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%) -- ◣▄ ▄◢□╳ .╴╴╴╴╴╴╴╴╴╴╴. ψLad╳□ ◤ ▼◥\▼ ◥╳ 夜々（やや） ◎機巧少女は傷つかない◎ ╳ ] ▊● ● [ ︱ (. ◙-*-. .=*￣*=- ︱ ▍ ◣ ﹀ ◢▋︱ [魔術迴路] 夜夜可愛いよ 夜夜～ `-.`-._上*`✽ ◙_╲ ︱ ▍ ◢╳◣ ▋╳ 夜夜可愛いよ ￣` 下☽)-╲╳╱)￣)‵╳ ▋╱▇▅▇╲▍□╳金剛力（こんごうりき） 世界一可愛いよ。 ︶ `︶︶ ︶╳□ --

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