Relational Data Model (Based on Chapter 7 in Fundamentals of Database Systems by Elmasri and Navathe, Ed. 4)

Relational Data Model

(Based on Chapter 7 in Fundamentals of Database Systems

by Elmasri and Navathe, Ed. 4)

The Relational Data Model 2

Outline1 Relational Model Concepts2 Characteristics of Relations3 Relational Integrity Constraints

3.1 Key Constraints3.2 Entity Integrity Constraints3.3 Referential Integrity Constraints

4 Update Operations on Relations5 Relational Algebra Operations

5.1 SELECT and PROJECT 5.2 Set Operations5.3 JOIN Operations

5.4 Additional Relational Operations


BASICS Data are stored in Relations(tables). A Relation is a mathematical concept

based on the ideas of sets. Relational Model has a strong theoret

ic foundation. The model was first proposed by Dr. E.

F. Codd of IBM in 1970.


INFORMAL DEFINITIONS RELATION:

A table of values A relation may be thought of as a set of rows. A relation may alternatively be though of as

a set of columns. Each row of the relation may be given an ide

ntifier( 一筆資料的代號 ). Each column is called by its column name or

column header or attribute name.



FORMAL DEFINITIONS A Relation may be defined in

multiple ways.

The Schema of a Relation:R (A1, A2, .....An)

Relation R is defined over attributes A1, A2, ..…,An

The Schema of a Relation:R (A1, A2, .....An)

Relation R is defined over attributes A1, A2, ..…,An


FORMAL DEFINITIONS (contd.) For Example

CUSTOMER (Cust-id, Cust-name, Address, Phone#)

Here, CUSTOMER is a relation defined over the four attributes Cust-id, Cust-name, Address, Phone#, each of which has a domain or a set of valid values. For example, the domain of Cust-id is 6 digit numbers.


FORMAL DEFINITIONS (contd.) A tuple is an ordered set of values Each value is derived from an appropr

iate domain(合乎型態 ). Each row in the CUSTOMER table may

be called as a tuple in the table and would consist of four values.


FORMAL DEFINITIONS (contd.) For example: <632895, "John Smith", "101 Main St.

Atlanta, GA 30332", "(404) 894-2000"> is a tuple belonging to the CUSTOMER relation.


FORMAL DEFINITIONS (contd.) A relation may be regarded as a set of

tuples (rows). Columns in a table are also called as a

ttributes of the relation.


FORMAL DEFINITIONS (contd.) The relation is formed over the Cartesian pr

oduct of the sets; each set has values from a domain; that domain is used in a specific role which i

s conveyed by the attribute name. For example, attribute Cust-name is define

d over the domain of strings of 25 characters. The role these strings play in the CUSTOMER relation is that of the name of customers.


FORMAL DEFINITIONS (contd.) Formally,

Given R(A1, A2, .........., An)r(R) is subset-of dom (A1) X dom (A2) X ....X dom(An)

R: schema of the relation r of R: a specific "value" of population of R.

Formally,Given R(A1, A2, .........., An)r(R) is subset-of dom (A1) X dom (A2) X ....X dom(An)

R: schema of the relation r of R: a specific "value" of population of R.

R is also called the intension of a R is also called the intension of a relationrelation r is also called the extension of a r is also called the extension of a relationrelation


FORMAL DEFINITIONS (contd.) For example:

Let S1 = {0,1}Let S2 = {a,b,c}Let R subset-of S1 X S2r(R) = {<0,a> , <0,b> , <1,c> }


DEFINITION SUMMARY

Informal Terms Table

ColumnRow

Value in a columnTable Definition

Populated Table( 內容 )

Formal Terms Relation

Attribute / DomainTuple

DomainSchema of a

relationExtension


Notes: Whereas languages like SQL use the

informal terms of TABLE (e.g. CREATE TABLE), COLUMN (e.g. SYSCOLUMN variable), the relational database textbooks present the model and operations on it using the formal terms(I.e., relation).


2 Characteristics of Relations Ordering of tuples in a relation

r(R): The tuples are not considered to be

ordered, even though they appear to be in the tabular form.


2 Characteristics of Relations Ordering of attributes in a relation sch

ema R (and of values within each tuple): We will consider the attributes in R(A1, A2,

..., An) and the values in t=<v1, v2, ..., vn> to be ordered .

