課程名稱︰離散數學
課程性質︰資工系選修
課程教師︰陳健輝
開課學院：電資
開課系所︰資工系
考試日期（年月日）︰2018/05/30
考試時限（分鐘）：2hr
試題 :
Examination #2 (範圍: Algebra)
1. For R = {s,t,x,y}, define + and *, making R into a ring, as follows.
+ s t x y * s t x y
s s t x y s s s s s
t t s y x t s t x y
x x y s t x s t x y
y y x t s y s ? s s
(a) Determine the entry marked by ?. (5%)
(b) Does R have zero divisors/ Find them if R does. (5%)
2. Let C be the set of complex numbers and S be the set of real matrices
of the following forms.
┌ ┐
│ a b │
│-b a │
└ ┘
(for every a+bi in C)
Then, (C,+,*) and (S,+,*) are two isomorphic rings, where + and * on
C(S) are ordinary addtion and multiplication for complex numbers
(matrices). Is it possible to obtain the sum and product of two complex
numbers without performing + and * on C? Why? (10%)
3. Suppose that G=<a> is a cyclic group and H(≠{e}) is a subgroup of G.
It is known that H is also cyclic. Find a generator of H. (10%)
4. Let A = {1,2,3}×{1,2,3,4,5}, and define R on A by (x1,y1)R(x2,y2) iff
x1+y1=x2+y2.
(a) Verify that R is an equivalence relation on A. (5%)
(b) Find the partition of A induced by R. (5%)
5. Let A be a set of n elements and R be an antisymmetric relation on A.
How many elements at most are there in R? Explain your answer. (10%)
6. Solve x≡8(mod 11), x≡9(mod 12), x≡10(mod 13), given the following
equalities (hints: you need to compute a1M1x1+a2M2x2+a3M3x3) (5%).
Also verify your answer. (5%)
[2]^(-1) = [6] in Z11, [11]^(-1) = [11] in Z12, [2]^(-1) = [7] in Z13,
11*12 = 132, 11*13 = 143, 12*13 = 156, 11*12*13 = 1716.
7. Prove that in Zn, [a] is a unit if and only if gcd(a,n)=1. (10%)
8. Prove that a unit in a ring R cannot be a zero divisor in R. (10%)
9. Suppose that G is a group and H is a subgroup of G. if |G|=2p and
1 < |H| < 2p, where p is a prime number, then H is cyclic. Why? (10%)
10. Prove by contradiction that 3^(1/2) is an irrational number. (10%)