relation algebra unit ii notes and their queries

CHAPTER 8
The Relational Algebra
Slide 8- 2

Copyright © 2016 Ramez Elmasri and Shamkant B. Navathe Slide 8- 3
Chapter Outline
 Relational Algebra
 Unary Relational Operations
 Relational Algebra Operations From Set Theory
 Binary Relational Operations
 Additional Relational Operations
 Examples of Queries in Relational Algebra
 Relational Calculus
 Tuple Relational Calculus
 Domain Relational Calculus
 Example Database Application (COMPANY)
 Overview of the QBE language (appendix D)

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)

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

Brief History of Origins of Algebra
 Muhammad ibn Musa al-Khwarizmi (800-847 CE) – from
Morocco wrote a book titled al-jabr about arithmetic of
variables
 Book was translated into Latin.
 Its title (al-jabr) gave Algebra its name.
 Al-Khwarizmi called variables “shay”
 “Shay” is Arabic for “thing”.
 Spanish transliterated “shay” as “xay” (“x” was “sh” in Spain).
 In time this word was abbreviated as x.
 Where does the word Algorithm come from?
 Algorithm originates from “al-Khwarizmi"
 Reference: PBS (http://www.pbs.org/empires/islam/innoalgebra.html)

Relational Algebra Overview
 Relational Algebra consists of several groups of operations
 SELECT (symbol:  (sigma))
 PROJECT (symbol:  (pi))
 RENAME (symbol:  (rho))
 UNION (  ), INTERSECTION (  ), DIFFERENCE (or MINUS, – )
 CARTESIAN PRODUCT ( x )
 JOIN (several variations of JOIN exist)
 DIVISION
 OUTER JOINS, OUTER UNION
 AGGREGATE FUNCTIONS (These compute summary of
information: for example, SUM, COUNT, AVG, MIN, MAX)

Database State for COMPANY
 All examples discussed below refer to the COMPANY database
shown here.

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 filter
 Keeps only those tuples that satisfy the qualifying condition
 Tuples 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)

 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 operation

(continued)
 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

The following query results refer to this
database state

Unary Relational Operations: PROJECT
 PROJECT Operation is denoted by  (pi)
 This operation keeps certain columns (attributes)
from a relation and discards the other columns.
 PROJECT creates a vertical partitioning
 The list of specified columns (attributes) is kept in
each tuple
 The other attributes in each tuple are discarded
 Example: To list each employee’s first and last
name and salary, the following is used:
LNAME, FNAME,SALARY(EMPLOYEE)

(cont.)
 The general form of the project operation is:
<attribute list>(R)
  (pi) is the symbol used to represent the project
operation
 <attribute list> is the desired list of attributes from
relation R.
 The project operation removes any duplicate
tuples
 This is because the result of the project operation
must be a set of tuples
 Mathematical sets do not allow duplicate elements.

(contd.)
 PROJECT Operation Properties
 The number of tuples in the result of projection
<list>(R) is always less or equal to the number of
tuples in R
 If the list of attributes includes a key of R, then the
number of tuples in the result of PROJECT is equal
to the number of tuples in R
 PROJECT is not commutative
  <list1> ( <list2> (R) ) =  <list1> (R) as long as <list2>
contains the attributes in <list1>

Examples of applying SELECT and
PROJECT operations

Relational Algebra Expressions
 We may want to apply several relational algebra
operations one after the other
 Either we can write the operations as a single
relational algebra expression by nesting the
operations, or
 We can apply one operation at a time and create
intermediate result relations.
 In the latter case, we must give names to the
relations that hold the intermediate results.

Single expression versus sequence of
relational operations (Example)
 To retrieve the first name, last name, and salary of all
employees who work in department number 5, we must
apply a select and a project operation
 We can write a single relational algebra expression as
follows:
 FNAME, LNAME, SALARY( DNO=5(EMPLOYEE))
 OR We can explicitly show the sequence of operations,
giving a name to each intermediate relation:
 DEP5_EMPS   DNO=5(EMPLOYEE)
 RESULT   FNAME, LNAME, SALARY (DEP5_EMPS)

Unary Relational Operations: RENAME
 The RENAME operator is denoted by  (rho)
 In some cases, we may want to rename the
attributes of a relation or the relation name or
both
 Useful when a query requires multiple
operations
 Necessary in some cases (see JOIN operation
later)