(However, a more general alternative definition of relation does not require this ordering).


2 Characteristics of Relations Values in a tuple:

All values are considered atomic (indivisible)( 不可分的 ).

A special null value is used to represent values that are unknown or inapplicable to certain tuples.


Notation( 表示法 ): We refer to component values of a tup

le t by t[Ai] = vi (the value of attribute Ai for tuple t).

Similarly, t[Au, Av, ..., Aw] refers to the subtuple of t containing the values of attributes Au, Av, ..., Aw, respectively.


3 Relational Integrity Constraints

Constraints( 限制 ) are conditions that must hold on all valid relation instances.

There are three main types of constraints: Domain constraints( 欄位值合乎型態 ) Key constraints, ( 要有鍵值欄位 ) Entity integrity constraints, and ( 鍵值欄位不可

為空值 Null) Referential integrity constraints( 參考時必須參考

的到 )


3.0 Domain Constraints Domain constraints specify that the v

alue of each attribute A must be atomic value from domain dom(A).( 值必須與型態一致 )


3.1 Key Constraints Superkey of R:

A set of attributes SK of R such that no two tuples in any valid relation instance r(R) will have the same value for SK. That is, for any distinct tuples t1 and t2 in r(R), t1[SK] t2[SK].

Key of R: A "minimal" superkey; that is, a superkey K such

that removal of any attribute from K results in a set of attributes that is not a superkey.


3.1 Key Constraints

Example: The CAR relation schema:

CAR(State, Reg#, SerialNo, Make, Model, Year) has two keys

Key1 = {State, Reg#}, Key2 = {SerialNo},

which are also superkeys. {SerialNo, Make} is a superkey but not a key.


3.1 Key Constraints If a relation has several candidate

keys, one is chosen arbitrarily to be the primary key.

The primary key attributes are underlined.


Entity Integrity: t[PK] <> null for any tuple t in r(R)

The primary key attributes PK of each relation schema R in S cannot have null values in any tuple of r(R).

This is because primary key values are used to identify the individual tuples.


Note: Other attributes of R may be similarly

constrained to disallow null values, even though they are not members of the primary key.


3.3 Referential Integrity A constraint involving two relations (t

he previous constraints involve a single relation).

Used to specify a relationship among tuples in two relations: the referencing relation( 參考表格 ) and the referenced relation( 被參考表格 ).


Foreign key Tuples in the referencing relation R1 have a

ttributes FK (called foreign key attributes) that reference the primary key attributes PK of the referenced relation R2.( 欄位的值為另一個表格的鍵值欄位 )

A tuple t1 in R1 is said to reference a tuple t2 in R2 if t1[FK] = t2[PK].(t1 參考 t2)

A referential integrity constraint can be displayed in a relational database schema as a directed arc from R1.FK to R2.


Referential Integrity Referential integrity states that: For a

ny tuple t1 in R1 there exists a tuple t2 in R2 such that t1[FK] = t2[PK], otherwise, t1[FK] = NULL

簡言之，不能參考的到一個不存在的值




4 Update Operations on Relations

There are three basic update operations on relations: INSERT a tuple. DELETE a tuple. MODIFY a tuple.

Integrity constraints should not be violated by the update operations.


Several update operations may have to be grouped together. (transactions)

Updates may propagate to cause other updates automatically. This may be necessary to maintain integrity constraints.


In case of integrity violation, several actions can be taken: cancel the operation that causes the violation

(REJECT option) perform the operation but inform the user of

the violation trigger additional updates so the violation is

corrected (CASCADE option, SET NULL option) execute a user-specified error-correction

routine


5 The Relational Algebra( 關聯代數 ) 一組表示查詢的符號，用以表示查詢的

過程 Operations to manipulate relations. Used to specify retrieval requests (qu

eries). Query result is in the form of a relatio

n.


5 The Relational Algebra Relational Operations: 5.1 SELECT and PROJECT

operations. 5.2 Set operations:

These include UNION ∪, INTERSECTION ∩, DIFFERENCE － , CARTESIAN PRODUCT .

5.3 JOIN operations . 5.4 Other relational operations:

DIVISION, OUTER JOIN, AGGREGATE FUNCTIONS.


