1. Relational Algebra

2. What do we mean by an "algebra"?
"Ordinary" algebra works on numbers, and contains operators that work on numbers: 2+4, 6-3, 95/5, 8*7.
Relational Algebra works on relations, and has operators which work on relations (tables).

3. Relational Algebra
Algebra is a formal structure consisting of sets and operations on those sets.
Relational algebra is a formal system for manipulating relations.

4. Relational Algebra
A language based on operators and a domain of values
Operators map values taken from the domain into other domain values
Hence, an expression involving operators and arguments produces a value in the domain
When the domain is a set of all relations (and the operators are as described later), we get the relational algebra
We refer to the expression as a query and the value produced as the query result

5. Relational Algebra-Terminology
Domain: set of relations
Basic operators: select, project, union, set difference, Cartesian product and rename
Derived operators: set intersection, division, join
Procedural: Relational expression specifies query by describing an algorithm (the sequence in which operators are applied) for determining the result of an expression

6. Select Operation() Syntax for the Select operator is  (relation)
predicate

7. What does the Select operator do?
The Select operator, s , selects tuples (rows) from the relation which satisfy (make TRUE) the specified predicate. So, we start with a given relation, and produce a new relation which is a (horizontal) subset of the old one, depending upon which rows are extracted from the old relation.

8. The STUDENT Relation The query s (STUDENT) STUID = 241504 STUNAME
MAJOR
CREDITS
151234
Jones,Ted
MGMT 47
241504
Smith, Jane CS
101
310927
Kool, Bill
MATH 68
428541
Lan, Jackie IS 93
529624
Witt, Stu
COE 75
629632
Kahn, Fran CS 65
The query s (STUDENT)
STUID =
STUID
STUNAME
MAJOR
CREDITS
241504
Smith, Jane CS
101

9. s (Student) STUNAME MAJOR CREDITS 151234 Jones,Ted MGMT 47 241504
STUID
STUNAME
MAJOR
CREDITS
151234
Jones,Ted
MGMT 47
241504
Smith, Jane CS
101
310927
Kool, Bill
MATH 68
428541
Lan, Jackie IS 93
529624
Witt, Stu
COE 75
629632
Kahn, Fran CS 65
s (Student)
MAJOR = 'CS'
STUID
STUNAME
MAJOR
CREDITS
241504
Smith, Jane CS
101
629632
Kahn, Fran CS 65

10. The 'Usual' Comparisons are Allowed in the Predicate
We may include the "usual" comparison operators ( < , > , =, etc.) in the predicate.
We can form "compound" predicates with an AND ( L ) and the OR ( V ).

11.  ( employee) - it will select rows having salary > 500
Selection ( )
eno
ename
sal
desig
IT1
ALI
500
TUTOR
BUS2
AHMED
1000
HEAD
IT2
SABA
400
CLERK
IT3
SALEH
BUS1
BADER
650
 ( employee) it will select rows having salary > 500
sal > 500
eno
ename
sal
desig
BUS2
AHMED
1000
HEAD
BUS1
BADER
650
TUTOR

12. Selection Condition Operators: <, , , >, =, 
Simple selection condition:
<attribute> operator <constant>
<attribute> operator <attribute>
<condition> AND <condition>
<condition> OR <condition>
NOT <condition>

13. s (Student) STUID STUNAME MAJOR CREDITS 428541 Lan, Jackie IS 93
151234
Jones,Ted
MGMT 47
241504
Smith, Jane CS
101
310927
Kool, Bill
MATH 68
428541
Lan, Jackie IS 93
529624
Witt, Stu
COE 75
629632
Kahn, Fran CS 65
s (Student)
((MAJOR = 'CS' ) L (CREDITS < 100))V(MAJOR='IS')
STUID
STUNAME
MAJOR
CREDITS
428541
Lan, Jackie IS 93
629632
Kahn, Fran CS 65

14. The Project Operator p a unary operator (works on a single relation)
produces a vertical subset of a relation, extracting the values of the attributes we specify, eliminating duplicates, and placing the values in a new relation

15. p (relation) The Project Operator, p
attribute(1),attribute(2),...attribute(n)
This will produce a list of the attribute values given, for all entities in the specified relation.

