可好了，不会出现下不来的问题
<P 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><B>第一章部分作业解答</B><B><p></p></B></P>
<P 0cm 0cm 0pt; TEXT-INDENT: 24.1pt; mso-char-indent-count: 2.0; mso-char-indent-size: 12.05pt"><B>作业</B><B><FONT face="Times New Roman">1</FONT></B><B>：</B><FONT face="Times New Roman">P18 ex4<p></p></FONT></P>
<P 0cm 0cm 0pt; TEXT-INDENT: 24pt; mso-char-indent-count: 2.0; mso-char-indent-size: 12.0pt">设<v:shapetype><FONT face="Times New Roman"> <v:stroke joinstyle="miter"></v:stroke><v:formulas><v:f eqn="if lineDrawn pixelLineWidth 0"></v:f><v:f eqn="sum @0 1 0"></v:f><v:f eqn="sum 0 0 @1"></v:f><v:f eqn="prod @2 1 2"></v:f><v:f eqn="prod @3 21600 pixelWidth"></v:f><v:f eqn="prod @3 21600 pixelHeight"></v:f><v:f eqn="sum @0 0 1"></v:f><v:f eqn="prod @6 1 2"></v:f><v:f eqn="prod @7 21600 pixelWidth"></v:f><v:f eqn="sum @8 21600 0"></v:f><v:f eqn="prod @7 21600 pixelHeight"></v:f><v:f eqn="sum @10 21600 0"></v:f></v:formulas><v:path connecttype="rect" gradientshapeok="t" extrusionok="f"></v:path><lock aspectratio="t" v:ext="edit"></lock></FONT></v:shapetype><v:shape><v:imagedata></v:imagedata></v:shape>是从集合<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>到集合<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>的一个关系。证明下列条件等价：<p></p></P>
<P 0cm 0cm 0pt 43.5pt; TEXT-INDENT: -19.5pt; tab-stops: list 43.5pt; mso-list: l1 level1 lfo1"><FONT face="Times New Roman">(1) </FONT>对于任意<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><p></p></P>
<P 0cm 0cm 0pt 24pt; TEXT-ALIGN: center" align=center><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><p></p></P>
<P 0cm 0cm 0pt 43.5pt; TEXT-INDENT: -19.5pt; tab-stops: list 43.5pt; mso-list: l1 level1 lfo1"><FONT face="Times New Roman">(2) </FONT>对于任意<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<p></p></P>
<P 0cm 0cm 0pt 24pt"><FONT face="Times New Roman"> </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape>。</P>
<P 0cm 0cm 0pt; TEXT-INDENT: 24pt"><B>证明</B>：<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>令<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，由<FONT face="Times New Roman">(1)</FONT>有<p></p></P>
<P 0cm 0cm 0pt 24pt"><FONT face="Times New Roman"> </FONT><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape>。</P>
<P 0cm 0cm 0pt">于是<FONT face="Times New Roman">(2) </FONT>成立。<p></p></P>
<P 0cm 0cm 0pt; TEXT-INDENT: 24pt"><v:shape><v:imagedata></v:imagedata></v:shape><FONT face="Times New Roman"> </FONT>对于任意<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，由关系的性质（定理<FONT face="Times New Roman">1.3.2(2)</FONT>）知<p></p></P>
<P 0cm 0cm 0pt; TEXT-ALIGN: center" align=center><v:shape><v:imagedata></v:imagedata></v:shape>，<p></p></P>
<P 0cm 0cm 0pt">于是只要证<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。事实上，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，则<p></p></P>
<P 0cm 0cm 0pt"><v:shape><v:imagedata></v:imagedata></v:shape>，有<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，若<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，则由<FONT face="Times New Roman">(2)</FONT>知<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，这与<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>矛盾。所以<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。从而，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，故<p></p></P>
<P 0cm 0cm 0pt"><v:shape><v:imagedata></v:imagedata></v:shape>。于是，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。<p></p></P>
<P 0cm 0cm 0pt"><FONT face="Times New Roman"> <p></p></FONT></P>
<P 0cm 0cm 0pt; TEXT-INDENT: 24.1pt; mso-char-indent-count: 2.0; mso-char-indent-size: 12.05pt"><B>作业</B><B><FONT face="Times New Roman">2</FONT></B><B>：</B><FONT face="Times New Roman">P25 ex1<p></p></FONT></P>
<P 0cm 0cm 0pt; TEXT-INDENT: 24pt">设<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>和<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是两个集合，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，证明：<p></p></P>
<P 0cm 0cm 0pt 43.5pt; TEXT-INDENT: -19.5pt; tab-stops: list 43.5pt; mso-list: l0 level1 lfo2"><FONT face="Times New Roman">(1) </FONT>对于任意<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><p></p></P>
<P 0cm 0cm 0pt 43.5pt; TEXT-INDENT: -19.5pt; tab-stops: list 43.5pt; mso-list: l0 level1 lfo2"><FONT face="Times New Roman">(2) </FONT>对于任意<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><p></p></P>
