RDBMS Algebra

Mathematical Relations
For two sets D1 and D2, the Cartesian product, D1 X D2 ,
is the set of all ordered pairs in which the first element
is from D1 and the second is from D2
Example
D1 = {1,3} and D2 = {a,b,c}
D1 X D2 = {(1,a), (1,b), (1,c), (3,a), (3,b), (3,c)}
A relation is any subset of the Cartesian productA relation is any subset of the Cartesian product
Example
R = {(x,y) | x ∈ D1, y ∈ D2, and y = a}
R = {(1,a), (3,a)}
Could form Cartesian product of 3 sets; relation is any
subset of the ordered triples so formed
Could extend to n sets, using n-tuples

Database Relations
• A relation schema, R, is a set of attributes A1, A2,…,An with
their domains D1, D2,…Dn
• A relation r on relation schema R is a set of mappings from
the attributes to their domains
• r is a set of n-tuples (A1:d1, A2:d2, …, An:dn) such that d1ε D1,
d2 εD2 , …, dn εDn
– Mapping the data in domain Dn to the attribute An
• In a table to represent the relation, list the Ai as column
headings, and let the (d1, d2, …dn) become the n-tuples, the
rows of the table
• The name of the attribute becomes the name of the
column

Relational Algebra
• Theoretical language with operators that
apply to one or two relations to produce
another relation
• Both operands and results are tables• Both operands and results are tables
• Can assign name to resulting table (rename)
• SELECT, PROJECT, JOIN allow many data
retrieval operations

SELECT Operation
• Applied to a single table, returns rows that meet a specified
predicate, copying them to new table
• Returns a horizontal subset of original table
SELECT tableName WHERE condition [GIVING newTableName]SELECT tableName WHERE condition [GIVING newTableName]
Symbolically, [newTableName = ] σ predicate (table-name)
• Predicate is called theta-condition, as in σθ(table-name)
• Result table is horizontal subset of operand
• Predicate can have operators <, <=, >, >=, =, <>, ∧(AND),
∨(OR), ¬ (NOT)

SELECT Example
SELECT Student WHERE stuId = ‘S1013’ GIVING Result
Result = σ STDID=‘S1013’ (Student)
Student ResultStudent
stuId lastName firstName major credits
S1001 Smith Tom History 90
S1002 Chin Ann Math 36
S1005 Lee Perry History 3
S1010 Burns Edward Art 63
S1013 McCarthy Owen Math 0
S1015 Jones Mary Math 42
S1020 Rivera Jane CSC 15
Result

SELECT Example
SELECT Class WHERE room = ‘H225’ GIVING Answer
Answer= σ ROOM=‘H225’ (Class)
Class AnswerClass
classNumber facId schedule room
ART103A F101 MWF9 H221
CSC201A F105 TuThF10 M110
CSC203A F105 MThF12 M110
HST205A F115 MWF11 H221
MTH101B F110 MTuTh9 H225
MTH103C F110 MWF11 H225
Answer
MTH103C F110 MWF11 H225

PROJECT Example
PROJECT Student OVER major GIVING Temp
Temp = Π major (Student)
Student
Temp
major
History
MathS1002 Chin Ann Math 36
Math
Art
CSC

PROJECT Example
PROJECT Class OVER (facId, room)
Π facId, room (Class)
Class
facId room
F101 H221
F105 M110CSC201A F105 TuThF10 M110
HST205A F115 MWF11 H221
MTH103C F110 MWF11 H225
F105 M110
F115 H221
F110 H225

PROJECT and SELECT Combination Example
Want the names and student IDs of History majors
SELECT Student WHERE major = ‘History’ GIVING Temp
PROJECT Temp OVER (lastName, firstName, stuId) GIVING Result
Student
Temp
Temp
Result
lastName firstName stuId
Smith Tom S1001
Lee Perry S1005

