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?