Relational Algebra Guide for Database Systems

Rushdi Shams, Dept of CSE, KUET 1
Database SystemsDatabase Systems
Relational AlgebraRelational Algebra
Version 1.0Version 1.0

2Rushdi Shams, Dept of CSE, KUET
What is Relational AlgebraWhat is Relational Algebra
 It is a collection of operations on relationsIt is a collection of operations on relations
 If you apply relational algebra on some relations,If you apply relational algebra on some relations,
the result will be another relationthe result will be another relation
 You can say a relation- a table.You can say a relation- a table.
 So, operation on tables=another tableSo, operation on tables=another table
 That is what relational algebra does!That is what relational algebra does!

Memory RefreshmentMemory Refreshment
 A relation is a set of attributes with values for eachA relation is a set of attributes with values for each
attribute such that:attribute such that:
 Each attribute value must be a single value onlyEach attribute value must be a single value only
(atomic).(atomic).
 All values for a given attribute must be of the same typeAll values for a given attribute must be of the same type
(or domain).(or domain).
 Each attribute name must be unique.Each attribute name must be unique.
 The order of attributes is insignificantThe order of attributes is insignificant
 No two rows (tuples) in a relation can be identical.No two rows (tuples) in a relation can be identical.
 The order of the rows (tuples) is insignificant.The order of the rows (tuples) is insignificant.

2 types of operation2 types of operation
1.1. Set theory operations:Set theory operations:
Union, Intersection, Difference and CartesianUnion, Intersection, Difference and Cartesian
product.product.
2.2. Specific Relational Operations:Specific Relational Operations:
Selection, Projection, Join, DivisionSelection, Projection, Join, Division
We will take a look into set theory operations,We will take a look into set theory operations,
and then we will dig into specific relationaland then we will dig into specific relational
operationsoperations

Our example tablesOur example tables
 There are 2 tables- R andThere are 2 tables- R and
S, we will play withS, we will play with
them!!them!!

UnionUnion
 RUS means, a unionRUS means, a union
operation on R and Soperation on R and S
that will produce anotherthat will produce another
table which will have alltable which will have all
the rows of R and Sthe rows of R and S
except the duplicatesexcept the duplicates
 There are 2 Sally in RThere are 2 Sally in R
and S, so, just take 1and S, so, just take 1
from themfrom them

DifferenceDifference
 R-S means, a differenceR-S means, a difference
operation on R and Soperation on R and S
that will produce anotherthat will produce another
table which will have alltable which will have all
the rows which is in Rthe rows which is in R
but not in S.but not in S.
 As Sally is in R and in S,As Sally is in R and in S,
so, she has been omitted.so, she has been omitted.

IntersectionIntersection
 R∩S means, anR∩S means, an
intersection operation onintersection operation on
R and S that will produceR and S that will produce
another table which willanother table which will
have all the rows whichhave all the rows which
are in both R and S.are in both R and S.
 As Sally is in R and in S,As Sally is in R and in S,
so, she has been selected.so, she has been selected.

More UnionMore Union

More IntersectionMore Intersection

More DifferenceMore Difference

ConstraintsConstraints
 You see tables, and start performing set theoryYou see tables, and start performing set theory
operations on them- MISTAKE!! You shouldoperations on them- MISTAKE!! You should
remember the following constraints before youremember the following constraints before you
perform set theory operations on tables.perform set theory operations on tables.
1.1. Attributes of relations needAttributes of relations need notnot be identical tobe identical to
perform union, intersection and differenceperform union, intersection and difference
operations.operations.
2.2. However, theyHowever, they must havemust have the same number ofthe same number of
attributes and the domains for correspondingattributes and the domains for corresponding
attributes must be identical.attributes must be identical.

Other things to rememberOther things to remember
 Union, Intersection and difference operatorsUnion, Intersection and difference operators
may only be applied to Union Compatiblemay only be applied to Union Compatible
relations.relations.
 Union and Intersection are commutativeUnion and Intersection are commutative
operationsoperations
RUS= SURRUS= SUR
R∩S = S∩RR∩S = S∩R
 Difference operation isDifference operation is NOTNOT commutative.commutative.
R - S not equal S - RR - S not equal S - R

Class PerformanceClass Performance
 Find out RUTFind out RUT
 Find out R∩TFind out R∩T
 Find out R-TFind out R-T
 Find out T-RFind out T-R

Cartesian ProductCartesian Product
 It provides all the combinations of rows from two tables.It provides all the combinations of rows from two tables.
 If A and B are 2 tables and A has 3 rows and B has 4 rows, then AIf A and B are 2 tables and A has 3 rows and B has 4 rows, then A×B is-×B is-
A1-B1A1-B1
A1-B2A1-B2
A1-B3A1-B3
A1-B4A1-B4
A2-B1A2-B1
A2-B2A2-B2
A2-B3A2-B3
A2-B4A2-B4
A3-B1A3-B1
A3-B2A3-B2
A3-B3A3-B3
A3-B4A3-B4

Cartesian Product (continued)Cartesian Product (continued)

More ProductMore Product

SelectionSelection
 Selection and Projection areSelection and Projection are unaryunary operators.operators.
 The selection operator isThe selection operator is sigma (sigma (σσ))
 The selection operation acts like a filter on aThe selection operation acts like a filter on a
relation byrelation by returning only a certain number ofreturning only a certain number of
rows.rows.
 TheThe resulting relation will have the sameresulting relation will have the same
number of attributesnumber of attributes as the original relation.as the original relation.