(continued)
 The general RENAME operation  can be
expressed by any of the following forms:
 S (B1, B2, …, Bn )(R) changes both:
 the relation name to S, and
 the column (attribute) names to B1, B1, …..Bn
 S(R) changes:
 the relation name only to S
 (B1, B2, …, Bn )(R) changes:
 the column (attribute) names only to B1, B1, …..Bn

(continued)
 For convenience, we also use a shorthand for
renaming attributes in an intermediate relation:
 If we write:
• RESULT   FNAME, LNAME, SALARY (DEP5_EMPS)
• RESULT will have the same attribute names as
DEP5_EMPS (same attributes as EMPLOYEE)
• If we write:
• RESULT (F, M, L, S, B, A, SX, SAL, SU, DNO)
 RESULT (F.M.L.S.B,A,SX,SAL,SU, DNO)(DEP5_EMPS)
• The 10 attributes of DEP5_EMPS are renamed to
F, M, L, S, B, A, SX, SAL, SU, DNO, respectively
Note: the  symbol is an assignment operator

Example of applying multiple operations
and RENAME

Relational Algebra Operations from
Set Theory: UNION
 UNION Operation
 Binary operation, denoted by 
 The result of 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
 The two operand relations R and S must be “type
compatible” (or UNION compatible)
 R and S must have same number of attributes
 Each pair of corresponding attributes must be type
compatible (have same or compatible domains)

Set Theory: UNION
 Example:
 To retrieve the social security numbers of all employees who
either work in department 5 (RESULT1 below) or directly
supervise an employee who works in department 5 (RESULT2
below)
 We can use the UNION operation as follows:
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

Figure 8.3 Result of the UNION
operation RESULT ← RESULT1 ∪
RESULT2.
Slide 8- 25

Set Theory
 Type Compatibility of operands is required for the binary
set operation UNION , (also for INTERSECTION , and
SET DIFFERENCE –, see next slides)
 R1(A1, A2, ..., An) and R2(B1, B2, ..., Bn) are type
compatible if:
 they have the same number of attributes, and
 the domains of corresponding attributes are type compatible
(i.e. dom(Ai)=dom(Bi) for i=1, 2, ..., n).
 The resulting relation for R1R2 (also for R1R2, or R1–
R2, see next slides) has the same attribute names as the
first operand relation R1 (by convention)

Relational Algebra Operations from Set
Theory: INTERSECTION
 INTERSECTION is denoted by 
 The result of the operation R  S, is a
relation that includes all tuples that are in
both R and S
 The attribute names in the result will be the
same as the attribute names in R
 The two operand relations R and S must be
“type compatible”

Theory: SET DIFFERENCE (cont.)
 SET DIFFERENCE (also called MINUS or
EXCEPT) is denoted by –
 The result of R – S, is a relation that includes all
tuples that are in R but not in S
 The attribute names in the result will be the
same as the attribute names in R
 The two operand relations R and S must be
“type compatible”

Example to illustrate the result of UNION,
INTERSECT, and DIFFERENCE

Some properties of UNION, INTERSECT,
and DIFFERENCE
 Notice that both union and intersection are commutative
operations; that is
 R  S = S  R, and R  S = S  R
 Both union and intersection can be treated as n-ary
operations applicable to any number of relations as both
are associative operations; that is
 R  (S  T) = (R  S)  T
 (R  S)  T = R  (S  T)
 The minus operation is not commutative; that is, in
general
 R – S ≠ S – R

Theory: CARTESIAN PRODUCT
 CARTESIAN (or CROSS) PRODUCT Operation
 This operation is used to combine tuples from two relations
in a combinatorial fashion.
 Denoted by R(A1, A2, . . ., An) x S(B1, B2, . . ., Bm)
 Result is a relation Q with degree n + m attributes:
 Q(A1, A2, . . ., An, B1, B2, . . ., Bm), in that order.
 The resulting relation state has one tuple for each
combination of tuples—one from R and one from S.
 Hence, if R has nR tuples (denoted as |R| = nR ), and S has
nS tuples, then R x S will have nR * nS tuples.
 The two operands do NOT have to be "type compatible”

