数据库复习2. Relational Algebra 关系代数

Relational Algebra 关系代数

Relational Algebra is the language to describe operations in relational DBMS helps understanding of Query execution,and particularly Query optimisation.

关系代数是描述关系DBMS中操作的语言，有助于理解查询执行，特别是查询优化。
It is basic set of relational operations to retrieve, filter and restructure relations.

它是一组用于检索、筛选和重组关系的基本关系操作。
Result of operations is always a new relation *.

原表经关系代数操作后的结果是一个新表

给定下列几张表作为操作实例用表：

STUDENT

FName	LName	ReaNum	BDate	Address	Gender	DName
Bart	Simpson	1111	1991-10-10	10 Evergreen Terrace, Springfield	M	Computer Science
Lisa	Simpson	2222	1993-06- 14	10 Evergreen Terrace, Springfield	F	Physics
Milhouse	Van Houten	3333	1991-09-07	Mill House, Springfield	M	History
Ralph	Wiggum	4444	1993- 12-22	Wiggum House, Springfield	M	History
Todd	Flanders	5555	1992-06-28	9 Evergreen Terrace, Springfield	M	Religious Studies
Rod	Flanders	6666	1994-02-27	9 Evergreen Terrace, Springfield	M	Religious Studies
A pu	Naha…	7777	1968-03-17	Kwik E Mart, Springfield	M	Computer Science
Monty	Burns	8888	1898-11-11	Nuclear Power Plant, Springfield	M	Physics

Department

DName	HoD	NoO/Emps
Physics	Homer Simpson	22
Computer Science	Edna Krabappel	34
Religious Studies	Rev. Lovejoy	12
History	Abe Simpson	18

DEPT_LOCATIONS

DName	DLocation
Physics	Science Building
Computer Sc ience	Science Building
Computer Sc ience	Computer Centre
Religious Studies	Main Building
History	Main Building

The SELECT operation 选择操作

Definition: Selects subset of tuples from relation on basis of a selection condition. 定义：根据选择条件从表中选出元组的子集的操作。
Denoted by σ（sigma) as the form σ<selection condition>(R): 用σ符号表示成σ<selection condition>(R)的形式：
- <Selection condition>: 选择条件：
  1. Selection condition is formed by arbitrarily connecting clauses using boolean operators AND,OR and NOT
    
    选择条件由使用布尔运算符“AND”、“OR”和“NOT”的任意连接子句构成。
  2. Selection condition is a boolean expression made up of a number of clauses of form:
    
    选择条件可以是一个布尔表达式，以下列形式表示：
```
<attribute name> <comparison op> <constant value>
or
<attribute name> <comparison op> <attribute name><comparison op> `is normally one of ｛ =，≠，＜，＞，≤ ，≥ ｝
```
- (R): R is just name of a database relation R是被操作表的表名

Note:

Result of SELECT operation is always a relation (as with all other relational algebra operations).

SELECT操作后的结果依然是一张表（这一点和其他关系操作符一样）
Relation resulting from SELECT has same attributes as R

SELECT操作后得到的新表与原表R有相同的表头
Number of tuples in resulting relation always less than or equal to number of tuples in R

新表的行数小于等于原表
SELECT operation is commutative 交换律

σ<cond1>(σ<cond2>(R)) = σ<cond2>(σ<cond1>(R) )

A sequence of SELECTs can be applied in any order, or can be converted to a single SELECT operation asσ＜condl> AND <cond2> (R)

多个SELECT语句可以由任意顺序应用，或者可以由布尔操作符连接成一个SELECT语句，例如：σ＜condl> AND <cond2> (R)
SELECT Examples: 下面来看SELECT语句的实例：

σ(LName ＝’Simpson’AND Gender = ’M’)OR(DName = ’Religious Studies’)(STUDENT)

FName	LName	ReaNum	BDate	Address	Gender	DName
Bart	Simpson	1111	1991-10-10	10 Evergreen Terrace, Springfield	M	Computer Science
Todd	Flanders	5555	1992-06-28	9 Evergreen Terrace, Springfield	M	Religious Studies
Rod	Flanders	6666	1994-02-27	9 Evergreen Terrace, Springfield	M	Religious Studies

The PROJECT operation. OPRATION操作

Definition: PROJECT selects some of columns on basis of an attribute list from attributes of a relation R

PROJECT操作按照属性列表选取R中的对应列组成新表
Denoted by π（ pi)， it takes the form π＜attribute list>(R)

用π（ pi)表示，形式为 π＜attribute list>(R)
- Where R is a relational algebra expression (whose result is a relation), its simplest case R is just name of database relation.R是关系代数表达式，它的最简形式是被操作表的表名。

Note:

Result of PROJECT operation is always a relation (as with all other relational algebra operations)

PROJECT操作后的结果依然是一张表（这一点和其他关系操作符一样）
Relation resulting from PROJ ECT has only attributes specified in<attribute list> and Attributes in same order as in list.