Selection (continued)Selection (continued)
 The resulting relationThe resulting relation maymay have fewer rows thanhave fewer rows than
the original relation.the original relation.
 The rows to be returned are dependent on aThe rows to be returned are dependent on a
condition that is part of the selection operator.condition that is part of the selection operator.
 σσCC (R) Returns only those rows in R that satisfy(R) Returns only those rows in R that satisfy
conditioncondition CC
 A condition C can be made up of anyA condition C can be made up of any
combination of comparison or logical operatorscombination of comparison or logical operators
that operate on the attributes of R.that operate on the attributes of R.

Selection ExampleSelection Example
 We have a table named EMP.We have a table named EMP.
 We will play with this table named EMP for a while!We will play with this table named EMP for a while!

Selection Example (continued)Selection Example (continued)
 Select only those Employees in the CS departmentSelect only those Employees in the CS department
σσ Dept=‘CS’Dept=‘CS’ (EMP)(EMP)

 Select only those Employees with last name Smith whoSelect only those Employees with last name Smith who
are assistant professorsare assistant professors
σσ Name=‘Smith’Name=‘Smith’ ΛΛ Rank=‘Assistant’Rank=‘Assistant’ (EMP)(EMP)

 Select only those Employees who are either AssistantSelect only those Employees who are either Assistant
Professors or in the Economics departmentProfessors or in the Economics department
σσ Dept =‘Econ’V Rank=‘Assistant’Dept =‘Econ’V Rank=‘Assistant’ (EMP)(EMP)

 Select only those Employees who are not in the CSSelect only those Employees who are not in the CS
department or Adjunctsdepartment or Adjuncts
σσ ¬¬(Dept=‘CS’ V Rank=‘Adjunct’)(Dept=‘CS’ V Rank=‘Adjunct’) (EMP)(EMP)

More SelectionMore Selection

 σσ(Rank='Adjunct'(Rank='Adjunct' ΛΛ Dept ='CS')Dept ='CS') (EMP)(EMP)
 σσ Rank= 'Associate'Rank= 'Associate' ΛΛ Dept ='CS'Dept ='CS' EMP )EMP )
 σσ Dept= 'CS'Dept= 'CS' VV Rank='Associate'Rank='Associate' EMP )EMP )
 σσ Age>26Age>26 (R U S)(R U S)

ProjectionProjection
 Projection is also aProjection is also a UnaryUnary operator.operator.
 The Projection operator isThe Projection operator is pi (pi (ππ))
 Projection limits the columns that will be returned from theProjection limits the columns that will be returned from the
original relation.original relation.
 The general syntax is:The general syntax is: ππ attributesattributes RR
Where attributes is the list of attributes to be displayed and R isWhere attributes is the list of attributes to be displayed and R is
the relation.the relation.
 The resulting table will have the same number of rows as theThe resulting table will have the same number of rows as the
original relation (unless there are duplicate rows produced).original relation (unless there are duplicate rows produced).
 The number of columns of the resulting relation may be equal toThe number of columns of the resulting relation may be equal to
or less than that of the original relation.or less than that of the original relation.

Projection ExampleProjection Example
 Project only the names and departments of theProject only the names and departments of the
employeesemployees
ππ name,deptname,dept (EMP)(EMP)

Combining selection &Combining selection &
projectionprojection
 Show the names of all employees working in the CSShow the names of all employees working in the CS
departmentdepartment
ππ namename ((σσ Dept ='CS'Dept ='CS' (EMP) )(EMP) )

Combining selection & projectionCombining selection & projection
(continued)(continued)
 Show the name and rank of those Employees who areShow the name and rank of those Employees who are
not in the CS department or Adjunctsnot in the CS department or Adjuncts
ππ name, rankname, rank (( σσ ¬¬(Rank='Adjunct' V Dept='CS')(Rank='Adjunct' V Dept='CS') (EMP) )(EMP) )

More ProjectionMore Projection

 ππ name, rankname,rank (( σσ ¬¬(Rank='Adjunct'(Rank='Adjunct' ΛΛ Dept ='CS')Dept ='CS') (EMP) )(EMP) )
 ππ fname,agefname,age ((σσ Age>22Age>22 (R U S) )(R U S) )
 σσ office>300office>300 (( ππ name,rankname,rank (EMP))(EMP))

Aggregate FunctionsAggregate Functions
 We can also apply Aggregate functions to columnsWe can also apply Aggregate functions to columns
and rowsand rows
1.1. SUMSUM
2.2. MINIMUMMINIMUM
3.3. MAXIMUMMAXIMUM
4.4. AVERAGE, MEAN, MEDIANAVERAGE, MEAN, MEDIAN
5.5. COUNTCOUNT
 Aggregate functions are sometimes written using theAggregate functions are sometimes written using the
Projection operator or theProjection operator or the Tau character (Tau character (ττ))

