilistic sum , and the product .
(2)   Calculate the Hamming index of , the Euclidean index of .
3.    What is the excluded middle law in the classical set theory? Let the universe , and the fuzzy sets

Verify the excluded middle law doesn’t satisfy in fuzzy set theory.
4.   If a union operator  satisfies the law of the excluded middle then  is not idempotent.
5.   Let A be a fuzzy matrix, then
(1)    The transitive closure of A, .
(2) The transitive closure of ,.
6.   Let  and  be a fuzzy relation on . Give the partition tree of X with respect to relation R.