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 AB, 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 B1. 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 poset6. 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 idempotentExercise D1. 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 setsVerify the excluded middle law doesnt 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.