chapters 3-6 relational data models, relational constraints, and relational algebra

CHAPTERS 3-6RELATIONAL DATA MODELS, RELATIONAL CONSTRAINTS, AND RELATIONAL ALGEBRA

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Flat file: A two dimensional array of attributes or data items

ProductX 1 Bellaire 5ProductY 2 Sugarland 5ProductZ 3 Houston 5Computerization 10 Stafford 4Reorganization 20 Houston 1Newbenefits 30 Stafford 4

Database Management Systems (DBMS): A generalized software system that is used to create, manage, and protect data bases

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Attribute: A name characteristic or property of an entity

= column header

Entity: A “thing” in the real world with an independent

existence physical existence: person, student, car

Domain - The valid set of atomic value for an attribute in a relation

e.g. SSN set of 9 digits GPA: 0<= GPA <= 4.0

Atomic - each value in the domain is indivisible

Name (Fname, Minit, Lname) – not atomic

Fname -- atomic Minit -- atomic Lname -- atomic

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RELATIONAL MODEL CONCEPTS

A Relation is a mathematical concept based on the ideas of sets

The model was first proposed by Dr. E.F. Codd of IBM Research in 1970 in the following paper:"A Relational Model for Large Shared Data

Banks," Communications of the ACM, June 1970

The above paper caused a major revolution in the field of database management and earned Dr. Codd the coveted ACM Turing Award 5

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INFORMAL DEFINITIONS

Informally, a relation looks like a table of values.

A relation typically contains a set of rows.

The data elements in each row represent certain facts that correspond to a real-world entity or relationship In the formal model, rows are called tuples

Each column has a column header that gives an indication of the meaning of the data items in that column In the formal model, the column header is called

an attribute name (or just attribute) 6

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FORMAL DEFINITIONS - SCHEMA

The Schema (or description) of a Relation: Denoted by R(A1, A2, .....An) R is the name of the relation The attributes of the relation are A1, A2, ..., An

Example:

CUSTOMER (Cust-id, Cust-name, Address, Phone#) CUSTOMER is the relation name Defined over the four attributes: Cust-id, Cust-

name, Address, Phone# Each attribute has a domain or a set of valid

values. For example, the domain of Cust-id is 6 digit

numbers.7

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FORMAL DEFINITIONS - TUPLE

A tuple is an ordered set of values (enclosed in angled brackets ‘< … >’)

Each value is derived from an appropriate domain.

A row in the CUSTOMER relation is a 4-tuple and would consist of four values, for example: <632895, "John Smith", "101 Main St. Atlanta,

GA 30332", "(404) 894-2000"> This is called a 4-tuple as it has 4 values A tuple (row) in the CUSTOMER relation.

A relation is a set of such tuples (rows) 8

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FORMAL DEFINITIONS - DOMAIN A domain has a logical definition:

Example: “USA_phone_numbers” are the set of 10 digit phone numbers valid in the U.S.

A domain also has a data-type or a format defined for it. The USA_phone_numbers may have a format: (ddd)ddd-

dddd where each d is a decimal digit. Dates have various formats such as year, month, date

formatted as yyyy-mm-dd, or as dd mm,yyyy etc.

The attribute name designates the role played by a domain in a relation: Used to interpret the meaning of the data elements

corresponding to that attribute Example: The domain Date may be used to define two

attributes named “Invoice-date” and “Payment-date” with different meanings

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FORMAL DEFINITIONS - STATE The relation state is a subset of the

Cartesian product of the domains of its attributeseach domain contains the set of all

possible values the attribute can take. Example: attribute Cust-name is defined over

the domain of character strings of maximum length 25dom(Cust-name) is varchar(25)

The role these strings play in the CUSTOMER relation is that of the name of a customer.

