Vinh university college of education mathematics department
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- Điều hướng trang này:
- -Threorem (*)
- 2. Exercise Ex 2.1.
- Ex 2.5.
| Problem 7. For every and . Then
Proof. Let F be the family of all subsets with at most pn elements of the set . If is the proportion of the number of subsets of F containing I then easily seen for every i. On the other hand, . By problem 6, we get (vietnamese) Giải Gọi F là họ tất cả các tập con có nhiều nhất pn phần tử của tập Nếu là tỷ số số tập con của F chứa i thì rõ ràng với mọi i. Do . Theo problem 6, ta có -Threorem (*) Let be a random variable gets value in . If G is family subsets of and for any belongs to least k elements of G then Problem 8. . Let S be a finite set and F be a family subsets of S. Suppose is a family subsets and for any element of S belongs to at least k elements of G. For every , sign Prove that, Proof. Suppose sử and sign Let be a random variable takes the value in S defined by in there, is a feature vector of A.By theorem (*), We have On gthe other hand, and . So The proof is complete. (Vietnamese) Giải. Giả sử và ký hiệu . Gọi là biến ngẫu nhiên nhận giá trị trong S được xác định bởi trong đó ký hiệu là vecto đặc trưng của tâp A ( vecto n chiều). Ta có Mặt khác, và . Từ đó Vậy, ta có điều cần chứng minh. 2. Exercise Ex 2.1. Let F be a family n-set , such that a, for every b, for every What is the maximum possible value of n? Ex 2.2. Let F be a family subsets of n-set X. Suppose for any two elements A, B of F such that . What is the maximun possible value of |F| ? Ex 2.3. Let be a family subsets of N sao Prove that Ex 2.4. Let F be a family subsets of m-set N. Such that For every then . For every such that Prove that, Ex 2.5. For every . Prove that, a family sets is at least points in Euclid space such that for every angle determined by the three points in this set is less than . Ex 2.6. Let be family consists of three partition of finite set N. Suppose for any we have Prove that . Since , show the partition such that the equality sign occurs. REFERNCES Ravi Boppana (2004), Unexpected Uses of Probability. Alon (1986), Explicit construction of esponential sized families of k-independent sets, Discrete Math 58, 191-193. Noga Alon, Joel Spencer (2000), Probabilitic Methods in Extremal Finite Set Theory, Wiley. F.R.K. Chung, P. Frankl, R.L. Graham and J.B. Shearer (1986), Some intersection theorems for ordered sets and graphs, J. Comninatorial Theory, Ser. A 43, 23-37. tải về 120.21 Kb. Chia sẻ với bạn bè của bạn:
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