5.1 SELECT operation (denoted

by ): Selects the tuples (rows) from a

relation R that satisfy a certain selection condition c

Form of the operation: c(R) The condition c is an arbitrary

Boolean expression on the attributes of R


5.1 SELECT operation (denoted

by ): Resulting relation has the same

attributes as R Resulting relation includes each

tuple in r(R) whose attribute values satisfy the condition c


Boolean expression The Boolean expression specified in <selection con

dition> is made up of a number of clauses of the form:( 條件式的寫法 )

<attribute name> <comparison op> <constant value>, or <attribute name> <comparison op> <attribute name>

Where <attribute name> is the name of the attribute of R,

<comparison op> is normally one of the operators {=, <, , >, , }, and

<constant value> is a constant value from attribute domain.


Boolean expression Clauses can be arbitrarily

connected by the Boolean operators AND, OR, and NOT to form a general selection condition.


Examples: DNO=4(EMPLOYEE) SALARY>30000(EMPLOYEE) (DNO=4 AND SALARY>25000) OR (EMPLOYEE)

(DNO=5 AND SALARY>30000)


5.1PROJECT operation (denoted by

):

Keeps only certain attributes (columns) from a relation R specified in an attribute list L( 所要取出的欄位 )

Form of operation: L(R) Resulting relation has only those attri

butes of R specified in L NAME,LNAME,SALARY(EMPLOYEE)


Eliminates duplicate tuples Duplicate tuples are eliminated by the

operation. 重複資料刪除 The PROJECT operation eliminates du

plicate tuples in the resulting relation so that it remains a mathematical set (no duplicate elements)


Example: SEX,SALARY(EMPLOYEE) If several male employees have

salary 30000, only a single tuple <M, 30000> is kept in the resulting relation.




Sequences of operations: Several operations can be combined t

o form a relational algebra expression (query)

將運算組合成為查詢式子


Example: Retrieve the names and salaries of em

ployees who work in department 4: FNAME,LNAME,SALARY (DNO=4 (EMPLOYEE) ) Relational algebra expression


Sequences of operations Alternatively, we specify explicit inter

mediate relations for each step: DEPT4_EMPS DNO=4(EMPLOYEE) R FNAME,LNAME,SALARY(DEPT4_EMPS) 以多步驟表示查詢


Sequences of operations Attributes can optionally be renamed

in the resulting left-hand-side relation :( 重新命名欄位 )

(this may be required for some operations that will be presented later) DEPT4_EMPS DNO=4(EMPLOYEE) R(FIRSTNAME,LASTNAME,SALARY)

FNAME,LNAME,SALARY (DEPT4_EMPS)



5.2 Set Operations( 集合運算 ) Binary operations from mathematical

set theory:( 將 tuple 看成集合元素 ) UNION: R1 R2, INTERSECTION: R1 R2, SET DIFFERENCE: R1 R2, CARTESIAN PRODUCT: R1 R2.


Union compatibility For , , , the operand relations R1(A1, A2,

..., An) and R2(B1, B2, ..., Bn) must have the same number of attributes, and the domains of corresponding attributes must b

e compatible; that is, dom(Ai)=dom(Bi) for i=1, 2, ..., n.

This condition is called union compatibility. 類似的資料才能作集合運算


Union compatibility The resulting relation for , , or

has the same attribute names as the first operand relation R1 (by convention).



CARTESIAN PRODUCT( 卡式積 ) CARTESIAN PRODUCT

R(A1, A2, ..., Am, B1, B2, ..., Bn) R1(A1, A2, ..., Am) X R2 (B1, B2, ..., Bn)

A tuple t exists in R for each combination of tuples t1 from R1 and t2 from R2 such that: t[A1, A2, ..., Am]=t1 and t[B1, B2, ..., Bn]=t2

If R1 has n1 tuples and R2 has n2 tuples, then R will have n1*n2 tuples.

瞭解關連 join 動作的媒介


CARTESIAN PRODUCT CARTESIAN PRODUCT is a meaningles

s operation ( 沒有語意 ) on its own. It can combine related tuples from tw

o relations if followed by the appropriate SELECT operation .( 這就是 join 動作 )


CARTESIAN PRODUCT Example:

Combine each DEPARTMENT tuple with the EMPLOYEE tuple of the manager.