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FORMAL DEFINITIONS - SUMMARY Formally,

Given R(A1, A2, .........., An) r(R) dom (A1) X dom (A2) X ....X dom(An)

R(A1, A2, …, An) is the schema of the relation R is the name of the relation A1, A2, …, An are the attributes of the relation r(R): a specific state (or "value" or “population”) of

relation R – this is a set of tuples (rows) r(R) = {t1, t2, …, tn} where each ti is an n-tuple ti = <v1, v2, …, vn> where each vj element-of

dom(Aj)

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FORMAL DEFINITIONS - EXAMPLE Let R(A1, A2) be a relation schema:

Let dom(A1) = {0,1} Let dom(A2) = {a,b,c}

Then: dom(A1) X dom(A2) is all possible combinations:{<0,a> , <0,b> , <0,c>, <1,a>, <1,b>, <1,c> }

The relation state r(R) dom(A1) X dom(A2) For example: r(R) could be {<0,a> , <0,b> , <1,c> }

this is one possible state (or “population” or “extension”) r of the relation R, defined over A1 and A2.

It has three 2-tuples: <0,a> , <0,b> , <1,c> 12

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DEFINITION SUMMARY

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Informal Terms Formal Terms

Table Relation

Column Header Attribute

All possible Column Values

Domain

Row Tuple

Table Definition Schema of a Relation

Populated Table State of the Relation

SUPER KEY: AN ATTRIBUTE OR A SET OF ATTRIBUTES THAT IDENTIFIES AN ENTITY UNIQUELY (MAY NOT BE MINIMAL SET) SSN SSN, NAME SSN, NAME, MAJOR

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CANDIDATE KEY: A SUPER KEY SUCH THAT NO PROPER SUBSET OF ITS ATTRIBUTES IS ITSELF A SUPER KEY. SO CANDIDATE KEYS MUST HAVE A MINIMAL IDENTIFIER.

STUIDSSN

PRIMARY KEY: THE CANDIDATE KEY THAT IS CHOSENOR THE CANDIDATE KEY THAT IS USED TO IDENTIFY TUPLES IN A RELATION

-- UNIQUE, MUST EXIST ALTERNATE KEY: A CANDIDATE KEY IN A RELATION THAT IS NOT SELECTEDE.G. IF PRIMARY KEY IS SSN THEN STUID IS A ALTERNATE KEY 15

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CONCATENATED (COMPOSITE) KEY: A PRIMARY KEY THAT IS COMPRISED OF TWO OR MORE ATTRIBUTES OR DATA ITEMS

G RADE_REPORT(STUID, COURSE#, GRADE)

FOREIGN KEY: A NON-KEY ATTRIBUTE IN ONE RELATION THAT APPEARS AS THE PRIMARY KEY (OR PART OF THE KEY) IN ANOTHER RELATION

EMPLOYEE(SSN, FNAME, MINIT, DNO)

DEPARTMENT(DNUMBER, DNAME, MANAGER)

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SECONDARY KEY: A FIELD THAT CAN HAVE DUPLICATE VALUES, AND THAT CAN BE USED AS SEARCH PATH BY THE USERS

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Referential Integrity Constraints for COMPANY database

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RELATIONAL ALGEBRA OVERVIEW Relational algebra is the basic set of operations

for the relational model These operations enable a user to specify

basic retrieval requests (or queries) The result of an operation is a new relation,

which may have been formed from one or more input relations This property makes the algebra “closed” (all

objects in relational algebra are relations)

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RELATIONAL ALGEBRA OVERVIEW (CONTINUED) The algebra operations thus produce new

relations These can be further manipulated using

operations of the same algebra A sequence of relational algebra operations

forms a relational algebra expressionThe result of a relational algebra

expression is also a relation that represents the result of a database query (or retrieval request)

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RELATIONAL ALGEBRA OVERVIEW Relational Algebra consists of several groups of

operations Unary Relational Operations

SELECT (symbol: (sigma)) PROJECT (symbol: (pi))

Relational Algebra Operations From Set Theory UNION ( ), INTERSECTION ( ), DIFFERENCE (or MINUS,

– ) CARTESIAN PRODUCT ( x )

Binary Relational Operations JOIN (several variations of JOIN exist) DIVISION

Additional Relational Operations OUTER JOINS, OUTER UNION AGGREGATE FUNCTIONS (These compute summary of

information: for example, SUM, COUNT, AVG, MIN, MAX)

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Unary Relational Operations: SELECT

The SELECT operation (denoted by (sigma)) is used to select a subset of the tuples from a relation based on a selection condition.The selection condition acts as a filterKeeps only those tuples that satisfy the

qualifying conditionTuples satisfying the condition are selected

whereas the other tuples are discarded (filtered out)