Product and Theta-join
• Binary operations – apply to two tables
• Product: Cartesian product – cross-product of
A and B – A TIMES B, written A x B
– all combinations of rows of A with rows of B– all combinations of rows of A with rows of B
– Degree of result is deg of A + deg of B
– Cardinality of result is (card of A) * (card of B)
• THETA join: TIMES followed by SELECT
A |x|θ B = σθ(A x B)

PRODUCT Example
Student X Enroll (Produces 63 tuples)
Student
Enroll
stuId classNumber grade
S1001 ART103A A
S1001 HST205A C
S1002 ART103A D
S1002 CSC201A F
S1002 MTH103C B
S1010 ART103A
S1010 MTH103C
X
S1020 Rivera Jane CSC 15 S1010 MTH103C
S1020 CSC201A B
S1020 MTH101B A
Student.stuId lastName firstName major credits Enroll.stuId classNumber grade
S1001 Smith Tom History 90S1001 ART103A A
S1001 Smith Tom History 90S1001 HST205A C
S1001 Smith Tom History 90S1002 ART103A D
S1001 Smith Tom History 90S1002 CSC201A F
S1001 Smith Tom History 90S1002 MTH103C B
S1001 Smith Tom History 90S1010 ART103A
S1001 Smith Tom History 90S1010 MTH103C
S1001 Smith Tom History 90S1020 CSC201A B
S1001 Smith Tom History 90S1020 MTH101B A
S1002 Chin Ann Math 36S1001 ART103A A
… … … … …… … …

JOIN Example
Faculty JOIN Class or Faculty |X| Class
Faculty
facId name department rank
F101 Adams Art Professor
F105 Tanaka CSC Instructor
F110 Byrne Math Assistant
F115 Smith History Associate
F221 Smith CSC Professor
Class
HST205A F115 MWF11 H221
|X|
MTH103C F110 MWF11 H225
facId name department rank class No schedule room
F101 Adams Art Professor ART103A MWF9 H221
F105 Tanaka CSC Instructor CSC203A MThF12 M110
F105 Tanaka CSC Instructor CSC201A TuThF10 M110
F110 Byrne Math Assistant MTH103C MWF11 H225
F110 Byrne Math Assistant MTH101B MTuTh9 H225
F115 Smith History Associate HST205A MWF11 H221

NATURAL JOIN Operation
• Requires two tables with common column(s)
(at least with same domain)
• combines the matching rows-rows of the two
tables that have the same values in thetables that have the same values in the
common column(s)
• symbolized by |x| as in
– [newTableName = ] Student |x| Enroll, or
– Student JOIN Enroll [GIVING newTableName]

Equi-join and Natural Join
• EQUIJOIN formed when θ is equality on
column common to both tables (or
comparable columns)
• The common column appears twice in the• The common column appears twice in the
result
• Eliminating the repeated column in the result
gives the NATURAL JOIN, usually called just
JOIN
A |x| B

NATURAL JOIN Example
Student JOIN Enroll or Student |X| Enroll
Equivalent to:
Student TIMES Enroll GIVING Temp3
SELECT Temp3 WHERE Student.stuId = Enroll.stuId GIVING Temp4
PROJECT Temp4 OVER (Student.stuId, lastName, firstName, major, credits, classNumber,
grade)
Student
Enroll
S1001 ART103A A
S1001 HST205A C
S1002 ART103A D
S1002 CSC201A F
JOINS1010 Burns Edward Art 63
S1002 CSC201A F
S1002 MTH103C B
S1010 ART103A
S1010 MTH103C
S1020 CSC201A B
S1020 MTH101B A
JOIN
stuId lastName firstName major credits classNumber grade
S1001 Smith Tom History 90HST205A C
S1001 Smith Tom History 90ART103A A
S1002 Chin Ann Math 36MTH103C B
S1002 Chin Ann Math 36CSC201A F
S1002 Chin Ann Math 36ART103A D
S1010 Burns Edward Art 63MTH103C
S1010 Burns Edward Art 63ART103A
S1020 Rivera Jane CSC 15MTH101B A
S1020 Rivera Jane CSC 15CSC201A B