DEP_EMP DEPARTMENT EMPLOYEE

DEPT_MANAGER MGRSSN=SSN(DEP_EMP)



5.3 JOIN Operations THETA Join( 條件是中不限等號 ) Equijoin( 條件式只能用等號 ) Outer Join( 不匹配的資料也選入 )


THETA JOIN: Similar to a CARTESIAN PRODUCT foll

owed by a SELECT. The condition c is called a join conditi

on. R(A1, A2, ..., Am, B1, B2, ..., Bn)

R1(A1, A2, ..., Am) c R2 (B1, B2, ..., Bn)


JOIN condition A join condition is of the form:

<condition> AND <condition> AND … AND <condition>

Where each condition is of the form Ai Bj , Ai is an attribute of R, Bj is an attribute of S, Ai and Bj have the same domain, and is one of the comparison operators {=, <, , >,

, }


EQUIJOIN: The join condition c includes one or m

ore equality comparisons involving attributes from R1 and R2.

That is, c is of the form:(Ai=Bj) AND ... AND (Ah=Bk); 1<i,h<m, 1<j,k<n


EQUIJOIN: In the above EQUIJOIN operation:

Ai, ..., Ah are called the join attributes of R1

Bj, ..., Bk are called the join attributes of R2


Example of using EQUIJOIN: Retrieve each DEPARTMENT's name a

nd its manager's name: T DEPARTMENT MGRSSN=SSN EMPLOYEE RESULT DNAME,FNAME,LNAME(T)


NATURAL JOIN (*): In an EQUIJOIN R R1 c R2, the join attri

bute of R2 appear redundantly in the result relation R.

In a NATURAL JOIN, the redundant join attributes of R2 are eliminated from R.

The equality condition is implied and need not be specified.

R R1 *(join attributes of R1),(join attributes of R2) R2


Example: Retrieve each EMPLOYEE's name and

the name of the DEPARTMENT he/she works for: T EMPLOYEE *(DNO),(DNUMBER) DEPARTMENT RESULT FNAME,LNAME,DNAME(T)


NATURAL JOIN (*): If the join attributes have the same na

mes in both relations, they need not be specified and we can write R R1 * R2.



the name of his/her SUPERVISOR: SUPERVISOR(SUPERSSN,SFN,SLN)

SSN,FNAME,LNAME(EMPLOYEE) T EMPLOYEE * SUPERVISOR RESULT FNAME,LNAME,SFN,SLN(T)


Note: In the original definition of NATURAL

JOIN, the join attributes were required to have the same names in both relations.


Note: There can be a more than one set of jo

in attributes with a different meaning between the same two relations.


For example:JOIN ATTRIBUTES

EMPLOYEE.SSN= DEPARTMENT.MGRSSN

EMPLOYEE.DNO= DEPARTMENT.DNUMBER

RELATIONSHIP

EMPLOYEE manages the

DEPARTMENT

EMPLOYEE works forthe

DEPARTMENT



the name of the DEPARTMENT he/she works for: T EMPLOYEE DNO=DNUMBER DEPARTMENT RESULT FNAME,LNAME,DNAME(T)


Recursive closure operation

JOIN ATTRIBUTES

EMPLOYEE(1).SUPERSSN = EMPLOYEE(2).SSN

RELATIONSHIP

EMPLOYEE(2) supervises EMPLOYEE(1)

A relation can have a set of join attributes to join it with itself :


Recursive closure operation

One can think of this as joining two distinct copies of the relation, although only one relation actually exists

In this case, renaming can be useful


Example: Retrieve each EMPLOYEE's name and th

e name of his/her SUPERVISOR: SUPERVISOR(SSSN,SFN,SLN)

SSN,FNAME,LNAME(EMPLOYEE) T EMPLOYEE SUPERSSN=SSSN SUPERVISOR RESULT FNAME,LNAME,SFN,SLN(T)


Complete Set of Relational Algebra Operations: 只要這些動作就夠了 All the operations discussed so far can be d

escribed as a sequence of only the operations SELECT, PROJECT, UNION, SET DIFFERENCE, and CARTESIAN PRODUCT.

Hence, the set { , , , , } is called a complete set of relational algebra operations.

Any query language equivalent to these operations is called relationally complete.


