課程名稱︰分析導論上
課程性質︰必修
課程教師︰王藹農
開課學院：理學院
開課系所︰數學系
考試日期（年月日）︰2016/11/10
考試時限（分鐘）：110分鐘
試題 :
1. For every rational numberp, p^2<3, can you find another rational number q,
q>p, q^2<3? If yes, express q in terms of p. q=q(p)=?
2. Suppose a,b屬於R^2 find c屬於R^2 and r>0 so that 2|x-a|=3|x-b| if and only
if |x-c|=r
3.Can you find a one-to-one and onto map f from the set of natural numbers
N={1,2,3,...} to the set of all its subsets. If yes, f(n)=?
4. is sqrt(2)+sqrt(3)an algebraic number or a transcendental number?
5. Let (X,d(x,y)) be a metric space. Define g(x,y)=arctan(d(x,y)). Is(X,g(x,y))
a matric space too?
6.In the field C of complex numbers, define a+bi>c+di if a>c, or if a=c and b>d
Is (C,>) an ordered field?
7. If we do not assume a Dedekind cut to have no maximal element, then a
Dedekind cut may or may not have a maximal element. In thi ordered field
slightly larger than R, find the additive identity element
8.In the topological space R of real numbers, define an equivalence relarion
by x~y if x-y is an integer. Denote by R/~ the set of equivalence classes with
the quotient topology. Is R/~ a compact topological space?
9.Generalized Cantor Set C is obtained by first removing the middle 1/4 from
the interval [0,1] so the remaining set is [0,3/8]U[5/8,1]. Then remove
sub-intervals of length 1/4^2 from the middle of each of the two remaining
intervals.Continuing indefinitely with this removal.
C is the set of points that one never removed.Find its length and dimension
10.Find a function f(x) which is continuous at rational points and discontinuous
at irrational points. Find a function g(x) which is continuous at irrational
points and discontinuous at rational points.