EQUIJOIN Example
Student EQUIJOIN Enroll (Produce 9 tuples)
Equivalent to:
Student TIMES Enroll GIVING Temp3
SELECT Temp3 WHERE Student.stuId = Enroll.stuId
Student
Enroll
S1001 ART103A A
S1001 HST205A C
S1002 ART103A D
S1002 CSC201A F
EQUIJOIN
Student X Enroll
produces 63 tuples
S1002 CSC201A F
S1002 MTH103C B
S1010 ART103A
S1010 MTH103C
S1020 CSC201A B
S1020 MTH101B A
EQUIJOIN
stuId lastName firstName major credits Enroll.stuId classNumber grade
S1001 Smith Tom History 90S1001 HST205A C
S1001 Smith Tom History 90S1001 ART103A A
S1002 Chin Ann Math 36S1002 MTH103C B
S1002 Chin Ann Math 36S1002 CSC201A F
S1002 Chin Ann Math 36S1002 ART103A D
S1010 Burns Edward Art 63S1010 MTH103C
S1010 Burns Edward Art 63S1010 ART103A
S1020 Rivera Jane CSC 15S1020 MTH101B A
S1020 Rivera Jane CSC 15S1020 CSC201A B
Final result
Just 9 tuples

More Complicated Queries
Find classes and grades of student Ann Chin
SELECT Student WHERE lastName=‘Chin’ AND
firstName=‘Ann’ GIVING Temp1
Temp1 JOIN Enroll GIVING Temp2
PROJECT Temp2 OVER (classNo, grade) GIVING Answer
ΠclassNo,grade((σlastName=‘Chen’ ^ firstName=‘Ann’(Student))|X|Enroll)

More Complicated Queries (cont.)
SELECT Student WHERE lastName=‘Chin’ AND firstName=‘Ann’ GIVING Temp1
Student
Temp1
Temp1 JOIN Enroll GIVING Temp2
Enroll
S1001 ART103A A
S1001 HST205A CS1001 HST205A C
S1002 ART103A D
S1002 CSC201A F
S1002 MTH103C B
S1010 ART103A
S1010 MTH103C
S1020 CSC201A B
S1020 MTH101B A
Temp2
stuId lastName firstName major credits classNumber grade
S1002 Chin Ann Math 36ART103A D
S1002 Chin Ann Math 36CSC201A F
S1002 Chin Ann Math 36MTH103C B
PROJECT Temp2 OVER (classNo, grade) GIVING
Answer
Answer
classNumber grade
ART103A D
CSC201A F
MTH103C B

More Complicated Queries (cont.)
Since we only need the stuId from Temp1, we can do a
PROJECT on Temp1 before we make the JOIN
ΠclassNo,grade(ΠstuId(σlastName=‘Chen’ ^ firstName=‘Ann’(Student))|X|Enroll)
A third way is:A third way is:
JOIN Student, Enroll GIVING Tempa
SELECT Tempa WHERE lastName=‘Chin’ AND firstName = ‘Ann’
GIVING Tempb
PROJECT Tempb Over (classNo, grade)
But it requires more work by the JOIN (produces 54 tubles), so is
less efficient

Semijoins and Outerjoins
• Left semijoin A|x B is formed by finding
A|x|B and projecting result onto attributes of A
• Result is tuples of A that participate in the join
• Right semijoin A x| B defined similarly; tuples of B that
participate in join
• Outerjoin is formed by adding to the join those tuples that• Outerjoin is formed by adding to the join those tuples that
have no match, adding null values for the attributes from the
other table e.g.
A OUTER- EQUIJOIN B
consists of the equijoin of A and B, supplemented by the
unmatched tuples of A with null values for attributes of B and
the unmatched tuples of B with null values for attributes of A
• Can also form left outerjoin or right outerjoin

