课程名称︰分析导论优一 课程性质︰数学系大二必修 课程教师︰王振男 开课学院：理学院 开课系所︰数学系 考试日期（年月日）︰2014/09/30 考试时限（分钟）：40 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 1. (10%) A real number is called algebraic if it is a root of an algebraic n equation f(x) = 0, where f(x) = a + a x + ... + a x is a polynomial with 0 1 n integer coefficients. Prove that the set of all polynomials with integer coefficients is countable and deduce that the set of algebraic numbers is also countable. 2. (10%) (a) Find the fallacy in the following "proof" of the statement "The set of all intervals of positive length is countable". Let {x , x , ...} denote the countable set of rational numbers and let I be 1 2 any interval of positive length. Then I contains infinitely many rational points x , but among these there will be one with smallest index n. Define n a function F by means of the equation F(I) = n, if x is the rational n number with smallest index in the interval I. This function establishes a one-to-one correspondence between the set of all intervals and a subset of the positive integers. Hence the set of all intervals is countable. (b) Can you revise the statement so that the proof given above proves the statement you revise? -- 第01话 似乎在课堂上听过的样子 第02话 那真是太令人绝望了 第03话 已经没什麽好期望了 第04话 被当、21都是存在的 第05话 怎麽可能会all pass 第06话 这考卷绝对有问题啊 第07话 你能面对真正的分数吗 第08话 我，真是个笨蛋 第09话 这样成绩，教授绝不会让我过的 第10话 再也不依靠考古题 第11话 最後留下的补考 第12话 我最爱的学分 --

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