Theory: CARTESIAN PRODUCT (cont.)
 Generally, CROSS PRODUCT is not a
meaningful operation
 Can become meaningful when followed by other
operations
 Example (not meaningful):
 FEMALE_EMPS   SEX=’F’(EMPLOYEE)
 EMPNAMES   FNAME, LNAME, SSN (FEMALE_EMPS)
 EMP_DEPENDENTS  EMPNAMES x DEPENDENT
 EMP_DEPENDENTS will contain every combination of
EMPNAMES and DEPENDENT
 whether or not they are actually related

Theory: CARTESIAN PRODUCT (cont.)
 To keep only combinations where the
DEPENDENT is related to the EMPLOYEE, we
add a SELECT operation as follows
 Example (meaningful):
 FEMALE_EMPS   SEX=’F’(EMPLOYEE)
 EMPNAMES   FNAME, LNAME, SSN (FEMALE_EMPS)
 EMP_DEPENDENTS  EMPNAMES x DEPENDENT
 ACTUAL_DEPS   SSN=ESSN(EMP_DEPENDENTS)
 RESULT   FNAME, LNAME, DEPENDENT_NAME (ACTUAL_DEPS)
 RESULT will now contain the name of female employees
and their dependents

Figure 8.5 The CARTESIAN PRODUCT
(CROSS PRODUCT) operation.
continued on next slide
Slide 8- 34

Figure 8.5 (continued) The CARTESIAN
PRODUCT (CROSS PRODUCT) operation.
Slide 8- 35

Figure 8.5 (continued) The CARTESIAN
PRODUCT (CROSS PRODUCT) operation.
Slide 8- 36

Binary Relational Operations: JOIN
 JOIN Operation (denoted by )
 The sequence of CARTESIAN PRODECT followed by
SELECT is used quite commonly to identify and select
related tuples from two relations
 A special operation, called JOIN combines this sequence
into a single operation
 This operation is very important for any relational database
with more than a single relation, because it allows us
combine related tuples from various relations
 The general form of a join operation on two relations R(A1,
A2, . . ., An) and S(B1, B2, . . ., Bm) is:
R <join condition>S
 where R and S can be any relations that result from general
relational algebra expressions.

Binary Relational Operations: JOIN (cont.)
 Example: Suppose that we want to retrieve the name of the
manager of each department.
 To get the manager’s name, we need to combine each
DEPARTMENT tuple with the EMPLOYEE tuple whose SSN
value matches the MGRSSN value in the department tuple.
 We do this by using the join operation.
 DEPT_MGR  DEPARTMENT MGRSSN=SSN EMPLOYEE
 MGRSSN=SSN is the join condition
 Combines each department record with the employee who
manages the department
 The join condition can also be specified as
DEPARTMENT.MGRSSN= EMPLOYEE.SSN

Figure 8.6 Result of the JOIN operation
DEPT_MGR ← DEPARTMENT|X|
Mgr_ssn=SsnEMPLOYEE.
Slide 8- 39

Some properties of JOIN
 Consider the following JOIN operation:
 R(A1, A2, . . ., An) S(B1, B2, . . ., Bm)
R.Ai=S.Bj
 Result is a relation Q with degree n + m attributes:
 Q(A1, A2, . . ., An, B1, B2, . . ., Bm), in that order.
 The resulting relation state has one tuple for each
combination of tuples—r from R and s from S, but only if
they satisfy the join condition r[Ai]=s[Bj]
 Hence, if R has nR tuples, and S has nS tuples, then the join
result will generally have less than nR * nS tuples.
 Only related tuples (based on the join condition) will appear
in the result

Some properties of JOIN
 The general case of JOIN operation is called a
Theta-join: R S
theta
 The join condition is called theta
 Theta can be any general boolean expression on
the attributes of R and S; for example:
 R.Ai<S.Bj AND (R.Ak=S.Bl OR R.Ap<S.Bq)
 Most join conditions involve one or more equality
conditions “AND”ed together; for example:
 R.Ai=S.Bj AND R.Ak=S.Bl AND R.Ap=S.Bq

Binary Relational Operations: EQUIJOIN
 EQUIJOIN Operation
 The most common use of join involves join
conditions with equality comparisons only
 Such a join, where the only comparison operator
used is =, is called an EQUIJOIN.
 In the result of an EQUIJOIN we always have one
or more pairs of attributes (whose names need not
be identical) that have identical values in every
tuple.
 The JOIN seen in the previous example was an
EQUIJOIN.

