課程名稱︰離散數學
課程性質︰系必修
課程教師︰陳健輝
開課學院：電資學院
開課系所︰資訊系
考試日期（年月日）︰2009/01/14
考試時限（分鐘）：120
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
(範圍: 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%)
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◆ From: 140.112.248.143
※ 編輯: syuusyou 來自: 140.112.248.143 (01/14 23:07)