数学分析原理习题第二章

1. prove that the empty set is a subset of every set.

for a element x in a empty set, x is null , so x is a element of every other set since x is null. so the empty set is a sub set of every set .

2. prove that the set of all algebaric numbers is countable.

( a complex number z is said to be algebraic if there are integers, not all zero ,such that

$a_{0}z^{n}+a_{1}z^{n-1}+ ... +a_{n-1}z^{0}+a_{n}=0$ )

hint: for

$n + |a_{0}| + |a_{1}| + ... + |a_{n}| = N$ ,

for every positive N, this equation is finite.

we collect these equations:

$n + |a_{0}| + |a_{1}| + ... + |a_{n}| = N$

as set $C_{N}$ , since these equations are finite, so $C_{N}$ is countable , so $\cup C_{N}$ are countable. for every algebraic number ,we can form a equation in $C_{N}$ , so all algebraic numbers are countable.

3. prove that there exist real numbers which are not algebraic.

since R is not countable , if all real numbers are algebraic , then R is countable, a contradiction.

4. is the set of all irrational real numbers countable?

if irrational real numbers is countable , then R-Q is countable , then R = (R-Q) $\cup$ Q is countable ,a contradiction . thus all irrational real numbers is not countable.

5. construct a bouded set of real numbers with exactly three limit points.

6. let $E^{'}$ be the set of all limit points of a set E. prove that $E^{'}$ is closed.

for any point p that is in $E^{'c}$ , p is not a limit point of E,so there exists a neighborhood of p such that q is not in E , so p is a interior point of $E^{'c}$ ,so $E^{'c}$ is open, so $E^{'}$ is closed.

prove that $E$ and $\bar{E}$ have the same limit points.(hint : $\bar{E}$ = $E$ $\cup E^{'}$ )

if p is the limit point of $E$ , then p is the limit point of $\bar{E}$ ,since $\bar{E}$ = $E$ $\cup E^{'}$ .

if p is the limit point of $\bar{E}$ , then the neighborhood of p contains a point q that q $\in$ $\bar{E}$ , if q $\in$ $E$ , then we have p is a limit point of $E$ .

if q is not in $E$ ,

Do $E$ and $E^{'}$ always have the same limit points?

7.let $A_{1},A_{2},A_{3}$ ...be subsets of a metric space'

(a) if $B_{n}=\cup ^{n}_{i=1}A_{i}$ , prove that $\bar{B}_{n}=\cup ^{n}_{i=1}\bar{A}_{i}$ , for n=1,2,3,...

$E\subset F\Rightarrow (\bar{E}\subset \bar{F})$

so, $(\forall i\in I(E_{i}\subset F))$ $\Rightarrow (\cup _{i\in I}(\bar{E_{i}}\subset \bar{F}))$

in particular(in reserve )

8.is every point of every open set $E\subset R^{2}$ a limit point of $E$ ? answer the same question for cclosed sets in $R^{2}$ .

9.let $E^{o}$ denote the set of all interior points of a set $E$ .

(a) pove that $E^{o}$ is always open.

for $p\in E^{0}$ , there exists $N_{r}(p)\subset E$ ,

then for $q\in N_{o}(p)$ , there exists $N_{s}(q)\in N_{r}(p)$ where s=min{d(p,q), r-d(p,q)},

so $N_{s}(q)\in E$ , so q is a interior point of $E$ ,

so $N_{r}(p)\in E^{o}$ ,so $E^{o}$ is open.

(b) prove that $E$ is open if and only if $E^{o}=E$

this statement is pretty straightforward

p is an interior point of G since G is open.

$G\subset E$ , so $p\subset E=E^{o}$

so $G\subset E^{o}$

(d) prove that the complement of $E^{o}$ is the closure of the complement of $E$

(e) Do $E$ and $\bar{E}$ always have the same interiors

10.define

$d(p,q) =\begin{cases} 1 & (\text{if } p\ne q) \\ 0 & (\text{if } p = q) \end{cases}$

prove that this is a metric.

$\quad\quad\quad\quad d(x,y)= \begin{cases} 0\le d(x,z) + d(z,y) & (x = y)\\ 1\begin{cases}=d(x,z)+d(z,y) & (z\in\{x,y\})\\ < 2=d(x,z)+d(z,y) & (z\notin \{x,y\}) \end{cases} & (x\ne y) \end{cases}$

11. define

$\begin{align}d_1(x,y) & = (x-y)^2,\\ d_2(x,y) & = \sqrt{|x-y|},\\ d_3(x,y) & = |x^2 - y^2|,\\ d_4(x,y) & = |x - 2y|,\\ d_5(x,y) & = \frac{|x-y|}{1+|x-y|}.\end{align}$

determine ,for each of these , whether it is a metric or not.

this is pretty straightforward . use the definition of metric.

12. let $k\subset R^{1}$ consist of 0 and the numbers 1/n, for n=1,2,3,...prove that k is compact directly from the definition.

contstruct a finite subcover of k

13. construct a compact set of real numbers whose limit points form a countable set.

14. Give an example of an open cover of the segment (0,1),which has no finite subcover.

$\{(1/n,1)\}$

15.show the theorem 2.36 and ts corollary are wrong if the word "compact " is replace by "closed " or by "bounded "

for closed: $[n,\infty )$

for bounded : $(0,1/n)$

16. regard Q, the set of all rational numbers, as a metric space, with d(p,q）=|p-q|. let E be the set of all p $\in$ Q, such that $2<p^{2}<3$

show that E is closed an bounded in Q, but that E is not compact. is E open in Q.

E ={ $p\in Q:p\in S$ } where $S=(\sqrt{2},\sqrt{3})\cup (-\sqrt{3},-\sqrt{2})$ , so E is bounded in Q

since the set of rational numbers is dense, so Q is dense in R,every limit point of Q is in Q,so every limit point of E is in E ,so E is closed.

if E is compact, we construct a open couvering set { $G_{\alpha }$ } of E :

{ $G_{\alpha }$ }={ $N_{r}(p)|p\in E and (p-r,p+r)\subset S$ }

E is compact ,then there are finitely indices $\alpha _{1},\alpha _{2}...\alpha _{n}$ such that

$E\subset G_{\alpha 1}\cup G_{\alpha 2}\cup ...\cup G_{\alpha n}$

[p+r, $\sqrt{3}$ ] can not be covered since Q is dense ,a contradiction.

17. let E be the set of all $x\in [0,1]$ whose decimal expansion contains only the digits 4 and 7. is E countable ? is E dense in [0,1]? is E compact ? is E perfect ?

posted @ 2013-01-16 15:59 StanleyWu 阅读(1029) 评论(0) 编辑收藏举报

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数学分析原理习题第二章

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