Semijoins Example
Student LEFT-SEMIJOIN Enroll or Student |X Enroll
Student
Enroll
S1001 ART103A A
S1001 HST205A C
S1002 ART103A D
S1002 CSC201A F
S1002 MTH103C B
S1010 ART103A
S1010 MTH103C
S1020 CSC201A B
S1020 MTH101B A
|X
stuId lastName firstName major
S1001 Smith Tom History
S1002 Chin Ann Math
S1010 Burns Edward Art
S1020 Rivera Jane CSC

Outerjoin Example
Student OUTER-EQUIJOIN Faculty
Compare Student.lastName with Faculty.name
Student
Faculty
OUTERJOIN
stuId lastName firstName major credits facId name department rank
S1001 Smith Tom History 90 F221 Smith CSC Professor
S1001 Smith Tom History 90 F115 Smith History Associate

Left-Outer-Equijoin Example
Student LEFT-OUTER-EQUIJOIN Faculty
Student
Faculty
LEFT-OUTER-EQUIJOIN

Right-Outer-Equijoin Example
Student RIGHT-OUTER-EQUIJOIN Faculty
Student
Faculty
LEFT-OUTER-EQUIJOIN

Division
• Binary operator where entire structure of one
table (divisor) is a portion of structure of the
other (dividend)
• Result is projection onto those attributes of• Result is projection onto those attributes of
the dividend that are not in the divisor of the
tuples of dividend that appear with all the
rows of the divisor
• It tells which values of those attributes appear
with all the values of the divisor

Divide Example
Club DIVIDED BY Stu or Club ÷÷÷÷ Stu
Club
ClubName StuNumb StuLastName
Computing S1001 Smith
Computing S1002 Chin
Stu
StuNumb StuLastName
S1001 Smith
S1002 Chin
S1005 Lee
Computing S1002 Chin
Drama S1001 Smith
Drama S1002 Chin
Drama S1005 Lee
Karate S1001 Smith
Karate S1002 Chin
Karate S1005 Lee
Club
DIVIDED
BY Stu
ClubName
Drama
Karate

Set Operations
• Tables must be union compatible – have same
basic structure
– Have the same degree
– Attributes in corresponding position have same– Attributes in corresponding position have same
domain
• Third column in first relation must have same domain
as third column in second relation

Set Operations
• A UNION B: set of tuples in either or both of A
and B, written A ∪ B
• A INTERSECTION B: set of tuples in both A and
B simultaneously, wriƩen A ∩ BB simultaneously, wriƩen A ∩ B
• Difference or A MINUS B: set of tuples in A but
not in B, written A - B

Set Operations (cont.)
• A UNION B: set of tuples in either or both of A and B,
written A ∪ B
– Remove duplicates
MainFac
FacID name department rank
F101 Adams Art Processor
F221 Smith CSC Processor
BarnchFac
F110 Byre Math Assistant
∪
F221 Smith CSC Processor F115 Smith History Associate
MainFac UNION BarnchFac

• A INTERSECTION B: set of tuples in both A and B
simultaneously, wriƩen A ∩ B
MainFac
BranchFac
∩
F221 Smith CSC Processor F115 Smith History Associate
MainFac INTERSECTION BranchFac

• Difference or A MINUS B: set of tuples in A but not in B, written A – B
• MainFac - BranchFac
MainFac
BranchFac
MainFac MINUS BranchFac
-
BranchFac - MainFac
MainFac
BranchFac
MainFac MINUS BranchFac
-

Relational Calculus
• Formal non-procedural language
• Two forms: tuple-oriented and domain-
oriented
• Based on predicate calculus• Based on predicate calculus

Tuple-oriented predicate calculus
• Uses tuple variables, which take tuples of
relations as values
• Query has form {S  P(S)}
– S is the tuple variable, stands for tuples of relation– S is the tuple variable, stands for tuples of relation
– P(S) is a formula that describes S
– Means “Find the set of all tuples, s, such that P(S)
is true when S=s.”
• Limit queries to safe expressions – test only
finite number of possibilities