Binary Relational Operations:
NATURAL JOIN Operation
 NATURAL JOIN Operation
 Another variation of JOIN called NATURAL JOIN —
denoted by * — was created to get rid of the second
(superfluous) attribute in an EQUIJOIN condition.
 because one of each pair of attributes with identical values is
superfluous
 The standard definition of natural join requires that the two
join attributes, or each pair of corresponding join attributes,
have the same name in both relations
 If this is not the case, a renaming operation is applied first.

Binary Relational Operations
NATURAL JOIN (continued)
 Example: To apply a natural join on the DNUMBER attributes of
DEPARTMENT and DEPT_LOCATIONS, it is sufficient to write:
 DEPT_LOCS  DEPARTMENT * DEPT_LOCATIONS
 Only attribute with the same name is DNUMBER
 An implicit join condition is created based on this attribute:
DEPARTMENT.DNUMBER=DEPT_LOCATIONS.DNUMBER
 Another example: Q  R(A,B,C,D) * S(C,D,E)
 The implicit join condition includes each pair of attributes with the
same name, “AND”ed together:
 R.C=S.C AND R.D.S.D
 Result keeps only one attribute of each such pair:
 Q(A,B,C,D,E)

Example of NATURAL JOIN operation

Complete Set of Relational Operations
 The set of operations including SELECT ,
PROJECT  , UNION , DIFFERENCE - ,
RENAME , and CARTESIAN PRODUCT X is
called a complete set because any other
relational algebra expression can be expressed
by a combination of these five operations.
 For example:
 R  S = (R  S ) – ((R - S)  (S - R))
 R <join condition>S =  <join condition> (R X S)

Binary Relational Operations: DIVISION
 DIVISION Operation
 The division operation is applied to two relations
 R(Z)  S(X), where X subset Z. Let Y = Z - X (and hence Z
= X  Y); that is, let Y be the set of attributes of R that are
not attributes of S.
 The result of DIVISION is a relation T(Y) that includes a
tuple t if tuples tR appear in R with tR [Y] = t, and with
 tR [X] = ts for every tuple ts in S.
 For a tuple t to appear in the result T of the DIVISION, the
values in t must appear in R in combination with every tuple
in S.

Example of DIVISION

Table 8.1 Operations of Relational
Algebra
Slide 8- 49

Table 8.1 Operations of Relational
Algebra (continued)
Slide 8- 50

Query Tree Notation
 Query Tree
 An internal data structure to represent a query
 Standard technique for estimating the work involved in
executing the query, the generation of intermediate results,
and the optimization of execution
 Nodes stand for operations like selection, projection, join,
renaming, division, ….
 Leaf nodes represent base relations
 A tree gives a good visual feel of the complexity of the
query and the operations involved
 Algebraic Query Optimization consists of rewriting the query
or modifying the query tree into an equivalent tree.
(see Chapter 15)

Example of Query Tree

Additional Relational Operations:
Aggregate Functions and Grouping
 A type of request that cannot be expressed in the basic
relational algebra is to specify mathematical aggregate
functions on collections of values from the database.
 Examples of such functions include retrieving the average
or total salary of all employees or the total number of
employee tuples.
 These functions are used in simple statistical queries that
summarize information from the database tuples.
 Common functions applied to collections of numeric
values include
 SUM, AVERAGE, MAXIMUM, and MINIMUM.
 The COUNT function is used for counting tuples or
values.