Examples: Select the EMPLOYEE tuples whose

department number is 4: DNO = 4 (EMPLOYEE)

Select the employee tuples whose salary is greater than $30,000:

SALARY > 30,000 (EMPLOYEE)25

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UNARY RELATIONAL OPERATIONS: SELECT

In general, the select operation is denoted by <selection condition>(R) where the symbol (sigma) is used to denote the select

operator the selection condition is a Boolean (conditional)

expression specified on the attributes of relation R tuples that make the condition true are selected

appear in the result of the operation tuples that make the condition false are filtered out

discarded from the result of the operation26

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UNARY RELATIONAL OPERATIONS: SELECT (CONTD.) SELECT Operation Properties

The SELECT operation <selection condition>(R) produces a relation S that has the same schema (same attributes) as R

SELECT is commutative: <condition1>( < condition2> (R)) = <condition2> ( < condition1> (R))

Because of commutativity property, a cascade (sequence) of SELECT operations may be applied in any order: <cond1>(<cond2> (<cond3> (R)) = <cond2> (<cond3> (<cond1> (

R))) A cascade of SELECT operations may be replaced

by a single selection with a conjunction of all the conditions: <cond1>(< cond2> (<cond3>(R)) = <cond1> AND < cond2> AND <

cond3>(R))) The number of tuples in the result of a SELECT

is less than (or equal to) the number of tuples in the input relation R

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Select Works on single table and takes rows that meet a specified condition, copy them into a new table

(Table name) Condition(s)

SQL (Structured Query language)

SELECT * FROM (table name) WHERE condition 1 AND condition 2 AND condition 3…

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Table

Condition(s)

Find employees who work for department number 5.

employee DNO = 5

SQL:SELECT * FROM employeeWHERE dno = 5;

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Employee

DNO=5

Query tree

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s(DNO=4 AND SALARY>25000) OR (DNO=5 AND SALARY>30000)(EMPLOYEE)

---------------------------------------------------------s<cond1>(s<cond2>(R)) = s<cond2>(s<cond1>(R))

s<cond1>(s<cond2>(. . .(s<condn> (R)) . . .)) = s<cond1> AND <cond2> AND . . .

AND <condn>(R)

Project Operates on a single table,

produces a vertical subset of the table, extract the values of specified columns

eliminate duplicate rows place the value in a new table

(table name)

column1, column2, column3, …

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SQL: SELECT column1, column2, column3, … FROM (table name)

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Table

column(s)

E.g. Show the names of all employees

employee fname, minit, lname

SELECT fname, minit, lname FROM employee;

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Employee

fname,minit,lname

Select & project

Show the names of all employees who work for department number 5

( employee)

fname, minit, lname dno = 5

SELECT fname, minit, lname FROM employee WHERE dno = 5;

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Employee

fname,minit,lname

DNO = 5

EXAMPLES OF APPLYING SELECT AND PROJECT OPERATIONS

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PRODUCT (or Cartesian product) R1 x R2

R1 X R2 is a table where width is the width of R1 plus the width of R2 and whose columns are the columns of R1 followed by the columns of R2

If R1 has X rows and M columnsR2 has Y rows and N columns

R1 X R2 = X * Y rows and M + N columns

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StudentID Fname Lname

101 Jim Smith102 Tim Brown103 Babara Houston

Credit_HoursSTUID Hours

101 60102 85

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Student X Credit_HoursID Fname Lname Stuid Hours

101 Jim Smith 101 60101 Jim Smith 102 85102 Tim Brown 101 60102 Tim Brown 102 85103 Babara Houston 101 60103 Babara Houston 102 85

Cartesian Product

QUERY TREE FOR CARTESIAN PRODUCT

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Table1 Table2

X

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Example of Query Tree

Theta JoinThe result of performing a SELECT operation using a comparison operator theta (=,<, <=, >, <=, <>) on the product

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101 60102 85

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Student X Credit_Hours (ID > STUID)ID Fname Lname Stuid Hours


Theta Join (>)




