玩转“关系代数“

短命女 2022-08-08 03:21 279阅读 0赞

用实例来说明关系代数：

![Center][]   ![Center 1][]

# 一、 基本关系代数运算 #

## 1、并 ##

R∪S=\{t|t∈R∨t∈S\}

上述例子进行并（R∪S）运算：

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">f</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话形容：取所有。

## 2、差 ##

R-S=\{t|t∈R∧t∉S\}

上述例子进行差（R-S）运算:

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话：取不同，R-S，在R中不在S中。

## 3、广义笛卡儿积 ##

R×S=\{t|t=<tn,tm>∧tn∈R∧tm∈S\}

上述例子进行广义笛卡儿积（R×S）运算:

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">R.A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">R.B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">R.C</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">S.A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">S.B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">S.C</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话：一一对应组合在一起。

## 4、投影 ##

πA(R) =\{t\[A\]|t∈R\}

上述例子进行投影（πA,C(R)）运算:

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话：取出它要求的列。

## 5、选择 ##

σF(R)=\{t|t∈R∧F(t)=True\}

上述例子进行选择（σA>B(R)）运算:

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话：选择满足条件（A>B）的行

# 二、扩展的关系代数运算 #

## 1、交 ##

R∩S=\{t|t∈R∧t∈S\}=R-(R-S)=S-(S-R)

上述例子进行交（R∩S）运算：

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话：在R中也在S中。

## 2、 连接 ##

### (1) θ连接 ###

上述例子进行θ连接（连接的条件是R.A<S.B）:

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">R.A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">R.B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">R.C</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">S.A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">S.B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">S.C</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话：先进行笛卡儿积运算，再按给定的条件投影。

### (2)自然连接 ###

公式比较复杂，在此省略。

举例：

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
 </tbody> 
</table>

关系R

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">D</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">C</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">d</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
 </tbody> 
</table>

关系S

进行自然连接：

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">D</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">a</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">g</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话：选择相同列，比较相同元素组保留，去重，组合。

## 3、除 ##

举例：

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">B</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">D</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">h</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">b</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">l</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">c</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">e</span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
  </tr> 
 </tbody> 
</table>

关系R

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">C</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">D</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">d</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">e</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">f</span></p> </td> 
  </tr> 
 </tbody> 
</table>

关系S

进行除（R÷S）运算：

<table> 
 <tbody> 
  <tr> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">A</span></strong></span></p> </td> 
   <td style="background:#9BBB59"> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">B</span></strong></span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">a</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">b</span></p> </td> 
  </tr> 
  <tr> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px"><strong><span style="color:black; background:white">c</span></strong></span></p> </td> 
   <td> <p><span style="font-family:KaiTi_GB2312; font-size:24px; color:black; background:white">k</span></p> </td> 
  </tr> 
 </tbody> 
</table>

自己的话：选择相同的行（R中一行对应S多行），去掉重复的列，留下剩余的列。

我最喜欢的学习方式，就是每学到一个新的东西，看完它的描述之后，我都会用自己的话来为自己描述一遍，可能是几个词，可能是几句话！能够帮助我们理解这个新的东西！

[Center]: https://img-blog.csdn.net/20150429202257409?watermark/2/text/aHR0cDovL2Jsb2cuY3Nkbi5uZXQvc2hpcWlqaWFtZW5namll/font/5a6L5L2T/fontsize/400/fill/I0JBQkFCMA==/dissolve/70/gravity/Center
[Center 1]: /images/20220805/5102794ef3d5443b91bee6d1a95b728c.png

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