16. p (Student)
STUNAME, MAJOR
Query states : What are the names and majors of all students?
STUNAME
MAJOR
Jones,Ted
MGMT
Smith, Jane CS
Kool, Bill
MATH
Lan, Jackie IS
Witt, Stu
COE
Kahn, Fran CS

17. p ( s (student)) STUNAME STUID Lan, Jackie 428541
Use the Relational Algebra to write a query which lists the names and id's of all students who are IS majors
p ( s (student))
STUNAME, STUID MAJOR='IS'
The innermost (select expression) produces:
STUID
STUNAME
MAJOR
CREDITS
428541
Lan, Jackie IS 93
The "project" is applied to this relation, yielding:
STUNAME
STUID
Lan, Jackie
428541

18. Set Operators
Relation is a set of tuples => set operations should apply
Result of combining two relations with a set operator is a relation => all its elements must be tuples having same structure
Hence, scope of set operations limited to union compatible relations

19. Union Compatible Relations
Two relations are union compatible if
Both have same number of columns
Type of attributes are the same in both
Attributes with the same name in both relations have the same domain
Union compatible relations can be combined using union, intersection, and set difference.

20. Examples of Union Compatibility and Non-Compatibility
Compatible (assuming expected domains):
INVENTORY (Inumber, Description, SalesPrice) ORDER (Partnumber, Description, Price)
And
EMPLOYEE (Enumber, Name, Dept) VACATION_SCHEDULE(Number, Name, Dept-To-Be-Billed)
Non-Compatible:
INVENTORY (Pnum, Desc, Quantity-In-Stock) ORDER (Pnum, Desc, Supplier)
And
INVENTORY (Part, Description) ORDER( Pnum, Desc, Supplier)

21. Union
Union Operator ... U
a binary operator, having usual definition for Union: A U B consists of tuples (rows) in either A or B or both (duplicate rows eliminated)

22. Union ( ) Union compatible rule :-
1.Two relations should have same degree
2. Two relation corresponding same type of column
domain.
Bus-Student (Degree = 4 Cardinality = 3
IT-Student (Degree = 4 Cardinality = 3)
Eno
Ename
addr
gsm i1
ali
ruwi
i25
bushra
nizwa
I34
saleh
seeb
Eno
Ename
addr
gsm
B45
bilal
wadi B2
saleh
seeb
aliya
jebroo

23. Union ( )
Union returns those tuples that are either or both of two relations.
IT-Student U Bus-Student (Degree = 4 Cardinality =6)
Eno
Ename
addr
gsm i1
ali
ruwi
i25
bushra
nizwa
I34
aliya
seeb
B45
bilal
wadi B2
saleh
jebroo

24. Example of Union Operator U
Consider the following relations and instances: M: MANUFACTURERS (Name, PeopleCount, Revenue)
C: COMPANY (Name, NumberEmployees, Sales)
IBM
50,000
9,000,000,000
Univac
25,000
500,000,000
Microsoft
35,000
1,500,000,000 M:
IBM
50,000
9,000,000,000
Univac
25,000
500,000,000 C:
IBM
50,000
9,000,000,000
Univac
25,000
500,000,000
Microsoft
35,000
1,500,000,000
M U C :

25. Intersection Represented by R1  R2
Produces a relation that includes all the tuples in both R1 and R2; R1 and R2 must be union compatible.

26. A B S# SNAME STATUS CITY S1 Smith 20 London S4 Clark S# SNAME STATUS

27. A  B S#
SNAME
STATUS
CITY S1
Smith 20
London S4
Clark

28. Example of Intersection Operator 
Consider the following relations and instances: M: MANUFACTURERS (Name, PeopleCount, Revenue)
C: COMPANY (Name, NumberEmployees, Sales)
IBM
50,000
9,000,000,000
Univac
25,000
500,000,000
Microsoft
35,000
1,500,000,000 M:
IBM
50,000
9,000,000,000
Univac
25,000
500,000,000 C:
M  C
IBM
50,000
9,000,000,000
Univac
25,000
500,000,000

29. Difference Represented by R1 - R2:
Produces a relation that includes all the tuples in R1 that are not in R2; R1 and R2 must be union compatible.

