Slide 6- 1
Chapter 6
The Relational Algebra and
Calculus
Relational Algebra

Algebra

Mathematical system consisting of:



Operands --- variables or values from which new
values can be constructed.
Operators --- symbols denoting procedures that
construct new values from given values.
Relational Algebra

An algebra whose operands are relations or
variables that represent relations.

The result can be used as a query language

to specify basic retrieval requests.
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Core Relational Algebra
Unary Operations:
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Selection : picking certain rows [tuples].
Projection : picking certain columns.
Renaming of relations and attributes.
Binary Operations:

Union , intersection , and difference - .


require both operands have the same relation
schema.
Products X and joins
 : compositions of relations.
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Relational Algebra Operations
© Pearson Education Limited 1995, 2005
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Selection (or Restriction)

SELECT operation is used to select a subset of the tuples
from a relation that satisfy a selection condition.

Example: To select the EMPLOYEE tuples whose
department number is four or those whose salary is
greater than $30,000 the following notation is used:

DNO = 4(EMPLOYEE)
SALARY > 30,000(EMPLOYEE)
Select operation is denoted by <selection condition>(R)
The symbol  (sigma) is used to denote the select operator


the selection condition is a Boolean expression specified on
the attributes of relation R
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Selection Example
(DNO=4 AND SALARY>25000) OR (DNO=5 AND SLARY>30000) (EMPLOYEE)
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Projection

PROJECT operation selects certain columns from the
table



Creates a vertical partitioning.
Example: To list each employee’s first and last name and
salary, the following is used:
LNAME, FNAME,SALARY (EMPLOYEE)
The general form of the project operation is <attribute
list>(R)



where  (pi) is the symbol used to represent the project
operation
<attribute list> is the desired list of attributes from the
attributes of relation R.
The project operation removes any duplicate tuples, if
any.
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Projection Example
LNAME, FNAME,SALARY (EMPLOYEE)
SEX, SALARY (EMPLOYEE)
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Renaming



We may want to apply several relational algebra operations one after the
other.
 we can write them as a single relational algebra expression by
nesting
 or we can apply one operation at a time and create intermediate
relations
In the latter case, we must give names to the relations that hold the
intermediate results.
Example: To retrieve the first name, last name, and salary of all
employees who work in department number 5:
 FNAME, LNAME, SALARY ( DNO=5 (EMPLOYEE))

OR We can explicitly show the sequence of operations, giving a name to
each intermediate relation:
TEMP  DNO=5 (EMPLOYEE)
R

FNAME, LNAME, SALARY
(TEMP)
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Renaming Example
(a) FNAME, LNAME, SALARY(DNO=5(EMPLOYEE))
(b) Using intermediate relations and renaming of attributes
.
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Renaming (cont.)


The rename operator is r
The general Rename operation can be expressed
by any of the following forms:
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r S (B1, B2, …, Bn ) ( R) is a renamed relation S based
on R with column names B1, B1,…..Bn.
r S ( R) is a renamed relation S based on R (which
does not specify column names).
r (B1, B2, …, Bn ) ( R) is a renamed relation with
column names B1, B2, …..Bn which does not
specify a new relation name.
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UNION Operation


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The result of this operation, denoted by R  S, is a
relation that includes all tuples that are either in R or in S
or in both R and S.
Duplicate tuples are eliminated.
Example: To retrieve the social security numbers of all
employees who either work in department 5 or directly
supervise an employee who works in department 5:
DEP5_EMPS  DNO=5 (EMPLOYEE)
RESULT1  SSN(DEP5_EMPS)
RESULT2(SSN)  SUPERSSN(DEP5_EMPS)
RESULT  RESULT1  RESULT2
The union operation produces the tuples that are in either
RESULT1 or RESULT2 or both. The two operands must
be “type compatible”.
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UNION Example1
RESULT RESULT1  RESULT2
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UNION Operation (cont.)

Type Compatibility
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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 be compatible; that is,
dom(Ai)=dom(Bi) for i=1, 2, ..., n.
The resulting relation for R1R2,R1  R2, or R1R2 has the same attribute names as the first
operand relation R.
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UNION Example2
STUDENTINSTRUCTOR
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INTERSECTION OPERATION



The result of this operation, denoted by R  S, is a
relation that includes all tuples that are in both R and S.
The two operands must be "type compatible"
Example: The result of the intersection operation (figure
below) includes only those who are both students and
instructors.
STUDENT INSTRUCTOR
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Set Difference (or MINUS) Operation



The result of this operation, denoted by R - S, is a relation
that includes all tuples that are in R but not in S.
The two operands must be "type compatible”.
Example: The figure shows the names of students who
are not instructors, and the names of instructors who are
not students.
STUDENT-INSTRUCTOR
INSTRUCTOR-STUDENT
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