Chapter on
Relational Algebra
Ritu CHaturvedi
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INTRODUCTION
 Relational algebra and relational calculus are formal
languages associated with the relational model.
 Informally, relational algebra is a (high-level)
procedural language and relational calculus a nonprocedural language.
 However, formally both are equivalent to one
another.
 A language that produces a relation that can be
derived using relational calculus is relationally
complete.
Ritu CHaturvedi
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Properties of Relational Algebra
 Procedural language
 Operators operate on one or more relations and the
result is always a relation without changing the
original relations.
 This property is called closure. (If we perform one or
more of the RA operations on a relational DB, we get
back a result set that belongs to the same DB =>
relational DB is closed with respect to the operations
of RA ).
 Both operands and results are relations, so output
from one operation can become input to another
operation.
 Allows expressions to be nested, just as in arithmetic.
Ritu CHaturvedi
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Properties of Relational Algebra
 5 basic operations in relational algebra:
Selection, Projection, Cartesian product,
Union, and Set Difference. (These perform
most of the data retrieval operations needed).
 Also have Join, Intersection, and Division
operations, which can be expressed in terms
of 5 basic operations.
Ritu CHaturvedi
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Unary Operators
 Select ()
(select in Class Notes )
 Project (π)
(project in Class Notes )
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Binary Operators
 Set Operations



Union
Intersection
Difference (minus)
 Cartesian Product (join)
 Join Operations




Theta Join
Equijoin
Natural Join
Outer Join
 Division
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Figure
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Figure
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Example :
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UNARY OPERATORS
Select ()
Selects a subset of tuples from a relation R that
satisfy a certain predicate (or a Boolean
condition).
R1 = <Predicate>(R)
Degree of R1 is the same as degree of R
Cardinality of R1 <= Cardinality of R
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Examples
 Select employees who work in department
number 4.
 Select employees who work in department
number 4 and whose salary is greater than
$70,000.
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Project ()
 Selects vertical subsets of a relation R
extracting the values of specified attributes of
R eliminating duplicates.
 R1 = <attribute-list>(R)
 Degree of R1  Degree of R
 Cardinality of R1  Cardinality of R (!!)
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Example:
 Retrieve the First Name, Last Name and
Salary of all employees.
 Retrieve the First Name, Last Name and
Salary of employees who work in department
4.
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Example:
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Example:
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Set operations
 Two relations R(A1,….An) and S(B1,…Bn) are
said to be Union Compatible if they have the
same degree n and
Dom(AI) = Dom(BI) where 1 I  n
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Union
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 being eliminated.
 If R and S have I and J tuples, respectively,
union is obtained by concatenating them into
one relation with a maximum of (I + J) tuples
 Example: List the names of all students and
instructors.
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Intersection
 R  S: is a relation that includes all tuples
that are in both R and in S.
 Expressed using basic operations:
R  S = R – (R – S)
 Example: List the names of those who are
both students and instructors.
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Difference
R  S : is a relation that includes all tuples that
are in R but not in S.
Examples:
 List the names of students who are not
instructors.
 List the names of instructors who are not
students.
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Example of set operators
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Cartesian Product
 R  S is a concatenation of every tuple of R with
every tuple of S.

If Degree(R ) = n and Degree(S) = m ,
then Degree(R  S) = n + m
 If Cardinality(R ) = t1 and Cardinality(S) = t2,
then Cardinality(R  S) = t1 * t2
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Example : R X S
R1
R2
R1 X R2
Al
No
Al
No
A
B
1
2
3
A
A
A
B
B
B
1
2
3
1
2
3
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Example : Cartesian Product
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Example (Decomposing complex relations)
List for all female employees, the names of their
dependents.
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Solution to Slide 23
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JOIN OPERATIONS
 Join is a derivative of Cartesian Product equivalent to
performing a Selection operation using the join
predicate as the selection, over the Cartesian product
of the two operand relations.
 Join is the most difficult operation to implement in an
RDBMS and is one of the reasons why relational
systems have intrinsic performance problems.
 See the figure in Slide 8
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Natural Join : The Process
 Links tables by selecting rows with common
values in common attribute(s)
 Three-stage process



Cartesian Product creates one table
Select yields appropriate rows
Project yields single copy of each attribute to
eliminate duplicate columns
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Natural Join
Information from two or more tables is
combined
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Natural Join: R
S
 The Natural Join is an Equijoin( next slide) of
the two relations R and S over all common
attributes x. One occurrence of each common
attribute is eliminated from the result.
 Degree of a natural join = (Sum of degrees of
R and S)  (number of attributes in x)
Example: Retrieve the names of the managers
of each department
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Other Joins
 EquiJOIN



Links tables based on equality condition that
compares specified columns of tables
Does not eliminate duplicate columns
Join criteria must be explicitly defined
 Theta JOIN

EquiJOIN that compares specified columns of
each table using operator other than equality
one
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Other Joins
 Outer JOIN



Matched pairs are retained
Unmatched values in other tables left null
Right and left
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Theta Join : R

S
 The Theta Join operation defines a relation
that contains tuples satisfying the predicate 
from the Cartesian Product of R and S
R
 S = (RS)
 Degree of Theta Join is the sum of degrees of
R and S
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Example : Theta Join
 Retrieve the names of the managers of each
department.
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Equijoin
When the predicate  contains only equality(=),
theta join is called Equijoin.
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Outer Join R
S
 The (left) outer join is a join in which tuples
from R that do not have matching values in
the common attributes of S are also included
in the result relation. Missing values in the
second relation are set to null.
(
 Example : Retrieve all employee names and
the department names they manage(if they
manage a department)
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Left Outer Join
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DIVISION OPERATOR R  S
 Let relation R be defined over attribute set A
and relation S be defined over attribute set B
such that B  A. Let C = A  B i.e. C is the set
of attributes of R That are not attributes of S.
 The Division operation defines a relation over
the attributes C that consists of the set of
tuples from R that match the combination of
every tuple in S.
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Example : Division
- Retrieve the names of employees who work
on all the projects that John Smith works on.
- Slide 8
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Example : Division
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More examples
 Queries (RA) (Questions 1 to 7 : Solutions to
be discussed in class)
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