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）︰2009/12/16
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.Which of the following relations are equivalence relations?Which are partial
orderings?For each equivalence relation,find the equivalence classes,and for
each partial ordering,find the maximal elements and minimal elements. (10%)
R1={(1,1),(2,2),(3,3),(4,4)}
R2={(1,1),(2,2),(3,3),(4,4),(1,2),(2,1)}
R3={(1,2),(2,1),(3,4),(4,3)}
R4={(1,2),(1,3),(2,3),(3,4)}
R5={(1,1),(2,2),(2,3),(3,4),(2,4)}
2.Find <e>,<a>,<b>,<c> for the following group.Is it a cyclic group?Why? (5%)
┌─┬────┐
│˙│e a b c │
├─┼────│
│e │e a b c │
│a │a e c b │
│b │b c e a │
│c │c b a e │
└─┴────┘
3.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%)
4.Prove that if (G,˙)is a cyclic group,then (G,˙)is abelian. (5%)
5.Prove that if G is a group and |G|>1 is a prime number,then G is cyclic (5%)
6.Let A={1,2,3,4,5,6,7}.Determine an equivalence relation R on A with |R|∈
{8,9},or explain why no such R exists. (10%)
7.Suppose that (G,˙)is a group and H is a finite nonempty of G.Prove that if
H is closed under the binary operation of G,then(H,˙)is also a group (10%)
8.Suppose that a is a generator of a group G and |G|=n.Prove that a^3is a
generator of G if and only if 3 is not a divisor of n. (10%)
____ _ _
9.Suppose that(K,˙,+)is a Boolean algebra.Prove that(a)a˙b=a + b and
_____ _ _
(b)a + b= a˙b for every a,b∈K. (10%)
10.Suppose that(K,˙,+)is a Boolean algebra and a is an atom of K.Prove that
a˙b=0 or a˙b=a for every b∈K. (5%)
11.Solve x≡5(mod 7),x≡6(mod 10),and x≡9(mod 13). (10%)
12.Suppose that(R,+,˙)and(S,⊕,ⓧ)are two rings with zero elements ZR and Zs,
respectively.Given a ring homomorphism f:R->S,let K={a∈R|f(a)=Zs}.
Prive that (a)(K,+,˙)is a ring,and(b) K is an ideal of R. (10%)
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