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Database Systems &Applications

Lec 11Relational Algebra

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Algebra

• Algebra is 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.

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Algebra

• Operands are relations or variables that represent

relations.

• Operators are designed to do the most common things

that we need to do with relations in a database.

– The result is an algebra that can be used as a query

language for relations.

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Relational Algebra

• There is a core relational algebra that has traditionally

been thought of as the relational algebra.

•• But there are several other operators we shall add to the

core in order to model better the SQL language --- the

principal language used in relational database systems.

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Core Relational Algebra

• Union, Intersection, and Difference.

– Usual set operations, but require both operands have

the same relation schema.

• Selection: picking certain rows.

• Projection: picking certain columns.

• Products and joins: compositions of relations.

• Renaming of relations and attributes.

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Set Operations

• Note: R and S same relation schema. (i.e.,

the structure of R and S is same)• Union: R S . The union of R and S, is the set of

elements that are in R or S or both.

Note: an element appears only once in the union even

it is present in both R and S.• Intersection: R S . The intersection of R and S, is the

set of elements that both in R and S.

• Difference: R – S . The difference of R and S, is the set

of elements that are in R but not in S.

Note: R – S is different from S – R. S – R is the set of

elements that are in S but not in R.

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scope of set operationslimited to union compatibleunion compatible

relationsrelations

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Union Compatible Relations

• Two relations are union compatibleunion compatible if

–Both have same number of columns

– Names of attributes are the same in both

– Attributes with the same name in both

relations have the same domain.

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Example

Tables: PersonPerson (P-ID, Name, Address, Hobby )

ProfessorProfessor (Id, Name, Office, Phone)

are not union compatible.

But

π Name

(PersonPerson) and π Name

(ProfessorProfessor)

are union compatible so

π Name

(PersonPerson) - π Name

(ProfessorProfessor)

makes sense.

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Example

• Student(ID,Name,DOB,Address)

• Faculty (psrn,Name,Courses)

π Name

(Student) U π Name

(Faculty) ?

π Name

(Student) π Name

(Faculty) ?

π Name

(Student) – π Name

(Faculty) ?

π Name

(Faculty) – π Name

(Student ) ?

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Selection

• R1 := σ C (R2)

– C is a condition (as in “if” statements) that

refers to attributes of R2.

– R1 is all those tuples of R2 that satisfy C .

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Example 1• Relation Registers

• 2004P7PS001Menu := σ st_id=“2004P7PS001” (Registers):

st_id c_no sem year

2009P7PS001 CSGC352 2 2009-10

2009P7PS020 CSGC352 2 2009-10

2009P7PS001 CSGC342 2 2009-10

2009P7PS005 CSGC352 2 2009-10

st_id c_no sem year2009P7PS001 CSGC352 2 2009-10

2009P7PS001 CSGC342 2 2009-10

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Example 2

•Produce table containing subset of rows of argument table satisfying condition

σcondition (relation)

Example:

PersonPerson σHobby =‘stamps’(PersonPerson)

1123 John 123 VSP stamps1123 John 123 VSP coins

5556 Mary 7 VJY reading

9876 Samu 5 HYD stamps

1123 John 123 VSP stamps9876 Samu 5 HYD stamps

Id Name Address Hobby Id Name Address Hobby

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Selection Condition

• Operators: <, ≤ , ≥ , >, =, ≠

• Simple selection condition:

– <attribute> operator <constant >

– <attribute> operator <attribute>

• <condition> AND <condition>

• <condition> OR <condition>

• NOT <condition>

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Selection Condition - Examples

∀σ Id>3000 OR Hobby=‘reading’ (PersonPerson)

∀σ Id>3000 AND Hobby=‘stamps’ (PersonPerson)

∀σ NOT(Hobby=‘reading’) (PersonPerson)

∀σ Hobby ≠ ‘reading’ (PersonPerson)

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Projection

• R1 := π L (R2)

– L is a list of attributes from the schema of R2.

– R1 is constructed by looking at each tuple of R2,

extracting the attributes on list L, in the order

specified, and creating from those components a

tuple for R1.

– Eliminate duplicate tuples, if any.