101 60102 85103 50

Student X Credit_Hours (ID > STUID)ID Fname Lname Stuid Hours

102 Tim Brown 101 60103 Babara Houston 101 60

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Theta Join (ID>STUID)

QUERY TREE FOR THETA JOIN

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Student Credit_Hours

X ID > STUID

Equijoin Product with “theta” is equality

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Student X Credit_Hours (ID = STUID)ID Fname Lname Stuid Hours


Equijoin




101 60102 85




101 60102 85

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101 Jim Smith 101 60102 Tim Brown 102 85

Equijoin

QUERY TREE FOR EQUIJOIN

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X ID = STUID

Natural Join |X| Is an equijoin which the repeated column is

eliminated

Usually join performs over column with the same names

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Credit HoursID Hours

101 60102 85103 50

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101 Jim Smith 101 60101 Jim Smith 102 85101 Jim Smith 103 50102 Tim Brown 101 60102 Tim Brown 102 85102 Tim Brown 103 50103 Babara Houston 101 60103 Babara Houston 102 85103 Babara Houston 103 50

Remove

Equi-join




101 60102 85103 50

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101 Jim Smith 101 60102 Tim Brown 102 85103 Babara Houston 103 50

Remove this column




101 60102 85103 50

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Student |X| Credit_Hours ID Fname Lname Hours

101 Jim Smith 60102 Tim Brown 85103 Babara Houston 50

QUERY TREE FOR NATURAL JOIN

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|X|

Semi-join: If R1 and R2 are tables

Semijoin of R1 and R2 is natural join of R1 and R2 and then projecting the result into the attributes of A

Semijoin is not cumulative

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Create tablescreate table student1

(id char(3) primary key, fname char(10), lname char(10));

insert into student1 values(‘101’,’Jim’,’Smith’);insert into student1 values(‘102’,’Tim’,’Brown’);insert into student1 values(‘103’,’Babara’,’Houston’);

----------------- ---- create table credit_hours (stuid char(3) primary key, hours number(3));

insert into credit_hours values(101,60);insert into credit_hours values(102,85);

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101 60102 85

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Student |X Credit_Hours ID Fname Lname

101 Jim Smith102 Tim Brown

Left Semi-Join




101 60102 85104 100

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Right Semi-Join

Student X| Credit_Hours ID Hours

101 60102 85

Outer Join:

Is an extension of a THETA JOIN, an EQUIJOIN, or a NATURAL JOIN

An outer join consists of all rows that appear in the usual theta join, plus an additional row for each of the tuples from the original tables that do not participate in the theta join.

In those rows that are unmatched original tuples, extend it by assigning null values to the other attributes.

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Left outer join unmatched rows from the first (left) table appear in the resulting table

Right outer join unmatched rows from the second (right) table appear in the resulting table

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101 60102 85104 100

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101 Jim Smith 60102 Tim Brown 85


101 Jim Smith 60102 Tim Brown 85103 Babara Houston

Left Outer Join Right Outer Join

Student |X| Credit_Hours ID Fname Lname Stuid Hours

101 Jim Smith 101 60102 Tim Brown 102 85

104 100

Outer Join -- OracleLeft-outer join

select * from student, credit_hours where id = stuid(+);

SELECT E.FNAME, E.LNAME, dependent_nameFROM EMPLOYEE E, DEPENDENT DWHERE E.SSN = D.ESSN(+);

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RIGHT-OUTER JOIN

select * from student, credit_hours where id(+) = stuid;

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Sample SQL create view: create view v_emp_dno as select fname, lname, dno from

employee;

select * from v_emp_dno;

create view v_department as select dnumber, dname from department;

select * from v_department;

Cartesian product:select * from v_emp_dno, v_department;

Natural join:

select * from v_emp_dno, v_department where dno = dnumber;

Left Outer join

select fname, lname, ssn, essn, dependent_name from employee, dependent where ssn = essn (+);

Right Outer joinselect essn, dependent_name, fname, lname, ssn from employee, dependent where essn (+) = ssn;

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Set operations: Union, Difference, Intersection, Division

Union (U) tables must be compatible - they must have same basic structure, both relations must have the same domains.

