[試題] 99下 陳健輝 離散數學 第二次期中考
課程名稱︰離散數學
課程性質︰選修
課程教師︰陳健輝
開課學院：電機資訊學院
開課系所︰資工系
考試日期（年月日）︰2011/5/24
考試時限（分鐘）：2小時
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
Examination #2
(範圍:Algebra)
1. Prove that if 3|n^2 then 3|n, where n is a positive integer, by the
methods of:
(a) p→q <=> ┐q→┐p; (5%)
(b) contradiction. (5%)
2. The following are some binary relations on A = {1, 2, 3, 4}.
R1 = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (1, 3), (1, 4), (2, 3),
(4, 2), (4, 3)}.
R2 = {(1, 1), (2, 2), (3, 3), (4, 4), (2, 3), (3, 2)}.
R3 = {(1, 2), (2, 1), (2, 3), (3, 2), (1, 4), (4, 1)}.
R4 = {(1, 1), (2, 2), (2, 3), (4, 4)}.
R5 = {(1, 2), (2, 1), (2, 3), (3, 3), (3, 4)}.
(a) Which are reflexive? (2%)
(b) Which are irreflexive? (2%)
(c) Which are symmetric? (2%)
(d) Which are antisymmetric? (2%)
(e) Which are transitive? (2%)
(f) Which are equivalence relations? (2%)
(g) For each of (f), find all equivalence classes. (2%)
(h) Which are partial orderings? (2%)
(i) For each of (h), how many possible topological orders on A are there?
(2%)
(j) Which are total orderings? (2%)
3. The following is a proof for a．0 = 0 in a Boolean algebra (K,．, +),
where a belongs to K.
_ _ _
a．0 = (a．0) + 0 = (a．0) + (a．a) = a．(0 + a) = a．a = 0.
Is it feasible to obtain a proof for a + 1 = 1 from the above by replacing
all occurrences of "+", "．", and "0" with "．", "+", and "1", respectively
? That is,
_ _ _
a + 1 = (a + 1)．1 = (a + 1)．(a + a) = a + (1．a) = a + a = 1.
Explain your answer. (10%)
4. For the commutative ring R = (Z, ⊕, ⊙), where Z is the set of integers
and a⊕b = a + b - 1, a⊙b = a + b - abfor any a, b belong to Z.
(a) find the identity for ⊕; (3%)
(b) find the inverse of 5 under ⊕; (3%)
(c) find the unity for ⊙l (3%)
(d) show that R is an integral domain; (5%)
(e) show that R is not a field; (5%)
(f) show that (Zodd, ⊕, ⊙) is a subring of R;
where Zodd is the set of odd integers; (6%)
(g) show that (Zodd, ⊕, ⊙) is an ideal; (5%)
5. Let C be the set of complex numbers and S be the set of real matrices of
the form [ a b] Define f: C→S be f(a+bi) = [ a b]
[-b a]. [-b a].
(a) Prove that f is a ring isomorphism from (C, +,．) to (S, ⊕, x),
where + and ． (⊕ and x) are ordinaryaddition and multiplication,
respectively, on complex numbers (matrices). (5%)
(b) How to compute (4+5i)．(2-3i) by using x? (5%)
6. Let G = <a> with o(a) = n. Prove that a^k, k belongs to Z+, generate G
if and only if gcd(k, n) = 1. (10%)
7. Prove that any group of prime order is cyclic. (10%)
8. (加分題)以下是有關本課程與課堂上發生的，何者為真？
(a) 老師的姓名是陳建輝 (2%)
(b) 老師習慣站著講課，但偶爾也會坐在椅子講課 (2%)
(c) 老師習慣帶一杯茶進教室，但也曾帶罐裝飲料進教室 (2%)
(d) 老師曾因趕課而延後20分鐘下課 (2%)
(e) 老師曾因事請助教代課 (2%)
(f) 上課曾遇有感地震 (2%)
(g) 上課曾遇投影設備故障 (2%)
(h) 老師曾提及張愛玲與金庸的小說 (2%)
(i) 老師曾提及曾國藩家訓 (2%)
(j) 老師曾提及妥瑞氏症 (2%)
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