Aggregate Function Operation
 Use of the Aggregate Functional operation ℱ
 ℱMAX Salary (EMPLOYEE) retrieves the maximum salary value
from the EMPLOYEE relation
 ℱMIN Salary (EMPLOYEE) retrieves the minimum Salary value
 ℱSUM Salary (EMPLOYEE) retrieves the sum of the Salary
 ℱCOUNT SSN, AVERAGE Salary (EMPLOYEE) computes the count
(number) of employees and their average salary
 Note: count just counts the number of rows, without removing
duplicates

Using Grouping with Aggregation
 The previous examples all summarized one or more
attributes for a set of tuples
 Maximum Salary or Count (number of) Ssn
 Grouping can be combined with Aggregate Functions
 Example: For each department, retrieve the DNO,
COUNT SSN, and AVERAGE SALARY
 A variation of aggregate operation ℱ allows this:
 Grouping attribute placed to left of symbol
 Aggregate functions to right of symbol
 DNO ℱCOUNT SSN, AVERAGE Salary (EMPLOYEE)
 Above operation groups employees by DNO (department
number) and computes the count of employees and
average salary per department

Figure 8.10 The aggregate function
operation.
a. ρR(Dno, No_of_employees, Average_sal)(Dno ℑ COUNT Ssn,
AVERAGE Salary (EMPLOYEE)).
b. Dno ℑ alary(EMPLOYEE).
c. ℑ COUNT Ssn, AVERAGE Salary(EMPLOYEE).
Slide 8- 56

Figure 7.1a Results of GROUP BY
and HAVING (in SQL). Q24.
Slide 8- 57

Additional Relational Operations
(continued)
 Recursive Closure Operations
 Another type of operation that, in general,
cannot be specified in the basic original
relational algebra is recursive closure.
 This operation is applied to a recursive
relationship.
 An example of a recursive operation is to
retrieve all SUPERVISEES of an EMPLOYEE
e at all levels — that is, all EMPLOYEE e’
directly supervised by e; all employees e’’
directly supervised by each employee e’; all
employees e’’’ directly supervised by each
employee e’’; and so on.

(continued)
 Although it is possible to retrieve employees at
each level and then take their union, we cannot,
in general, specify a query such as “retrieve the
supervisees of ‘James Borg’ at all levels” without
utilizing a looping mechanism.
 The SQL3 standard includes syntax for recursive
closure.

Figure 8.11 A two-level recursive
query.
Slide 8- 60

(continued)
 The OUTER JOIN Operation
 In NATURAL JOIN and EQUIJOIN, tuples without a
matching (or related) tuple are eliminated from the join
result
 Tuples with null in the join attributes are also eliminated
 This amounts to loss of information.
 A set of operations, called OUTER joins, can be used when
we want to keep all the tuples in R, or all those in S, or all
those in both relations in the result of the join, regardless of
whether or not they have matching tuples in the other
relation.

(continued)
 The left outer join operation keeps every tuple in
the first or left relation R in R S; if no matching
tuple is found in S, then the attributes of S in the
join result are filled or “padded” with null values.
 A similar operation, right outer join, keeps every
tuple in the second or right relation S in the result
of R S.
 A third operation, full outer join, denoted by
keeps all tuples in both the left and the right
relations when no matching tuples are found,
padding them with null values as needed.

Figure 8.12 The result of a LEFT
OUTER JOIN operation.
Slide 8- 63

(continued)
 OUTER UNION Operations
 The outer union operation was developed to take
the union of tuples from two relations if the
relations are not type compatible.
 This operation will take the union of tuples in two
relations R(X, Y) and S(X, Z) that are partially
compatible, meaning that only some of their
attributes, say X, are type compatible.
 The attributes that are type compatible are
represented only once in the result, and those
attributes that are not type compatible from either
relation are also kept in the result relation T(X, Y,
Z).

(continued)
 Example: An outer union can be applied to two relations
whose schemas are STUDENT(Name, SSN, Department,
Advisor) and INSTRUCTOR(Name, SSN, Department,
Rank).
 Tuples from the two relations are matched based on having the
same combination of values of the shared attributes— Name,
SSN, Department.
 If a student is also an instructor, both Advisor and Rank will
have a value; otherwise, one of these two attributes will be null.
 The result relation STUDENT_OR_INSTRUCTOR will have the
following attributes:
STUDENT_OR_INSTRUCTOR (Name, SSN, Department,
Advisor, Rank)

Inner join
Slide 16- 66

Outer join
 Left outer join
Slide 16- 67

 Right outer join
Slide 16- 68

 Full outer join
Slide 16- 69

Chapter Summary
 Relational Algebra
 Examples of Queries in Relational Algebra
 Relational Calculus
 Tuple Relational Calculus
 Domain Relational Calculus
 Overview of the QBE language (appendix C)

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