Example Tuple-oriented Relational Calculus
Example 1: Find the names of all faculty members who are Professors in CSC
Department
{F.name | F ∈ Faculty ^ F.department = ‘CSC’ ^ F.rank = ‘Professor’}
Example 2: Find the last names of all students enrolled in CSC201A
{S.lastName | S ∈ Student ^ EXISTS E (E ∈ Enroll ^ E.stuId = S.stuId ^
E.classNo = ‘CSC201A)}E.classNo = ‘CSC201A)}
Example 3: Find the first and last names of all students who are enrolled in at
least one class that meets in room H221.
{S.firstName, S.lastName | S ∈ Student ^ EXISTS E(E ∈ Enroll ^ E.stuId =
S.stuId ^ EXISTS C (C ∈ Class ^ C.classNo = E.classNo ^ C.room = ‘H221’))}

Domain-oriented predicate calculus
• Uses variables that take their values from domains
• Query has form
{<x1,x2,...,xn>  P(x1,x2,...,xm )}
– x1,x2,...,xn are domain variables
– P(x1,x2,...,xm) is a predicate– P(x1,x2,...,xm) is a predicate
– n<=m
– Means set of all domain variables x1,x2,...,xn for which
predicate P(x1,x2,...,xm) is true
• Predicate must be a formula
• Often test for membership condition, <x,y,z >∈ X

Example Domain-oriented Relational Calculus
• Example 1: Find the name of all faculty members who are professors in
CSC Department
– {LN | ∃ FI, DP, RK (<F1, LN, DP, RK> ∈ Faculty ^ RK = ‘Professor’ ^ DP = ‘CSC’)}
• Example 2: Find the last names of all students enrolled in CSC201A.
– {LN | ∃ SI, FN, MJ, CR (<SI, LN, FN, MJ, CR> ∈ Student ^ ∃ CN, GR (<SI, CN, GR>
∈ Enroll ^ CN = ‘CSC201A’))}
• Example 3: Find the first and last names of all students who are enrolled in
at least one class that meets in room H221.at least one class that meets in room H221.
– {FN, LN | ∃ SI, MJ, CR (<SI, LN, FN, MJ, CR> ∈ Student ^ ∃ CN, GR (<SI, CN, GR>
∈ Enroll ^ ∃ FI, SH, RM (<CN, FI, SH, RM> ∈ Class ^ RM = ‘H221’)))}
SI: stuId, LN: lastName, FN: firstName, MJ: major, CR: credits, CN: calssNo, FI: facId,
SH: schedule, RM: room, DP: department, RK: rank, GR: grade

Views
• External models in 3-level architecture are called
external views
• Relational views are slightly different
• Relational view is constructed from existing (base)
tablestables
• View can be a window into a base table (subset)
• View can contain data from more than one table
• View can contain calculated data
• Views hide portions of database from users
• External model may have views and base tables

Views (cont.)
• Original table
– Student(stuId, lastName, firstName, ssn, major, credits)
• Organize into two tables
– PersonalStu(stuId, lastName, firstName, ssn)
– AcademicStu(stuId, major, credits)
• Create the original table as a view using a natural join• Create the original table as a view using a natural join
of PersonalStu and AcademicStu
• Restrict updates to views
– View must contain primary key to be updatable
– Views constructed from summary data are not updatable

Mapping ER to Relational Model
• Each strong entity set becomes a table
• Non-composite, single-valued attributes become attributes of table
• Composite attributes: either make the composite a single attribute or use
individual attributes for components, ignoring the composite
• Multi-valued attributes: remove them to a new table along with the
primary key of the original table; also keep key in original table
• Weak entity sets become tables by adding primary key of owner entity• Weak entity sets become tables by adding primary key of owner entity
• Binary Relationships:
– 1:M-place primary key of 1 side in table of M side as foreign key
– 1:1- make sure they are not the same entity. If not, use either key as foreign
key in the other table
– M:M-create a relationship table with primary keys of related entities, along
with any relationship attributes
• Ternary or higher degree relationships: construct relationship table of
keys, along with any relationship attributes