30. Set-difference (–)
The minus operator returns only records which are only in the first relation.
students
Sports students
rollno
name
22045
saleh
22056
arif
22345
ahmed
23356
majid
23456
anwar
rollno
name
22045
saleh
22056
arif
23456
anwar
students-sports students
rollno
name
22345
ahmed
23356
majid

31. Example of Difference Operator -
Consider the following relations and instances: M: MANUFACTURERS (Name, PeopleCount, Revenue)
C: COMPANY (Name, NumberEmployees, Sales)
IBM
50,000
9,000,000,000
Univac
25,000
500,000,000
Microsoft
35,000
1,500,000,000 M:
IBM
50,000
9,000,000,000
Univac
25,000
500,000,000 C:
M – C :
Microsoft
35,000
1,500,000,000

32. Cartesian product Represented by R1 X R2:
Produces a relation that has the attributes of R1 and R2 and includes as tuples all possible combinations of tuples from R1 and R2.
Build a relation from two specified relations consisting of all possible combinations of tuples, one from each of the two relations.

33. CROSS PRODUCT (X) or CARTESIAN Product
It takes 2 relations and it returns new relation where every record of first relation is combined with every record of second relation.
EMP 
(Degree = 5 Cardinality =4)
DEPT
 (Degree = 3 Cardinality = 3)
Eno
Ename
sal
Desig
Dno E1
ali
500
tutor 10 E2
ahmed
1000
head E3
saleh
700 20 E4
sami
400
clerk 30
Dno
Dname
Loc 10
Business B2 20
Information Tech. B1 30
Foundation C1

34. EMPDEPT Degree = 5+3 = 8 Cardinality = 4*3 = 12 )
Eno
Ename
sal
Desig
Dno
Dname
Loc E1
ali
500
tutor 10
Business B2 20
Information Tech. B1 30
Foundation C1 E2
ahmed
1000
head E3
saleh
700 E4
sami
400
clerk

35. The Natural JOIN Operator in The Relational Algebra
"the heart of the relational algebra"
a binary operator
we "join" together two tables, based on a common attribute (the Join attribute)
the new relation will contain the columns of both tables which have been joined, and its rows will be the concatenation of a row from the first table and a row from the second table which matches the Join attribute

36. The Natural Join Symbol: Ä
A ÄB means the Natural Join of relations A and B
Note that if there is a tuple in relation A in which its attribute to be joined does not match any value of the attribute to be joined in relation B, it will not appear in the joined relation.

37. PC PC Ä EMPLOYEE would then be: EMPLOYEE COMPID EMPNUM 32808 M759 611
TAGNUM
COMPID
EMPNUM
EMPNUM
EMPNAME
32808
M759
611
124
Alvarez, R.
37691
B121
124
258
Lopez, M.
57772
C007
567
567
Feinstein, B.
59836
B221
124
77740
M759
567
611
Dinh, M.
80269
C007
852
PC Ä EMPLOYEE would then be:
TAGNUM
COMPID
EMPNUM
EMPNAME
32808
M759
611
Dinh. M.
37691
B121
124
Alvarez, R.
57772
C007
567
Feinstein, B.
59836
B221
124
Alvarez, R.
77740
M759
567
Feinstein, B.

38. List the tag number and computer ID together with the name of the employee to whom the PC is assigned.
p ( PC Ä EMPLOYEE )
TAGNUM, COMPID, EMPNAME

39. Answer to the Query TAGNUM COMPID EMPNAME 32808 M759 Dinh. M. 37691
B121
Alvarez, R.
57772
C007
Feinstein, B.
59836
B221
Alvarez, R.
77740
M759
Feinstein, B.