Additional Operations For database applications, additional

operations are needed that were not part of the original relational algebra.

These include: Aggregate functions and grouping.( 聚集

函數，資料倉儲中的重要運算 ) OUTER JOIN and OUTER UNION.


Division R(Z) ÷ S(X):

XZ Y= Z – X Result is in the form of T(Y), includes a tu

ple t if tuple tR appear in R with tR[Y]=t, and with tR[X] = tS for every tuple tS in S



Division Operation Example:

“Retrieve the names of employees who work on all the projects that ‘John Smith’ work on.”

SMITH FNAME=‘John’ AND LNAME=‘Smith’(EMPLOYEE) SMITH_PNOS PNO(WORK_ON * ESSN=SSN SMITH) SSN_PNOS PNO,ESSN(WORK_ON) SSNS(SSN) SSN_PNOS ÷ SMITH_PNOS RESULT FNAME,LNAME(SSNS * EMPLOYEE)


5.4 Additional Relational Operations

Aggregate functions and grouping. OUTER JOIN and OUTER UNION


AGGREGATE FUNCTIONS Functions such as SUM, COUNT, AVER

AGE, MIN, MAX are often applied to sets of values or sets of tuples in database applications<grouping attributes> F<function list> (R)

The grouping attributes are optional


AGGREGATE FUNCTIONS Example 1:

Retrieve the average salary of all employees (no grouping needed):

R(AVGSAL) F AVERAGE SALARY (EMPLOYEE)


AGGREGATE FUNCTIONS Example 2:

For each department, retrieve the department number, the number of employees, and the average salary (in the department):

R(DNO,NUMEMPS,AVGSAL)

DNO F COUNT SSN, AVERAGE SALARY (EMPLOYEE)

DNO is called the grouping attribute in the above example



OUTER JOIN In a regular EQUIJOIN or NATURAL JOIN op

eration, tuples in R1 or R2 that do not have matching tuples in the other relation do not appear in the result

Some queries require all tuples in R1 (or R2 or both) to appear in the result

When no matching tuples are found, nulls are placed for the missing attributes


OUTER JOIN LEFT OUTER JOIN:

R1 R2 lets every tuple in R1 appear in the result

RIGHT OUTER JOIN: R1 R2 lets every tuple in R2 appear in the res

ult FULL OUTER JOIN:

R1 R2 lets every tuple in R1 or R2 appear in the result



Example of Queries (1) Query 1

Retrieve the name and address of all employees who work for the ‘Research’ department.


Answers of queries (1) RESEARCH_DEPT DNAME=‘Research’(DEPARTMENT) RESEARCH_DEPT_EMPS (RESEARCH_DEPT

DNUMBER=DNO EMPLOYEE) RESULT FNAME,LNAME,ADDRESS(RESEARCH_DEPT_EMP

S)

RESEARCH_DEPT DNAME=‘Research’(DEPARTMENT) RESEARCH_DEPT_EMPS (RESEARCH_DEPT

DNUMBER=DNO EMPLOYEE) RESULT FNAME,LNAME,ADDRESS(RESEARCH_DEPT_EMP

S)



For every project located in ‘Stafford’, list the project number, the controlling department number, and the department manage’s last name, address, and birthdate.


Answers of queries (2) STAFFFORD_PROJS

PLOCATION=‘STAFFORD’ (PROJECT) CONTR_DEPT (STAFFORD_PROJS

DNUM=DNUMBER DEPARTMENT) PROJ_DEPT_MGR

(CONTR_DEPT MGRSSN=SSN EMPLOYEE) RESULT PNUMBER,DNUM,LNAME,ADDRESS,BDATE(PROJ_DEPT_

MGR)

STAFFFORD_PROJS PLOCATION=‘STAFFORD’ (PROJECT)

CONTR_DEPT (STAFFORD_PROJS DNUM=DNUMBER DEPARTMENT)

PROJ_DEPT_MGR (CONTR_DEPT MGRSSN=SSN EMPLOYEE)

RESULT PNUMBER,DNUM,LNAME,ADDRESS,BDATE(PROJ_DEPT_MGR)



Find the names of employees who work on all the projects controlled by department number 5.