<P 0cm 0cm 0pt 43.5pt; TEXT-INDENT: -19.5pt; tab-stops: list 43.5pt; mso-list: l0 level1 lfo2"><FONT face="Times New Roman">(3) </FONT><v:shape><v:imagedata></v:imagedata></v:shape>是一个满射当且仅当<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><p></p></P>
<P 0cm 0cm 0pt 43.5pt; TEXT-INDENT: -19.5pt; tab-stops: list 43.5pt; mso-list: l0 level1 lfo2"><FONT face="Times New Roman">(4) </FONT><v:shape><v:imagedata></v:imagedata></v:shape>是一个单射当且仅当<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape><p></p></P>
<P 0cm 0cm 0pt 24pt"><B>证明</B>：<FONT face="Times New Roman">(1)</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，则<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，于是<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，从而<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。<p></p></P>
<P 0cm 0cm 0pt 24pt"><FONT face="Times New Roman">(2)</FONT><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape><v:shape><v:imagedata><FONT face="Times New Roman"></FONT></v:imagedata></v:shape>，则<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，有<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，而<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，故<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<p></p></P>
<P 0cm 0cm 0pt">有<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，因<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是映射，从而<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，于是<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，从而<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。<p></p></P>
<P 0cm 0cm 0pt 24pt"><FONT face="Times New Roman">(3)</FONT>若<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是一个满射，则由<FONT face="Times New Roman">(2)</FONT>有<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，只要证<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。事实<p></p></P>
<P 0cm 0cm 0pt">上，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，则<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>有<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，于是<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，从而<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，故<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，所以<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。<p></p></P>
<P 0cm 0cm 0pt; TEXT-INDENT: 24pt">若<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，下证<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是一个满射。事实上，因<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是任意的，取<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>有<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，从而<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，有<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。故<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是一个满射。<p></p></P>
<P 0cm 0cm 0pt 24pt"><FONT face="Times New Roman">(4) </FONT>同理可证明。<p></p></P>
<P 0cm 0cm 0pt 24pt"><FONT face="Times New Roman"> <p></p></FONT></P>
<P 0cm 0cm 0pt 24pt"><B>作业</B><B><FONT face="Times New Roman">3</FONT></B><B>：</B><FONT face="Times New Roman">P39 ex1<p></p></FONT></P>
<P 0cm 0cm 0pt 24pt">证明：全体有理数构成的集合<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是一个可数集。<p></p></P>
<P 0cm 0cm 0pt 24pt"><B>证法一：</B><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>。令<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，则<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>可数，且<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，<p></p></P>
<P 0cm 0cm 0pt">从而<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>可数，同理<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>可数，于是全体有理数构成的集合<v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>是一个可数集。<p></p></P>
<P 0cm 0cm 0pt; TEXT-INDENT: 24pt"><B>证法二：</B><v:shape><FONT face="Times New Roman"> <v:imagedata></v:imagedata></FONT></v:shape>，作映射<p></p></P><v:shape><v:imagedata></v:imagedata></v:shape>，使得<v:shape> <v:imagedata></v:imagedata></v:shape>，则<v:shape> <v:imagedata></v:imagedata></v:shape>是满射，又从<v:shape> <v:imagedata></v:imagedata></v:shape>到<v:shape> <v:imagedata></v:imagedata></v:shape>之间存在一一映射，从而从<v:shape> <v:imagedata></v:imagedata></v:shape>到<v:shape> <v:imagedata></v:imagedata></v:shape>存在满射，由定理1.7.3知全体有理数构成的集合<v:shape> <v:imagedata></v:imagedata></v:shape>是一个可数集。
这是点拓答案，还有，要吗？到里面吧
<STRONG><FONT size=2>进来有奖</FONT></STRONG>
什么奖？
<P>可以告诉我怎么样才可以得到下载软件的金币吗？</P>
<P>我是 刚来!谢谢了</P>
我觉得应该再设一个推荐好友,就是通过在其他论坛等方式宣传本论坛,根据回馈的人数给予该ID一定的奖励,这样对论坛,对用户都有好处.
<P>一块可以下十个</P>
<P>还是很划算的！</P>
<P>这个转换系统有很多BUG
<p>恳请管理员及维护人员对起进行测试！
看起来倒是不错，可是我的都不能转换啊。