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<p style="line-height: 200%" align="center" ><b><font size="5">集合的运算</font></b></p>
<p style="line-height: 200%" >设 E为全集,<img border="0" src="image/kong.GIF" width="11" height="11">为空集,A,B为任意集合,且A<img src="image/baohan.gif" width="11" height="10">E,B<img src="image/baohan.gif" width="11" height="10">E</p>
<p style="line-height: 200%"><b>并</b> <br>
A<img border="0" src="image/bing.gif" width="14" height="14">B={x|x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="image/xiqu.gif" width="9" height="15">x<img border="0" src="Image/shuyu.gif" width="13" height="11">B}<br>
性质:A<img border="0" src="image/bing.gif" width="14" height="14">B=B<img border="0" src="image/bing.gif" width="14" height="14">A<br>
A<img border="0" src="image/bing.gif" width="14" height="14">A=A<br>
A<img border="0" src="image/bing.gif" width="14" height="14"><img border="0" src="image/kong.GIF" width="11" height="11">=A<br>
A<img border="0" src="image/bing.gif" width="14" height="14">E=E<br>
(A<img border="0" src="image/bing.gif" width="14" height="14">B)<img border="0" src="image/bing.gif" width="14" height="14">C=A<img border="0" src="image/bing.gif" width="14" height="14">(B<img border="0" src="image/bing.gif" width="14" height="14">C) </p>
<p style="line-height: 200%"><b>交</b> <br>
A<img src="image/jiao.gif" width="14" height="15">B={x|x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="Image/hequ.gif" width="9" height="11">x<img border="0" src="Image/shuyu.gif" width="13" height="11">B}<br>
性质:A<img src="image/jiao.gif" width="14" height="15">B=B<img src="image/jiao.gif" width="14" height="15">A<br>
A<img src="image/jiao.gif" width="14" height="15">A=A<br>
A<img src="image/jiao.gif" width="14" height="15"><img border="0" src="image/kong.GIF" width="11" height="11">=<img border="0" src="image/kong.GIF" width="11" height="11"><br>
A<img src="image/jiao.gif" width="14" height="15">E=A<br>
(A<img src="image/jiao.gif" width="14" height="15">B)<img src="image/jiao.gif" width="14" height="15">C=A<img src="image/jiao.gif" width="14" height="15">(B<img src="image/jiao.gif" width="14" height="15">C) <br>
A<img src="image/jiao.gif" width="14" height="15">(B<img border="0" src="image/bing.gif" width="14" height="14">C)=(A<img src="image/jiao.gif" width="14" height="15">B)<img border="0" src="image/bing.gif" width="14" height="14">(A<img src="image/jiao.gif" width="14" height="15">C)<br>
A<img border="0" src="image/bing.gif" width="14" height="14">(B<img src="image/jiao.gif" width="14" height="15">C)=(A<img border="0" src="image/bing.gif" width="14" height="14">B)<img src="image/jiao.gif" width="14" height="15">(A<img border="0" src="image/bing.gif" width="14" height="14">C) </p>
<p style="line-height: 200%"><b>相对补集</b> <br>
设 A 和 B 是任何两个集合，B 对
A 的相对补集 A-B，是由属于 A 的但不属于集合 B 的所有元素构成的集合。 </p>
<p style="line-height: 200%"> A-B={x|x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="Image/hequ.gif" width="9" height="11">x<img border="0" src="Image/bushuyu.gif" width="12" height="12">B} </p>
<p style="line-height: 200%"><b>补</b> <br>
给定全集 E，对于任何集合 A 来说，A
对 E 的相对补集，称为 A 的绝对补集，或简称为 A 的补集。并记作 ~A。 </p>
<p style="line-height: 200%"> 性质:A-<img border="0" src="image/kong.GIF" width="11" height="11">=A<br>
A-B<img src="image/baohan.gif" width="11" height="10">A<br>
A<img border="0" src="image/bing.gif" width="14" height="14">~A=E<br>
A<img src="image/jiao.gif" width="14" height="15">~A=<img border="0" src="image/kong.GIF" width="11" height="11"> <br>
~~A=A<br>
<b>定理:</b> 如果A=~B,则A<img border="0" src="image/bing.gif" width="14" height="14">B=E,A<img src="image/jiao.gif" width="14" height="15">B=<img border="0" src="image/kong.GIF" width="11" height="11"> </p>
<p style="line-height: 200%"> <b> 德·摩根定理</b> <br>
~(A<img border="0" src="image/bing.gif" width="14" height="14">B)=(~A<img src="image/jiao.gif" width="14" height="15">~B)<br>
~(A<img src="image/jiao.gif" width="14" height="15">B)=(~A<img border="0" src="image/bing.gif" width="14" height="14">~B) </p>
<p style="line-height: 200%"><b>环和(对称差)</b><img src="image/huanhe.gif" v:shapes="_x0000_i1026" width="13" height="11"> <br>
设 A 和 B 是任何两个集合，A 和
B 的环和是集合 A<img src="image/huanhe.gif" v:shapes="_x0000_i1026" width="13" height="11">B，集合
A<img src="image/huanhe.gif" v:shapes="_x0000_i1026" width="13" height="11">B 是由或属于 A 或属于 B，但不同时属于 A 和 B 的那些元素构成。对集合 A 和 B 的这种运算
