relation algebra

7/28/2019 Relation Algebra

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Relational Algebra

The Relational Algebra is used to define the

ways in which relations (tables) can be operated

to manipulate their data.

The basic set of operations for the relationalmodel is the relational algebra.

This Algebra is composed of Unary operations

(involving a single table) and Binary operations

(involving multiple tables).


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Relational Algebra Operators Relational algebra defines the theoretical way of

manipulating table contents using the relational functions

Operators SELECT PROJECT INTERSECT UNION

DIFFERENCE PRODUCT DIVIDE


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Relational AlgebraFundamental or Basic operators

select project

Cartesian product

union

set difference - set intersection

division


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Relational Algebra Operators

SELECTyields values forall attributes found ina table. It yields a horizontal subset of a table.

predicate(R)

PROJECTproduces a list of all values forselected attributes. It yields a vertical subset ofa table. a1an(R)

DIFFERENCEyields all rows in one table thatare not found in the other table; i.e., it subtractsone table from the other. The tables must beunion compatible. R - S


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A PRODUCTproduces a list of all

possible pairs of rows from two tables.

Default is CARTESIAN PRODUCT R X S

DIVIDE requires the use of one single-

column table and one two-column table.

RS


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UNIONcombines all rows from two

tables. The two tables must be union

compatible. R S INTERSECTproduces a listing that

contains only the rows that appear in

both tables. The two tables must beunion compatible. R S


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Relational Algebra: 5 Basic

Operations

Selection ( ) Selects a subset ofrows

fromrelation (horizontal).

Projection ( ) Retains only wanted columns

from relation (vertical).

Cross-product (x) Allows us to combine tworelations.

Set-difference () Tuples in r1, but not in r2.

Union ( ) Tuples in r1 and/or in r2.

Since each operation returns a relation, operations

can be composed! (Algebra is closed.)


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Unary Operations

Selection

Projection


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Selection

Gives a horizontal subset of a relation

A B

a1

a2

b1

b2

R

C D

c1

c2

d1

d2

(B=b2) (R) =

A B

b2

C D

c2 d2a2


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Relational Operators

Selection: horizontal subset of a table

Employee

E-No E-Name D-No

179 Silva 7

857 Perera 4

342 Dias 7

Sales Employee

E-No E-Name D-No

179 Silva 7

342 Dias 7

Sales-Emp = D-No=7 (Employee)


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Projection

Gives a vertical subset of a relation

A B

a1

a2

b1

b2

R

A (R) =A

a1

a2

Duplicate tuples are eliminated


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Projection

Can project more than one column

Duplicate tuples are eliminated

A B

a1

a2

b1

b2

R

C D

c1

c2

d1

d2

B,C (R) =B C

b1

b2

c1

c2


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Projection: vertical subset of a table

Employee

E-No E-Name D-No

179 Silva 7

857 Perera 4342 Dias 7

Employee Names

E-No E-Name

179 Silva

857 Perera342 Dias

Emp-Names = E-No, E-Name (Employee)


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Binary Operations

Cartesian Product


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Cartesian Product

Set of all tuples (r X s) where r tuples

belong to the first relation R and s tuples

belong to the second relation S. Concatenation of every tuple in R with

every tuple in S.


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Cartesian ProductA B

a1

a2

b1

b2

R

C D

c1

c2

c3

d1

d2

d3

SRX S =

A B

a1

a1

a1

a2

a2

a2

b1

b1

b1

b2

b2

b2

C D

c1

c2

c3

c1

c2

c3

d1

d2

d3

d1

d2

d3


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Cartesian Product: Creates a single table from two tables.

D-No D-Name M-No

4 Finance 857

7 Sales 179

DepartmentEmployee

E-No E-Name D-No

179 Silva 7857 Perera 4

342 Dias 7

Emp-Info

E-No E-Name D-No D-No D-Name M-No

179 Silva 7 4 Finance 857

857 Perera 4 4 Finance 857

342 Dias 7 4 Finance 857

179 Silva 7 7 Sales 179857 Perera 4 7 Sales 179

342 Dias 7 7 Sales 179

Emp-Info = Employee E.D-No=D.D-No Department


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Cartesian Product - Example

Students Coursesstud# name course course# name

100 Fred PH PH Pharmacy

200 Dave CM CM Computing

300 Bob CM

Students X Courses =

stud# Students.name course course# Courses.name

100 Fred PH PH Pharmacy

100 Fred PH CM Computing

200 Dave CM PH Pharmacy

200 Dave CM CM Computing

300 Bob CM PH Pharmacy

300 Bob CM CM Computing


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Set OperationsUnion

Intersection

Difference


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Union

Takes the set of rows in each table andcombines them, eliminating duplicates

Participating relations must be compatible, iehave the same number of columns, and the

same column names, domains, and data typesR S R S

A B

a1 b1

a2 b2

A B

a2 b2

a3 b3

A B

a1 b1

a2 b2a3 b3


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Union

S S1 2

S1

S2

sid sname rating age22 dustin 7 45.0

31 lubber 8 55.5

58 rusty 10 35.0

sid sname rating age

28 yuppy 9 35.0

31 lubber 8 55.5

44 guppy 5 35.0

58 rusty 10 35.0

sid sname rating age22 dustin 7 45.0

31 lubber 8 55.5

58 rusty 10 35.0

44 guppy 5 35.028 yuppy 9 35.0


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Intersection

Takes the set of rows that are common toeach relation

Participating relations must be compatible

R S R SA B

a1 b1

a2 b2

A B

a2 b2

a3 b3

A B

a2 b2


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Intersection

R n S =

A B

a1

a2

b1

b2

S

A B

a1

a2

b1

b2

A B

A1

a2a3

b1

b2b3

R


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DivisionA Ba

a

b

e

e

a

b

b

c

d

d

b

C D

c

e

e

c

e

d

d

f

f

d

f

e

R

C D

c

e

d

f

SA B

a

e

b

dRS =..-


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DivisionA B

a

a

b

e

e

a

b

b

c

d

d

b

C D

c

e

e

c

e

d

d

d

f

d

f

e

R

C

c

e

S

RS = bb3

A B D

d

d3

a

a3

b..-


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Intersection

S1

S2

S S1 2


31 lubber 8 55.558 rusty 10 35.0


22 dustin 7 45.0

31 lubber 8 55.5

58 rusty 10 35.0


28 yuppy 9 35.0

31 lubber 8 55.5

44 guppy 5 35.0

58 rusty 10 35.0


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Difference

Takes the set of rows in the first relationbut not the second

Participating relations must be compatible

R S R - SA B

a1 b1

a2 b2

A B

a2 b2

a3 b3

A B

a1 b1


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Set Difference

A B

a1

a2

b1

b2

S

A B

A1

a2

a3

b1

b2

b3

R

R- S =A B

a3 b3


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Set Difference

S1

S2

S S1 2

S2 S1


22 dustin 7 45.0

31 lubber 8 55.5

58 rusty 10 35.0


28 yuppy 9 35.0

31 lubber 8 55.5

44 guppy 5 35.0

58 rusty 10 35.0


22 dustin 7 45.0


28 yuppy 9 35.044 guppy 5 35.0


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relation algebra

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