Answers of queries (3) DEPT5_PROJS(PNO) PNUMBER(DNUM=5(PROJECT)) EMP_PROJ(SSN, PNO) ESSN,PNO(WORKS_ON) RESULT_EMP_SSNS EMP_PROJ DEPT5_PROJS RESULT LNAME,FNAME(RESULT_EMP_SSNS*EMPLOY

EE)

DEPT5_PROJS(PNO) PNUMBER(DNUM=5(PROJECT)) EMP_PROJ(SSN, PNO) ESSN,PNO(WORKS_ON) RESULT_EMP_SSNS EMP_PROJ DEPT5_PROJS RESULT LNAME,FNAME(RESULT_EMP_SSNS*EMPLOY

EE)



Make a list of project numbers for projects that involve an employee whose last name is ‘Smith’, either as a worker or a manager of the department that controls the project.


Answers of queries (4) SMITHS(ESSN) PUNMBER(LNAME=‘Smith’(EMPLOYEE)) SMITH_WORKER_PROJS PNO(WORKS_ON*SMITHS) MGRS LNAME,DNUMBER(EMPLOYEE SSN=MGRSSNDEPARTMENT) SMITH_MGRS LNAME=‘Smith’(MGRS) SMITH_MANAGED_DEPTS(DNUM) DNUMBER(SMITH_MG

RS) SMITH_MGR_PROJS(PNO) PNUMBER(SMITH_MANAGED_

DEPTS*PROJECT) RESULT (SMITH_WORKER_PROJSSMITH_MGR_PRO

JS)

SMITHS(ESSN) PUNMBER(LNAME=‘Smith’(EMPLOYEE)) SMITH_WORKER_PROJS PNO(WORKS_ON*SMITHS) MGRS LNAME,DNUMBER(EMPLOYEE SSN=MGRSSNDEPARTMENT) SMITH_MGRS LNAME=‘Smith’(MGRS) SMITH_MANAGED_DEPTS(DNUM) DNUMBER(SMITH_MG

RS) SMITH_MGR_PROJS(PNO) PNUMBER(SMITH_MANAGED_

DEPTS*PROJECT) RESULT (SMITH_WORKER_PROJSSMITH_MGR_PRO

JS)



List the names of all employees with two or more dependents.


Answers of queries (5)

T1(SSN, NO_OF_DEPS) ESSNFCOUNT DEPENDENT_NAME(DEPENDENT)

T2 NO_OF_DEPS2(T1) RESULT LNAME,FNAME(T2*EMPLOYEE)

T1(SSN, NO_OF_DEPS) ESSNFCOUNT DEPENDENT_NAME(DEPENDENT)

T2 NO_OF_DEPS2(T1) RESULT LNAME,FNAME(T2*EMPLOYEE)



Retrieve the names of employees who have no dependents.


Answers of queries (6) ALL_EMPS SSN(EMPLOYEE) EMPS_WITH_DEPS(SSN) ESSN(DEPENDENT) EMPS_WITHOUT_DEPS

(ALL_EMPS EMPS_WITH_DEPS) RESULT LNAME,FNAME(EMPS_WITHOUT_DEPS * EMPLOYEE)

ALL_EMPS SSN(EMPLOYEE) EMPS_WITH_DEPS(SSN) ESSN(DEPENDENT) EMPS_WITHOUT_DEPS

(ALL_EMPS EMPS_WITH_DEPS) RESULT LNAME,FNAME(EMPS_WITHOUT_DEPS * EMPLOYEE)



List the names of managers who have at least one dependent.


Answers of queries (7) MGRS(SSN) MGRSSN(DEPARTMENT) EMPS_WITH_DEPS(SSN) ESSN(DEPENDENT) MGRS_WITH_DEPS (MGRS EMPS_WITH_DEPS) RESULT LNAME,FNAME(MGRS_WITH_DEPS * EMPLOYEE)

MGRS(SSN) MGRSSN(DEPARTMENT) EMPS_WITH_DEPS(SSN) ESSN(DEPENDENT) MGRS_WITH_DEPS (MGRS EMPS_WITH_DEPS) RESULT LNAME,FNAME(MGRS_WITH_DEPS * EMPLOYEE)

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Relational Data Model (Based on Chapter 7 in Fundamentals of Database Systems by Elmasri and Navathe, Ed. 4)