[試題] 103上 王振男 分析導論優一 第一次小考

[xavier13540 (柊 四千)]

課程名稱︰分析導論優一
課程性質︰數學系大二必修
課程教師︰王振男
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2014/09/30
考試時限（分鐘）：40
是否需發放獎勵金：是
（如未明確表示，則不予發放）

試題 :


 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話 似乎在課堂上聽過的樣子          第02話 那真是太令人絕望了
第03話 已經沒什麼好期望了              第04話 被當、21都是存在的
第05話 怎麼可能會all pass              第06話 這考卷絕對有問題啊
第07話 你能面對真正的分數嗎            第08話 我，真是個笨蛋
第09話 這樣成績，教授絕不會讓我過的    第10話 再也不依靠考古題
第11話 最後留下的補考                  第12話 我最愛的學分

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