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Example

• Relation Registers

• courses := π c_no,sem,year (Registers):

st_id c_no sem year

2009P7PS001 CSGC352 2 2009-10

2009P7PS020 CSGC352 2 2009-10

2009P7PS001 CSGC342 2 2009-10

2009P7PS005 CSGC352 2 2009-10

c_no sem yearCSGC352 2 2009-10

CSGC342 2 2009-10

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Project Operator

•Produces table containing subset of columns of argument table

πattribute list (relation)

•Example

•PersonPerson πName,Hobby (PersonPerson)

1123 John 123 VSP stamps1123 John 123 VSP coins

5556 Mary 7 VJY reading

9876 Samu 5 HYD stamps

John stamps John coinsMary readingSamu stamps

Id Name Address Hobby Name Hobby

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Project Operator

1123 John 123 VSP stamps

1123 John 123 VSP coins5556 Mary 7 VJY reading9876 Samu 5 HYD stamps

John 123 VSPMary 7 VJYSamu 5 HYD

Result is a table (no duplicates); can have fewer tuples thanthe original

Id Name Address Hobby Name Address

• Example:

PersonPerson πName,Address(PersonPerson)

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Expressions

1123 John 123 VSP stamps1123 John 123 VSP coins5556 Mary 7 VJY reading9876 Samu 5 HYD stamps

1123 John5556 Mary

Id Name Address Hobby Id Name

PersonPerson

ResultResult

π Id, Name

(σ Hobby=’reading’ OR Hobby=’coins’

(PersonPerson) )

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Product

• R3 := R1 × R2

– Pair each tuple t1 of R1 with each tuple t2 of R2.

– Concatenation t1t2 is a tuple of R3.

– Schema of R3 is the attributes of R1 and then R2, in

order.

– But beware attribute A of the same name in R1 and

R2: use R1. A and R2. A.

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ExampleRegisters:(R) Students: (S)

st_id c_no sem year st_id st_name

2009P7PS001 CSGC342 2 2009-10 2009P7PS001 Amit

2009P7PS001 CSGC352 2 2009-10 2009P7PS020 Piyush

2009P7PS020 CSGC342 2 2009-10 2009P7PS050 Sunil

2009P7PS005 CSGC352 2 2009-10

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Example (cont.) T := R × S

R.st_id c_no sem year S.st_id st_name

2009P7PS001 CSGC342 2 2009-10 2009P7PS001 Amit2009P7PS001 CSGC342 2 2009-10 2009P7PS020 Piyush


2009P7PS001 CSGC352 2 2009-10 2009P7PS001 Amit

2009P7PS001 CSGC352 2 2009-10 2009P7PS020 Piyush2009P7PS001 CSGC352 2 2009-10 2009P7PS050 Sunil

2009P7PS020 CSGC342 2 2009-10 2009P7PS001 Amit



2009P7PS005 CSGC352 2 2009-10 2009P7PS001 Amit2009P7PS005 CSGC352 2 2009-10 2009P7PS020 Piyush


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Theta-Join

• R3 := R1 c R2

– Take the product R1 × R2.

– Then apply σ C to the result.

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2009P7PS001 CSGC342 2 2009-10 2009P7PS001 Amit



2009P7PS005 CSGC352 2 2009-10

StudentsInfo := Registers JOIN Registers.st_id = Student.st_id Students

StudentsInfo:

R.st_id c_no sem year S.st_id st_name2009P7PS001 CSGC342 2 2009-10 2009P7PS001 Amit

2009P7PS001 CSGC352 2 2009-10 2009P7PS001 Amit


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2009P7PS001 CSGC342 2 2009-10 2004P7PS001 Amit



2009P7PS005 CSGC352 2 2009-10

StudentsInfo := Registers JOIN StudentsStudentsInfo:

st_id c_no sem year st_name2009P7PS001 CSGC342 2 2009-10 Amit

2009P7PS001 CSGC352 2 2009-10 Amit

2009P7PS020 CSGC342 2 2009-10 Piyush

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Renaming

• The RENAME operator gives a new schema to a

relation.

• R1 := RENAMER1(A1,…,An)(R2) makes R1 be a relation with attributes A1,…,An and the same tuples as R2.

• Simplified notation: R1(A1,…,An) := R2.

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Example

• Courses:

• R(c_code, c_title, credits) := Courses

• R:

c_no c_name units

CSGC342 Operating Systems 3

CSGC352 Database Systems 3

c_code c_title creditsCSGC342 Operating Systems 3

CSGC352 Database Systems 3

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