作者syuusyou (syuusyou)
看板NTU-Exam
标题[试题] 97上 陈健辉 离散数学 期末考
时间Wed Jan 14 23:07:03 2009
课程名称︰离散数学
课程性质︰系必修
课程教师︰陈健辉
开课学院:电资学院
开课系所︰资讯系
考试日期(年月日)︰2009/01/14
考试时限(分钟):120
是否需发放奖励金:是
(如未明确表示,则不予发放)
试题 :
(范围: Algebra)
1. A partial ordering on A = {2, 3, 4, 6, 8, 12, 36, 60} is defined as
follows: iRj if and only if i|j.
(a) Draw the Hasse diagram induced by R. (5%)
(b) Why is A not a lattice? (5%)
2. Let A = {1, 2, 3, 4, 5, 6, 7}. Define R on A as follows: xRy if and only
if 3|(x-y).
(a) Show that R is an equivalence relation on A. (5%)
(b) Find the equivalence classes induced by R. (5%)
3. The following is a proof for a.(a+b) = a for every a, b belong to K, where
(K,., +) is a Bollean algebra. Prove a + (a.b) = a for every a, b belong
to K. (10%)
a.(a+b) = (a.a)+(a.b) = a+(a.b) = (a.1)+(a.b) = a.(1+b) = a.1 = a
4. Prove that a unit in a ring (R,+,.) cannot be a proper devisor of zero.
(10%)
5. Suppose that (R,+,.) is a ring. Prove that if S belong to R is finite,
then (S,+,.) is a ring if and only if for a, b belong to S, a+b belongs to
S and a.b belongs to S. (10%)
6. Solve x ≡ 8 (mod 11), x ≡ 9 (mod 12), and x ≡ 10 (mod 13). (10%)
7. Let (R,+,.) be a ring whose all possible operations are shown below.
┌─┬────┐ ┌─┬────┐
│+ │z u a b │ │.│z u a b │
├─┼────┤ ├─┼────┤
│z │z u a b │ │z │z z z z │
│u │u z b a │ │u │z u a b │
│a │a b z u │ │a │z a b u │
│b │b a u z │ │b │z b u a │
└─┴────┘ └─┴────┘
(a) Is R a field? Explain your reason? (5%)
(b) R' = {u,z} is a subring. Is R' an ideal? Explain your reason. (5%)
8. Suppose that f: G → H is a group homomorphism and f is onto, where (G,.)
and (H,。) are two group. Prove that if G is abelian, then H is abelian.
(10%)
9. Prove that <|a|> = Z12 if and only if gcd(a,12) = 1, where 0 <= a <= 11 and
(Z12,+) is a group. (10%)
10. Explain why any group of prime order is cyclic. (10%)
--
※ 发信站: 批踢踢实业坊(ptt.cc)
◆ From: 140.112.248.143
※ 编辑: syuusyou 来自: 140.112.248.143 (01/14 23:07)