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<p style="line-height: 150%" align="center"><b>笛卡儿乘积</b> </p>
<p style="line-height: 150%"><b><a name="content-3-3-2-dikaerchengji"></a>笛卡儿乘积</b></p>
<p style="line-height: 150%"> 给定任何两个集合 A 和 B 。如果序偶的第一个成员是
A 的一个元素，第二个成员是 B 的一个元素，则所有这样的序偶的集合，称为集合 A 和 B 的笛卡儿乘积，并记作 A<img src="image/cheng.GIF" width="13" height="14">B
，亦即：</p>
<p style="line-height: 150%" align="center">A<img src="image/cheng.GIF" width="13" height="14">B
={<x,y>|(x<img src="image/shuyu.gif" width="13" height="11">A)<img src="image/hequ.gif" width="9" height="11">(y<img src="image/shuyu.gif" width="13" height="11">B)}</p>
<p style="line-height: 150%" align="left"> 由定义可以看出，集合 A 和 B 的笛卡儿乘积仍是一个集合，这个集合的元素都是序偶，这些序偶的第一个成员取自集合
A，第二个成员取自集合 B。</p>
<p style="line-height: 150%" align="left">如: 设 A={a,b} B={c,d} c={2}</p>
<p style="line-height: 150%" align="left"> 则 A<img src="image/cheng.GIF" width="13" height="14">B={<a,c>,<a,d>,<b,c>,<b,d>}</p>
<p style="line-height: 150%" align="left"> (A<img src="image/cheng.GIF" width="13" height="14">B)<img src="image/cheng.GIF" width="13" height="14">C={<<a,c>,2>,<<a,d>,2>,<<b,c>,2>,<<b,d>,2>}<br>
={<a,c,2>,<a,d,2>,<b,c,2>,<b,d,2>}</p>
<p style="line-height: 150%" align="left"> A<img src="image/cheng.GIF" width="13" height="14">(B<img src="image/cheng.GIF" width="13" height="14">C)={<a,<c,2>>,<a,<d,2>>,<b,<c,2>>,<b,<d,2>>} </p>
<p style="line-height: 150%" align="left"> </p>
<p style="line-height: 150%" align="left"><b>运算律</b></p>
<blockquote>
<p style="line-height: 150%" align="left">1、因为:序偶是有序的,所以,笛卡尔积不满足交换律,结合律.</p>
<p style="line-height: 150%" align="left">2、A<img src="image/cheng.GIF" width="13" height="14">(B<img border="0" src="Image/bing.gif" width="14" height="14">C)=(A<img src="image/cheng.GIF" width="13" height="14">B)<img border="0" src="Image/bing.gif" width="14" height="14">(A<img src="image/cheng.GIF" width="13" height="14">C)</p>
<p style="line-height: 150%" align="left">3、A<img src="image/cheng.GIF" width="13" height="14">(B<img border="0" src="Image/jiao.gif" width="14" height="15">C)=( A<img src="image/cheng.GIF" width="13" height="14">B)<img border="0" src="Image/jiao.gif" width="14" height="15">(A<img src="image/cheng.GIF" width="13" height="14">C)</p>
<p style="line-height: 150%" align="left">4、(B<img border="0" src="Image/bing.gif" width="14" height="14">C)<img src="image/cheng.GIF" width="13" height="14"> A=(B<img src="image/cheng.GIF" width="13" height="14">A)<img border="0" src="Image/bing.gif" width="14" height="14">(C<img src="image/cheng.GIF" width="13" height="14">A)</p>
<p style="line-height: 150%" align="left">5、(B<img border="0" src="Image/jiao.gif" width="14" height="15">C)<img src="image/cheng.GIF" width="13" height="14">A=(B<img src="image/cheng.GIF" width="13" height="14">A)<img border="0" src="Image/jiao.gif" width="14" height="15">(C<img src="image/cheng.GIF" width="13" height="14">A)</p>
<p style="line-height: 150%" align="left"> </p>
</blockquote>
<p style="line-height: 150%" align="left"><b>证明</b>2：（定义法）</p>
<p style="line-height: 150%" align="left"> </p>
<p style="line-height: 150%" align="left"> </p>
<p style="line-height: 150%" align="left"><b>笛卡尔积的推广</b></p>
<p style="line-height: 150%" align="left"><b> </b>设<img border="0" src="image/a1.gif" width="12" height="15" >、<img border="0" src="image/a2.gif" width="14" height="15">、<img border="0" src="image/a3.gif" width="14" height="15">、…、<img border="0" src="image/an.gif" width="14" height="15">为集合，则</p>
<p style="line-height: 150%" align="center"><img border="0" src="image/a1.gif" width="12" height="15" ><img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/a2.gif" width="14" height="15"><img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/a3.gif" width="14" height="15"><img src="image/cheng.GIF" width="13" height="14">…<img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/an.gif" width="14" height="15">=(…((<img border="0" src="image/a1.gif" width="12" height="15" ><img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/a2.gif" width="14" height="15">)<img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/a3.gif" width="14" height="15">)<img src="image/cheng.GIF" width="13" height="14">…<img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/an.gif" width="14" height="15">)（左结合）</p>
<p style="line-height: 150%" align="left"> 若<img border="0" src="image/a1.gif" width="12" height="15" >=<img border="0" src="image/a2.gif" width="14" height="15">=<img border="0" src="image/a3.gif" width="14" height="15">=…=<img border="0" src="image/an.gif" width="14" height="15">=A
，则</p>
<p style="line-height: 150%" align="center"><img border="0" src="image/a1.gif" width="12" height="15" ><img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/a2.gif" width="14" height="15"><img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/a3.gif" width="14" height="15"><img src="image/cheng.GIF" width="13" height="14">…<img src="image/cheng.GIF" width="13" height="14"><img border="0" src="image/an.gif" width="14" height="15">
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