PROJECT操作后的新表只含有<attribute list>中的属性，且属性排列顺序和<attribute list>一致。
PROJECT operation is not commutative. PROJECT操作不具有交换律。
π＜listl> （ π＜list2>(R) ) ＝π＜listl>(R)，provided <list 2> contains all attributes in<list 1>otherwise left hand side is an invalid expression.提供的 <list 2> 要包含<list 1>中的所有属性，否则π＜listl>是无效表达式。
Number of tuples in resulting relation is always less than or equal to number of tuples in R

新表的行数小于等于原表
Remember: result is a relation so no duplicates.PROJECT的结果没有重复值。

Number of tuples in resulting relation is always less than or equal to number of tuples in R

新表的行数小于等于原表

The PROJECT operation举例：

The RENAME Operation 改名操作

将经过SELECT操作的STUDENT的结果表命名为DEPT_CS.
将经过PROJECT操作的DEPT_CS的结果表中的{Fname,Lname}属性命名为RESULT(FirstName, LastName).

UNION, INTERSECTION, DIFFERENCE Operatiion 并，交，补操作

Application Condition应用条件：

Two relations must have same type of tuples. 两张表必须行数相同。
Formally, two relationsR(A1,A2, ...,An ) and S( B1, B2, ...,Bn) are union compatible if They have same degree n and

dom( Ai) = dom( Bi). R和S可兼容前提是他们的列数相同且每个属性的定义域相同。

UNION 并
- Denoted by R∪S
- Results in relation that includes all tuples that are either in R, or in S, or in both R and S (Duplicate tuples are eliminated)

INTERSECTION 交
- Denoted by R∩S
- Results in a relation that includes all tuples that are in both R and S

DIFFERENCE 补
- Denoted byR - S
  
  一 Results in a relation that includes all tuples that are in R but not in S
UNION and INTERSECTION are commutative but DIFFERE NCE is not.并，交操作具有交换律但是补操作没有。

CARTESIAN PRODUCT Operation 笛卡尔积操作

Also a standard binary set operation. 二进制集合操作
Denoted by × 用×表示
Also known as CROSS PRODUCT or CROSS JOIN operation. 也被称为叉乘或交叉联合操作
Relations on which it is applied do not have to be union compatible. 叉乘的两张表不需要满足联合兼容（属性数同，属性定义域同）
Result of R( A1, A2,..., An )×S( B1, B2,,,,Bm) is relation Q with n + m attributes in that order i.e. Q( A1, A2, ..., An, S1, S2, ..., Sm),Q has one tuple for each combination of tuples: one from R and one from S

- Hence, if R has n tuples and S has m tuples then R×S will have n*m tuples.

笛卡尔积R( A1, A2,..., An )×S( B1, B2,,,,Bm)的结果表中有n+m列和n*m行

The JOIN Operation

Used to combine related tuples from two relations into single tuples. 被用来将两张表中相关的元组结合成一张表。
Denoted by ⋈ .用⋈表示。
Result of the JOIN is a relation Q with n + m attributes in that order. i.e. Q(A1,A2, ...,An, B1, B2, ..., Bm)

JOIN的结果表Q有n+m个属性且按照Q(A1,A2, ...,An, B1, B2, ..., Bm)的顺序排列。
However, unlike CARTESIAN PRODUCT resulting relation Q in a JOIN has one tuple for each combination of tuples whenever combination satisfies join condition.

JOIN的本质是先对A、B两张表做笛卡尔积，然后再按照 join condition选取合适的行。

举例：选取FEM_STUDS和DEPARTMENT表中StudDept和DName相等的元组并将新表命名为FEM_STUD_HODS.

先将FEM_STUDS和DEPARTMENT做笛卡尔积，得到：

再按照JOIN条件中StudDept=DName选出结果：

NATURAL JOIN

NATURAL JOIN was created to get rid of the additional redundant attribute in EQUIJOIN.

NATURAL JOIN在EQUIJOIN的基础上去掉了多余的属性。

Denoted by* 用*表示

AGGREGATE FUNCTION Operation 聚合操作

函数名称	描述
COUNT()	计数
SUM()	求和
AVG()	求平均值
MAX()	最大值
MIN()	平均值
…	…

举例：

OUTER JOIN Operation

Sometimes we might wa nt to keep all tuples in R or S or both in results of JOIN regardless of whether they satisfy join condition or not.

有时，我们可能希望在JOIN的结果中保持所有元组在R或S中，或两者都在，而不管它们是否满足连接条件。
There are 3 types of OUTER JOIN operations.
- LEFT OUTER JOIN 左外连接
- RIGHT OUTER JOIN 右外连接
- FULL OUTER JOIN 全连接

若右表没有对应的值，则将其取值为NULL

若左表没有对应的值，则将其取值为NULL

保留左右表所有的元组，无法对应取值为NULL

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