Aggregate Functions (continued)Aggregate Functions (continued)
 Find the minimum SalaryFind the minimum Salary
ττ MIN(salary)MIN(salary) (EMP)(EMP)
 Find the average SalaryFind the average Salary
ττ AVG(salaryAVG(salary) (EMP)) (EMP)
 Count the number of employees in the CS departmentCount the number of employees in the CS department
ττ COUNT(name)COUNT(name) (( σσ Dept='CS'Dept='CS' (EMP) )(EMP) )
 Find the total payroll for the Economics departmentFind the total payroll for the Economics department
ττ SUM(salary)SUM(salary) ((σσ Dept='Econ'Dept='Econ' (EMP) )(EMP) )

JoinJoin
 Join operations bring together two tables and combineJoin operations bring together two tables and combine
their rows and columns in a specific fashion.their rows and columns in a specific fashion.
 The generic join operator called theThe generic join operator called the Theta JoinTheta Join isis
 It takes as arguments the columns from the two tablesIt takes as arguments the columns from the two tables
that are to be joined.that are to be joined.
 The join condition can beThe join condition can be
 When the join condition operator is When the join condition operator is == then we call then we call
this an Equijointhis an Equijoin
 Note that the columns in common are repeated.Note that the columns in common are repeated.

Join (continued)Join (continued)
 Now, we have two tables Emp and depart.Now, we have two tables Emp and depart.
 We will play with these 2 tables for a while!We will play with these 2 tables for a while!

 Find all information on every employee including theirFind all information on every employee including their
department infodepartment info
EMPEMP emp.Dept=depart.Deptemp.Dept=depart.Dept DEPARTDEPART

 Take a look, one table has 5 rows, the other has 4 rows,Take a look, one table has 5 rows, the other has 4 rows,
the resultant has 5 rows!the resultant has 5 rows!

 Find all information on every employee including departmentFind all information on every employee including department
info where the employee works in an office numbered less thaninfo where the employee works in an office numbered less than
the department main officethe department main office
EMPEMP (emp.office<depart.mainoffice)(emp.office<depart.mainoffice) ΛΛ (emp.dept =depart.dept)(emp.dept =depart.dept) DEPARTDEPART

Natural JoinNatural Join
 Notice in the generic (Theta) join operation, anyNotice in the generic (Theta) join operation, any
attributes in common (such as dept above) areattributes in common (such as dept above) are
repeated.repeated.
 The Natural Join operation removes theseThe Natural Join operation removes these
duplicate attributes.duplicate attributes.
 The natural join operator is:The natural join operator is: **
 We can also assume usingWe can also assume using ** that the jointhat the join
condition will becondition will be == on the two attributes inon the two attributes in
common.common.

Natural Join (continued)Natural Join (continued)
 Emp * departEmp * depart
 There are 2 Dept columns, one is omitted!There are 2 Dept columns, one is omitted!

Outer JoinOuter Join
 In the Join operations so far, only those tuples from bothIn the Join operations so far, only those tuples from both
relations that satisfy the join condition are included in therelations that satisfy the join condition are included in the
output relation.output relation.
 The Outer join includes other rows as well according to a fewThe Outer join includes other rows as well according to a few
rules.rules.
 Three types of outer joins:Three types of outer joins:
1.1. Left Outer Join includes all rows in the left hand relation andLeft Outer Join includes all rows in the left hand relation and
includes only those matching rows from the right hand relation.includes only those matching rows from the right hand relation.
2.2. Right Outer Join includes all rows in the right hand relationRight Outer Join includes all rows in the right hand relation
and includes only those matching rows from the left handand includes only those matching rows from the left hand
relation.relation.
3.3. Full Outer Join includes all rows in the left hand relation andFull Outer Join includes all rows in the left hand relation and
from the right hand relation.from the right hand relation.

Outer Join (continued)Outer Join (continued)
 Now we have another 2 tables.Now we have another 2 tables.
 We will play with them for next few slides!We will play with them for next few slides!

Left Outer JoinLeft Outer Join
PEOPLEPEOPLE people.food=menu.foodpeople.food=menu.food MENUMENU

Right Outer JoinRight Outer Join

Full Outer JoinFull Outer Join

Outer UnionOuter Union
 In this case, just go from row of table 1 to row of table 2.In this case, just go from row of table 1 to row of table 2.
 Take common columns just once.Take common columns just once.
 if matching criteria don’t match, then put NULL, else include inif matching criteria don’t match, then put NULL, else include in
the tablethe table
 Repeat the same but this time go from row of table 2 to row ofRepeat the same but this time go from row of table 2 to row of
table 1table 1

ReferencesReferences
 Database Management Systems by Prof.Database Management Systems by Prof.
Holowczak, Zicklin School of Business,Holowczak, Zicklin School of Business,
Baruch College, City University of NewBaruch College, City University of New
YorkYork
 Database Systems: Design, ImplementationDatabase Systems: Design, Implementation
& Management by Rob & Coronel, 6& Management by Rob & Coronel, 6thth
EditionEdition

Relational Algebra Guide for Database Systems

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