ER Diagram
Example Department
deptCode
office
deptName
EmploysChairsHasMajor
Offers
Textbook
isbn
title
stuId
lastName
rank
Faculty-
Class-
Textbook
Teaches
publisher
Textbook
author
zip
credit
Student
stuId
major
lastName
firstName
address
number
city
state
street
Faculty
facId
firstName
phone

ER to Relations
Each strong entity set becomes a table
• Table name same as entity name
• Non-composite, single-valued attributes,
represented by single oval, become attributes
of the relationof the relation
• Department(deptName, office, deptCode)
Department
deptCode
office
deptName

ER to Relations
Composite attributes
• An attribute is atomic, it cannot be composite
• Two approaches
– Make each ER attribute an attribute in the relation
– Represent the total address with just one attribute in the
relation
• Cannot search of all students in particular zip• Cannot search of all students in particular zip
• Student(stuId, lastName, firstName, major,
credits, number, street, city, state, zip)
• Student(stuId, lastName, firstName, major,
credits, address)credit
Student
stuId
major
lastName
firstName
address
number
city
state
street
zip

ER to Relations
Multi-valued attributes
• Remove them to a new table along with the
primary key of the original table
• Keep key in original table
• Textbook(isbn, title, publisher)
• Author(isbn, lastName, firstName)
isbn
• Student(stuId, lastName, firstName, major1,
major2, credits, number, street, city, state, zip
Author(isbn, lastName, firstName)
Textbook
isbn
author
title
publisher
zip
credit
Student
stuId
major
lastName
firstName
address
number
city
state
street
• Student(stuId, lastName, firstName,
credits, number, street, city, state, zip
Author(isbn, lastName, firstName)
• StuMajor(stuId, major)

ER to Relations
Multi-valued attributes
• If multiple phone numbers for faculty
• First approach, multiple attributes in one relation
Faculty(facId, lastName, firstName, rank, phone, alternatePhone, street, city,
state, zip)
• Second approach, create a PhoneNumber relation
• Can have primary number, alternate number, cell number, spouse number,
home number, etc.
• Faculty(facId, lastName, firstName, rank, street, city, state, zip)
• PhoneNumber(facId, phoneNumber, typePhone)

ER to Relations
Weak entity
• Weak entity sets become tables by adding primary key of owner
entity
Faculty
• Faculty(facId, lastName, firstName, rank, phone,
alternatePhone, street, city, state, zip)
Evaluation
IsRated
• Evaluation(facId, date, rater, rating)

ER to Relations
Binary Relationships
– 1:M-place primary key of 1 side in table of M side as foreign key
– 1:1- make sure they are not the same entity. If not, use either key as foreign key in the other table
– M:M-create a relationship table with primary keys of related entities, along with any relationship attributes
Faculty
• Faculty(facId, lastName, firstName, rank, phone,
alternatePhone, street, city, state, zip)
Class
Teaches
• Teaches(calssNo, facId)
• Class(classNo, facId, schedule, room)
• Don’t need Teaches relation
1
M

ER to Relations
Ternary or higher degree relationships
– Construct relationship table of keys, along with any relationship
attributes
Faculty
• Faculty(facId, lastName, firstName, rank,
phone, alternatePhone, street, city, state, zip)
• Faculty-Class-Textbook(classNo, isbn, facId)
• Choose classNo, isbn as primary key, because for
each combination of classNo and isbn there is
Class
Faculty-
Class-
Textbook
each combination of classNo and isbn there is
only one faculty member1
M
Textbook
M

Convert ER to Relational Tables
• Department(deptName, office, deptCode)
• Student(stuId, lastName, firstName, major, credits, number,
street, city, state, zip)
• Student(stuId, lastName, firstName, major, credits, address)
//cannot search of all students in particular zip//cannot search of all students in particular zip
• Textbook(isbn, title, publisher)
• Author(isbn, lastName, firstName)
• Faculty(facId, office, deptCode, cellPhone, homePhone,
officePhone) //may have other phone, eg, spousePhone
• Faculty(facId, office, deptCode)
• PhoneNumber(facId, number, typePhone)

RDBMS Algebra

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