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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