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<p style="line-height: 150%" align="center"><b><font size="5">集合间的关系</font></b></p>
<p style="line-height: 150%"><b>包含</b> </p>
<p style="line-height: 150%"> 设 A 和 B 是任意的集合。如果集合
A 的每一个元素，都是集合 B 的一个元素，则称 A 是 B 的子集。或者 A 被包含于 B 中，或说 B 包含 A。记作 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，或
B<img src="image/shuyu_you.gif" v:shapes="_x0000_i1025" width="11" height="10">
A。另外，如果集合
B 不包含集合 A，则把它表示成 A<img src="image/bubaohan.gif" width="10" height="10">B。 </p>
<p style="line-height: 150%"> 谓词表示法: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/dengjia.gif" width="17" height="9"><img border="0" src="Image/renyi.gif" width="9" height="11">x(x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="Image/yunhan.gif" width="15" height="9">x<img border="0" src="Image/shuyu.gif" width="13" height="11">B) </p>
<p style="line-height: 150%"><b>真包含</b> </p>
<p style="line-height: 150%"> 设 A 和 B 是两个任意集合。如果
A 是 B 的子集且 B 中至少有一个元素不属于 A ,则称 A 是 B 的真子集，并记作 A<img src="image/shuyu_zhenziji.gif" width="10" height="8" >B,也称为
B 真包含 A。 </p>
<p style="line-height: 150%"> 谓词表示法: A<img src="image/baohan.gif" width="10" height="8">B<img border="0" src="Image/dengjia.gif" width="17" height="9"><img border="0" src="Image/renyi.gif" width="9" height="11">x(x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="Image/yunhan.gif" width="15" height="9">x<img border="0" src="Image/shuyu.gif" width="13" height="11">B)<img border="0" src="Image/hequ.gif" width="9" height="11"><img border="0" src="Image/cunzai.gif" width="7" height="11">x(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: 150%"><b>相等</b></p>
<p style="line-height: 150%"> 给定两个集合
A 和 B，当且仅当 A 的每一个元素都是 B 的一个元素，B 的每一个元素也都是 A
的一个元素,即A<img border="0" src="image/baohan.gif" width="10" height="8">B且B<img border="0" src="image/shuyu_you.gif" width="11" height="10">A，A 和 B 才是相等的，并记作 A=B，否则，称集合
A 和 B 是不相等的，并记作 A!=B。</p>
<p style="line-height: 150%"> 用谓词方法表示: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)<br>
<img border="0" src="Image/dengjia.gif" width="17" height="9"><img border="0" src="Image/renyi.gif" width="9" height="11">x(x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="Image/yunhan.gif" width="15" height="9">x<img border="0" src="Image/shuyu.gif" width="13" height="11">B)<img border="0" src="Image/hequ.gif" width="9" height="11"><img border="0" src="Image/renyi.gif" width="9" height="11">x(x<img border="0" src="Image/shuyu.gif" width="13" height="11">B<img border="0" src="Image/yunhan.gif" width="15" height="9">x<img border="0" src="Image/shuyu.gif" width="13" height="11">A)<br>
<img border="0" src="Image/dengjia.gif" width="17" height="9"><img border="0" src="Image/renyi.gif" width="9" height="11">x(x<img border="0" src="Image/shuyu.gif" width="13" height="11">A<img border="0" src="Image/dengtong.gif" width="16" height="9">x<img border="0" src="Image/shuyu.gif" width="13" height="11">B)</p>
<p style="line-height: 150%"><b>注意:</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>、<img border="0" src="Image/shuyu.gif" width="13" height="11">的区别</p>
<p style="line-height: 150%"> 如：A={a，b，{a}，{b}}</p>
<p style="line-height: 150%"> {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>A
且{a}<img border="0" src="Image/shuyu.gif" width="13" height="11">A</p>
<p style="line-height: 150%"> </p>
<p style="line-height: 150%"><b>定理1：</b>设A为任一集合，有（1）<img border="0" src="image/kong.GIF" width="11" height="11"><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，（2）A<span lang="EN-US" style="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"><span style="mso-text-raise:-4.0pt"><!--[if gte vml 1]><v:shapetype
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</span>A，（3）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>E。</p>
<p style="line-height: 150%">定理2：设A、B、C为集合，如果AB，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>C，则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>C。</p>
<p style="line-height: 150%">证明：用谓词证明法（略）<span lang="EN-US" style="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"><span style="mso-text-raise:-4.0pt"><!--[if gte vml 1]><v:shapetype
id="_x0000_t75" coordsize="21600,21600" o:spt="75" o:preferrelative="t"
path="m@4@5l@4@11@9@11@9@5xe" filled="f" stroked="f">
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<v:f eqn="sum @0 1 0"/>
<v:f eqn="sum 0 0 @1"/>
<v:f eqn="prod @2 1 2"/>
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<v:f eqn="prod @3 21600 pixelHeight"/>
<v:f eqn="sum @0 0 1"/>
<v:f eqn="prod @6 1 2"/>
<v:f eqn="prod @7 21600 pixelWidth"/>
<v:f eqn="sum @8 21600 0"/>
<v:f eqn="prod @7 21600 pixelHeight"/>
<v:f eqn="sum @10 21600 0"/>
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