Chapter Five
Relational Algebra
Relational Calculus
Objectives
Fundamental operations in RA
 Union
 Set difference
 Select
 Project
 Cartesian Product
 Relational Calculus

Query Languages
Procedural
1.

Relational Algebra (RA)
Non-Procedural
2.


Tuple Relational Calculus
Domain Relational Calculus
2
Fundamental Operations in RA

Binary Operations




UNION
MINUS
CARTESIAN PRODUCT
Unary Operations


SELECT
PROJECT
3
Union of R  S
1.
Union of R  S
a.
b.
All tuples in R or S
Union Compatible


Some degree
Attributes of R&S must be the same
4
Union of R  S
Faculty
Name ID
Salary
Smith
1
Name
ID
Salary
70,000
Smith
1
70,000
Nelson 3
85,000
Anderson
3
45,000
Larson 6
62,000
Faculty U Staff
Staff
Name
ID
Salary
Smith
1
70,000
Nelson
3
85,000
Larson
6
62,000
Anderson
3
45,000
5
Union of R  S
name (faculty)

name (staff)
6
Set Difference (MINUS) R - S


Set of tuples in R but not in S
Union Compatible
Faculty - Staff
Name
ID
Salary
Nelson
3
85,000
Larson
6
62,000
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Intersection: R  S
Set of of tuples belong to both R & S
 Union Compatible
Find the list of faculty members who are also
staff

Faculty  Staff
Name
ID
Salary
Smith
1
70,000
R  S = R – (R – S)
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Cartesian Product R x S

Set of (K1 + K2) tuples: The first K1 tuples are from R.
The last K2 tuples are from S
Semester
S_Num Year
Season
1
99
F
2
99
S
3
00
F
Semester_Course
C_Num
S_Num
200
1
250
1
300
2
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Cartesian Product R x S
RxS
S_Num
Year
Season
C_Num
S_Num
1
99
F
200
1
1
99
F
250
1
1
99
F
300
2
2
99
S
200
1
2
99
S
250
1
2
99
S
300
2
3
00
F
200
1
3
00
F
250
1
3
00
F
300
2
List of courses offered in year 99?
10
Selection (Unary Relation)




Select tuples that satisfy a given
predicate
 major = ‘COSC’ (Student)
Result is another relation
Conditions are relational operator (,,
,  , )
Logical operators AND (), OR(),
NOT()
11
Selection (Unary Relation)



Find all Faculty members which make less
than $45,000
 salary < 45,000(Faculty)
Find all staff who make less than $40,000 and
ID > 100
 salary < 40,000 AND ID > 100 (Staff)
List of number of courses offered in year 99
 S_Num = S_Num AND year = 99 (Semester x
Semester_Course)
12
Projection (Unary)




Select attributes
(Pi)
Find the number of faculty that teach
COSC courses
name( course=‘COSC’(Faculty))
13
Theta Join


R
R ij S
Theta join allows us to combine the selection & the cartesian
product into an operation
A
B
a
b
B
c
b
c
R
R BD S
A B
C
D E
a b
a
b c
a b
a
b d
a c
a
c d
b c
a
c d
C
D
E
a
b
c
a
b
d
b
e
c
a
c
d
If  is = It is called Equijoin
If the attributes have the same name, the join is called Natural Join
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Part Two
Relational Calculus (RC)



Non-procedural
Most commercial query language
Types:


Tuple RC: Variables represent tuples
Domain RC: Variables represent values of
domain
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Tuple Relational Calculus:
{ t | P(t)}
Examples
 List of students with GPA > 3
{ t | t Є student ^ t[GPA] > 3 }
 List of students name with GPA > 3
{ t | s Є student (t[NAME] = s[NAME] ^
t[GPA] > 3)}
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