The union of two relations is the set of tuples in either or both relations 71

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EXAMPLE TO ILLUSTRATE THE RESULT OF UNION, INTERSECT, AND DIFFERENCE

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SQL--UnionSelect ssn from employee where dno = 5Unionselect distinct(essn) from dependent;

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SSN --------- 123456789 333445555 666884444 453453453

4 rows selected

ESSN --------- 123456789 333445555 987654321

3 rows selected

SSN --------- 123456789 333445555 453453453 666884444 987654321

5 rows selected

U =

Difference (-) The difference between two relations is the set of tuples that belong to the first relation but not in the second relation.

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SQL--Minus Select ssn from employeeminusselect distinct(essn) from dependent;

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SSN --------- 123456789 333445555 999887777 987654321 666884444 453453453 987987987 888665555

8 rows selected

ESSN --------- 123456789 333445555 987654321

3 rows selected

SSN --------- 453453453 666884444 888665555 987987987 999887777

5 rows selected

- =

Intersection () The intersection of two relations is the set of tuples that belong to both relations simultaneously.

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Student1ID Fname Lname


Student2ID Fname Lname

101 Jim Smith102 Tim Brown105 Kim Lee110 Mike Moore

Chapters 5-8 77

Intersection

Student1 Student2ID Fname Lname

101 Jim Smith102 Tim Brown

Division () A binary operation that can be defined on two relations where the entire structure of one (the divisor) is a portion of the structure of the other (the dividen)

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SC1 C2a 1b 1c 2d 4a 3b 3e 6a 9b 9

RC1ab

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S /RC2139

Division

EXAMPLE OF DIVISION

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AGGREGATE FUNCTIONS AND GROUPING

Script F: (group attributes) <function, attribute> (R)

Functions = sum, average, maximum, minimum, count

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ALL EMPLOYEES (NO GROUP BY)

SELECT sum(salary), Max (salary), min(salary), avg(salary)

FROM employee;

SUM(SALARY) MAX(SALARY) MIN(SALARY) AVG(SALARY)

----------- ----------- ----------- ---------------------- ----------- ----------- -----------

281000 55000 25000 35125

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EXAMPLE: RETRIEVE THE DEPARTMENT NUMBER, NUMBER OF EMPLOYEES, AND AVERAGE SALARY IN THE DEPARTMENT – GROUP BY DNO

RESULT(DNO, NUMBER_OF_EMPLOYEES, AVG_SAL) count SSN, Average SALARY EMPLOYEE

SELECT dno, count(ssn), avg(salary)

FROM employee

GROUP BY dno

order by dno;

DNO COUNT(SSN) AVG(SALARY)

-------------------------------------- ---------- -----------

1 1 55000

4 3 31000

5 4 33250

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GROUP BY

SELECT dno, sum(salary), Max (salary), min(salary), avg(salary)

FROM employeeGROUP BY dno;

DNO SUM(SALARY) MAX(SALARY) MIN(SALARY) AVG(SALARY)

-------------------------------------- -----------

1 55000 55000 55000 55000

5 133000 40000 25000 33250

4 93000 43000 25000 3100084

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DNO count SSN, Average SALARY (EMPLOYEE)

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RESULTDNO COUNT_SSN AVERAGE_SALARY

5 4 332504 3 310001 1 55000

IF GROUPING ATTRIBUTES ARE NOT SPECIFIED

count SSN, Average SALARY (EMPLOYEE)

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RESULTCOUNT_SSN AVERAGE_SALARY

8 35125

SELECT sum(salary), Max (salary), min(salary), avg(salary)

FROM employee, departmentWHERE dno = dnumberAND dname = 'Research';

SUM(SALARY) MAX(SALARY) MIN(SALARY) AVG(SALARY)

----------- ----------- ----------- -----------

133000 40000 25000 33250

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ViewCreate View V_Dno5 as (select fname, lname, dno from employee where dno = 5)--------------view V_DNO5 created.-------------Select * from V_DNO5;FNAME LNAME DNO

--------------- --------------- -----------------

John Smith 5

Franklin Wong 5

Ramesh Narayan 5

Joyce English 588

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chapters 3-6 relational data models, relational constraints, and relational algebra

Documents

customer relation

domain of cust

domain date

formal model

relation namedefined

data elements

set of rows

relational data models