<img src="image/huanhe.gif" v:shapes="_x0000_i1026" width="13" height="11">，称为对称差。 </p>
<p style="line-height: 200%"> A<img src="image/huanhe.gif" v:shapes="_x0000_i1026" width="13" height="11">B=(A-B)<img border="0" src="image/bing.gif" width="14" height="14">(B-A)<br>
={x|(x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="Image/hequ.gif" width="9" height="11">x<img border="0" src="Image/bushuyu.gif" width="12" height="12">B)<img border="0" src="image/xiqu.gif" width="9" height="15">(x<img border="0" src="Image/shuyu.gif" width="13" height="11">B<img border="0" src="Image/hequ.gif" width="9" height="11">x<img border="0" src="Image/bushuyu.gif" width="12" height="12">A} </p>
<p style="line-height: 200%" align="center"><img border="0" src="image/huanhetu.gif" width="166" height="117"> </p>
<p style="line-height: 200%"><b>环积</b><img src="image/huanji.gif" v:shapes="_x0000_i1025" width="11" height="11"> </p>
<p style="line-height: 200%"> A<img src="image/huanji.gif" v:shapes="_x0000_i1025" width="11" height="11">B=~(A<img src="image/huanhe.gif" v:shapes="_x0000_i1026" width="13" height="11">B)<br>
={x|(x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="Image/hequ.gif" width="9" height="11">x<img border="0" src="Image/shuyu.gif" width="13" height="11">B)<img border="0" src="image/xiqu.gif" width="9" height="15">(x<img border="0" src="Image/bushuyu.gif" width="12" height="12">B<img border="0" src="Image/hequ.gif" width="9" height="11">x<img border="0" src="Image/bushuyu.gif" width="12" height="12">A} </p>
<p style="line-height: 200%" align="center"><img border="0" src="image/huanjitu.gif" width="177" height="112"> </p>
<p style="line-height: 200%"><b>幂集</b></p>
<p style="line-height: 200%"> 给定集合A，由A的所有子集构成的集合称为A的幂集，记为
<img src="image/lu.gif" width="10" height="10">(A).</p>
<p style="line-height: 200%">如:A={a,b,c}</p>
<p style="line-height: 200%">
<img src="image/lu.gif" width="10" height="10">(A)={<img border="0" src="image/kong.GIF" width="11" height="11">,{a},{b},{c},{a,b},{a,c},{c,d},{a,b,c}}</p>
<p style="line-height: 200%"> 此时: {a,b}
<img border="0" src="Image/shuyu.gif" width="13" height="11">
<img src="image/lu.gif" width="10" height="10">(A)</p>
<p style="line-height: 200%"> 一般地:若|A|=n 则 |
<img src="image/lu.gif" width="10" height="10">(A)|=<img border="0" src="Image/2den.gif" width="13" height="12"></p>
<p style="line-height: 200%"><b>例</b>:证明 A<img src="image/baohan.gif" width="11" height="10">B<img border="0" src="Image/dengjia.gif" width="17" height="9"><img src="image/lu.gif" width="10" height="10">(A)<img src="image/baohan.gif" width="11" height="10"><img src="image/lu.gif" width="10" height="10">(B)</p>
<p style="line-height: 200%"><b>例</b>: 求<img src="image/lu.gif" width="10" height="10">{a,b,{a}}</p>
<p style="line-height: 200%"> 求<img src="image/lu.gif" width="10" height="10">(<img src="image/lu.gif" width="10" height="10">(<img src="image/lu.gif" width="10" height="10">(<img border="0" src="image/kong.GIF" width="11" height="11">)))</p>
<p style="line-height: 200%"> </p>
<p style="line-height: 200%"> </p>
<p style="line-height: 200%"> </p>
<p style="line-height: 200%"><b>方法:证明两个集合相等</b> </p>
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<li>
<p style="line-height: 200%">用集合相等的定义证明,即 A=B<img border="0" src="Image/dengjia.gif" width="17" height="9">(A<span style="mso-text-raise: -4.0pt; font-size: 10.5pt; mso-bidi-font-size: 12.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA" lang="EN-US"><img src="image/baohan.gif" v:shapes="_x0000_i1025" width="11" height="10"></span>B<img border="0" src="Image/hequ.gif" width="9" height="11">B<span style="mso-text-raise: -4.0pt; font-size: 10.5pt; mso-bidi-font-size: 12.0pt; font-family: Times New Roman; mso-fareast-font-family: 宋体; mso-font-kerning: 1.0pt; mso-ansi-language: EN-US; mso-fareast-language: ZH-CN; mso-bidi-language: AR-SA" lang="EN-US"><img src="image/baohan.gif" v:shapes="_x0000_i1025" width="11" height="10"></span>A) </p>
</li>
<li>
<p style="line-height: 200%">用已知的集合运算率证明 </p>
</li>
<li>
<p style="line-height: 200%">用文氏图验证</p>
</li>
</ul>
<p style="line-height: 200%"><b>例1:</b>证明 (A-B)-C=A-(B-C)</p>
<p style="line-height: 200%"><b>例2:</b>证明 (A-B)<img border="0" src="image/bing.gif" width="14" height="14">(B-A)=(A<img border="0" src="image/bing.gif" width="14" height="14">B)-(A<img src="image/jiao.gif" width="14" height="15">B)</p>
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