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<p style="line-height: 200%" align="center"><b><font size="5">关系的逆</font></b> </p>
<p style="line-height: 200%"> 给定任意两个集合
X 和 Y,如果 R 是个从 X 到 Y 的<a href="content-4-1-1.htm#content-4-1-1-guanxi">关系</a>,则从
Y 到 X 的关系<br>
R<sup>-1</sup>={<y,x>|<x,y><img src="image/shuyu.gif" width="13" height="11">R}
称为 R 的逆关系,或记为:~R . </p>
<p style="line-height: 200%"><b>例:</b>设A={2,3,4} B={2,3,4,5,6} <br>
R={<2,2>,<2,4>,<2,6>,<3,3>,<3,6>}<br>
则:R<sup>-1</sup>={<2,2>,<4,2>,<6,2>,<3,3>,<6,3>}</p>
<p style="line-height: 200%"><b>注1:</b> 关系矩阵</p>
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<p align="center" style="line-height: 200%"><img src="Image/mr.gif" width="22" height="15">=</td>
<td width="25%" style="border-left-style: solid" align="center">
<p style="line-height: 200%">1</p>
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<td width="25%" align="center">
<p style="line-height: 200%">0</p>
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1
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0
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<td width="25%" style="border-right-style: solid" align="center">
<p style="line-height: 200%">1</p>
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<td width="25%" style="border-left-style: solid" align="center">
<p style="line-height: 200%">0</p>
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<td width="25%" align="center">
<p style="line-height: 200%">1</p>
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<td width="25%" align="center">
0
</td>
<td width="25%" align="center">
0
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<td width="25%" style="border-right-style: solid" align="center">
<p style="line-height: 200%">1</p>
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<td width="25%" style="border-left-style: solid" align="center">
<p style="line-height: 200%">0</p>
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<td width="25%" align="center">
<p style="line-height: 200%">0</p>
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0
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0
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<td width="25%" style="border-right-style: solid" align="center">
<p style="line-height: 200%">0</p>
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<p align="center" style="line-height: 200%"><img src="Image/mr.gif" width="22" height="15"><sup>-1</sup>=</td>
<td width="21%" style="border-left-style: solid" align="center" height="29">
<p style="line-height: 200%">1</p>
</td>
<td width="22%" align="center" height="29">
<p style="line-height: 200%">0</p>
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<td width="22%" style="border-right-style: solid" align="center" height="29">
<p style="line-height: 200%">0</p>
</td>
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<td width="21%" style="border-left-style: solid" align="center" height="29">
0
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<td width="22%" align="center" height="29">
1
</td>
<td width="22%" style="border-right-style: solid" align="center" height="29">
0
</td>
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<tr>
<td width="21%" style="border-left-style: solid" align="center" height="29">
<p style="line-height: 200%">1</p>
</td>
<td width="22%" align="center" height="29">
<p style="line-height: 200%">0</p>
</td>
<td width="22%" style="border-right-style: solid" align="center" height="29">
<p style="line-height: 200%">0</p>
</td>
</tr>
<tr>
<td width="21%" style="border-left-style: solid" align="center" height="29">
0
</td>
<td width="22%" align="center" height="29">
0
</td>
<td width="22%" style="border-right-style: solid" align="center" height="29">
0
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<td width="21%" style="border-left-style: solid" align="center" height="29">
<p style="line-height: 200%">1</p>
</td>
<td width="22%" align="center" height="29">
<p style="line-height: 200%">1</p>
</td>
<td width="22%" style="border-right-style: solid" align="center" height="29">
<p style="line-height: 200%">0</p>
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<tr><td colspan="2"><p align="center" style="line-height: 200%"><br>
<img src="Image/mr.gif" width="22" height="15">-1=(<img src="Image/mr.gif" width="22" height="15">)<sup>T</sup>
(原矩阵的转置)<br>
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<p style="line-height: 200%"> <b>注2:</b>关系图中,每条弧线的箭头反响.</p>
<p style="line-height: 200%"> </p>
<p style="line-height: 200%"><b>关系的逆运算的性质:</b> </p>
<p style="line-height: 200%"> 设
R 是从集合
X 到 Y 的<a href="content-4-1-1.htm#content-4-1-1-guanxi">关系</a>，并且 S 是从集合
Y 到 Z 的关系。于是有： </p>
<blockquote>
<ol>
<li>
<p style="line-height: 200%" align="left">(R<img src="image/juhao.GIF" width="8" height="13">S)<sup>-1</sup>=S<sup>-1</sup><img src="image/juhao.GIF" width="8" height="13">R<sup>-1</sup></li>
<li>
<p style="line-height: 200%" align="left">(R<sup>-1</sup>)<sup>-1</sup>=R</li>
<li>
<p style="line-height: 200%" align="left">(R<img border="0" src="image/bing.gif" width="14" height="14">S)<sup>-1</sup>=R<sup>-1</sup><img border="0" src="image/bing.gif" width="14" height="14">S<sup>-1</sup></li>
<li>
<p style="line-height: 200%" align="left">(R<img border="0" src="image/jiao.gif" width="14" height="15">S)<sup>-1</sup>=R<sup>-1</sup><img border="0" src="image/jiao.gif" width="14" height="15">S<sup>-1</sup> </li>
<li>
<p style="line-height: 200%" align="left">(A<img border="0" src="image/cheng.GIF" width="13" height="14">B)<sup>-1</sup>=B<img border="0" src="image/cheng.GIF" width="13" height="14">A</li>
<li>
<p style="line-height: 200%" align="left"><img src="image/kong.GIF" width="11" height="11"><sup>-1</sup>=<img src="image/kong.GIF" width="11" height="11"> </li>
<li>
<p style="line-height: 200%" align="left">(A<img border="0" src="image/cheng.GIF" width="13" height="14">B-R)<sup>-1</sup>=B<img border="0" src="image/cheng.GIF" width="13" height="14">A-R<sup>-1</sup></li>
<li>
<p style="line-height: 200%" align="left">(R-S)<sup>-1</sup>=R<sup>-1</sup>-S<sup>-1</sup></li>
<li>
<p style="line-height: 200%" align="left">R=S<img border="0" src="image/dengjia.gif" width="17" height="9">R<sup>-1</sup>=S<sup>-1</sup></li>
<li>
<p style="line-height: 200%" align="left">R<img border="0" src="image/baohan.gif" width="11" height="10">S<img border="0" src="image/dengjia.gif" width="17" height="9">R<sup>-1</sup><img border="0" src="image/baohan.gif" width="11" height="10">S<sup>-1</sup></li>
</ol>
</blockquote>
<p style="line-height: 200%" align="left">证明(1)(略) </p>
<p style="line-height: 200%"><b>
定理:</b> 设 R 是集合
X 中的二元关系， R=R<sup>-1</sup>
当且仅当关系 R 是对称的。 </p>
<p style="line-height: 200%">证明:(略) </p>
<p style="line-height: 200%"> </p>
<p style="line-height: 200%"> </p>
<p style="line-height: 200%"> </p>
<p style="line-height: 200%"> </p>
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<p style="line-height: 200%"> <img src="IMAGE/yuancheng_2.gif" width="100" height="69"></div>
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<p style="line-height: 200%" align="center"> </p>
<p style="line-height: 150%"> </p><p align="right"><b><a href="contentFrame-mulu.htm"><<back</a></b>
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