1. Lecture 15: Relational Algebra
CSE 480: Database Systems
Lecture 15: Relational Algebra
Reference:
Read Chapter of the textbook

2. What is Relational Algebra?
A formal way to express a query in relational model
A query consists of relational expressions describing the sequence of operators applied to obtain the query result
Example :
 LName, Sex ( DNo=4 OR Salary> (EMPLOYEE) )
 and  are the operators

3. Why Study Relational Algebra?
Query processing in DBMS
SELECT A.Name, B.Grade FROM A, B WHERE A.Id = B.Id
 Name, Grade (Id,Name(A  B))

4. Relational Algebra Operator
Output of a relational algebra expression is a relation
Types of operators in a relational algebra expression
Unary: op(Relation)
Binary: Relation op Relation
Relational algebra operators
select, project, union, set difference, Cartesian product

5. Select Operator () Id Name Address Status
condition (R) (NOT THE SAME AS SELECT in SQL)
Resulting relation
contains only tuples in R that satisfy the given condition
has the same attributes (columns) as the original relation
Example:
Student Status=‘Junior’(Student)
Id Name Address Status
Id Name Address Status
John Main Junior
Lee Main Senior
Mary 7 Lake Dr Freshman
Bart Pine St Junior
John Main Junior
Bart Pine St Junior

6. Selection Condition select_condition (relation)
Selection condition acts as a filter to the rows in relation
Selection condition is a Boolean expression
Simple selection condition:
<attribute> operator <constant>
<attribute> operator <attribute>
where operator: <, , , >, =, 
<condition> AND <condition>
<condition> OR <condition>
NOT <condition>

7. Selection Condition - Examples
Student
John Main Junior
Lee Main Senior
Mary 7 Lake Dr Freshman
Bart Pine St Junior
Id Name Address Status
Selectivity
Fraction of tuples selected by a selection condition
 Id>3000 OR Status=‘Freshman’ (Student) selectivity = 2/4
 Id>3000 AND Id <3999 (Student) selectivity = 0/4
 NOT(Status=‘Senior’) (Student) selectivity = 3/4
 Status‘Senior’ (Student) selectivity = 3/4

8. Selection Condition – Incorrect Queries
Find professors who taught both CS315 and CS305
 CrsCode=‘CS315’ AND CrsCode=‘CS305’ (TEACHING) X
Find courses taught by CS professors
 PROFESSOR.Id=Teaching.ProfId AND PROFESSOR.DeptId=‘CS’ (TEACHING) X

9. Properties of SELECT operator
Commutative:
 <condition1>( < condition2> (R)) =  <condition2> ( < condition1> (R))
Cascade of SELECT operations may be applied in any order:
<cond1>(<cond2> (<cond3> (R)) = <cond2> (<cond3> (<cond1> ( R)))
Cascade of SELECT operations may be replaced by a single selection with a conjunction of all the conditions:
<cond1>(< cond2> (<cond3>(R)) =  <cond1> AND < cond2> AND < cond3>(R)))

10. Project Operator
Produces a relation containing a subset of columns of its input relation
attribute list(R)
Example:
Student Name,Status(Student)
Id Name Address Status Name Status
John Main Junior
Lee 123 Main Freshman
Mary 7 Lake Dr Senior
Bart 5 Pine St Junior
John Junior
Lee Freshman
Mary Senior
Bart Junior

11. Project Operator
Resulting relation has no duplicates; therefore it can have fewer tuples than the input relation
Example:
Student Address(Student)
Id Name Address Status
Address
123 Main
7 Lake Dr
5 Pine St
John Main Junior
Lee Main Freshman
Mary 7 Lake Dr Senior
Bart 5 Pine St Junior

12. Properties of PROJECT Operator
Number of tuples in the result of projection <list>(R) is always less or equal to the number of tuples in R
If list of projected attributes includes a superkey, the resulting relation has the same number of tuples as the input relation
PROJECT operator is not commutative
 <list1> ( <list2> (R) )   <list2> ( <list1> (R) )

13. Relational Algebra Expressions
Find the Ids and names of junior undergraduate students
Student
John Main Junior
Lee 123 Main Freshman
Mary 7 Lake Dr Senior
Bart 5 Pine St Junior
Id Name Address Status
John
Bart
Result
Id Name
 Id, Name ( Status=’Junior’ (Student) )

14. Relational Algebra Expressions
We may want to apply several relational algebra operations one after the other
We can write the operations as a single relational algebra expression by nesting the operations, or
We can apply one operation at a time and create intermediate result relations.
In the latter case, we must give names to the relations that hold the intermediate results.

15. Single versus Sequence of operations
Query: Find the first name, last name, and salary of all employees who work in department number 5
We can write a single relational algebra expression:
FNAME, LNAME, SALARY( DNO=5(EMPLOYEE))
OR We can explicitly show the sequence of operations, giving a name to each intermediate relation:
DEP5_EMPS   DNO=5(EMPLOYEE)
RESULT   FNAME, LNAME, SALARY (DEP5_EMPS)

16. RENAME operator The RENAME operator is denoted by 
We may rename the attributes of a relation or the relation name or both
S (B1, B2, …, Bn )(R)
Change the relation name from R to S, and
Change column (attribute) names to B1, B2, …..Bn
S(R) :
Change the relation name only to S
(B1, B2, …, Bn )(R) :
Change the column (attribute) names only to B1, B2, …..Bn

17. Example
Change relation name to Emp and attributes to First_name, Last_name, and Salary
Emp(First_name, Last_name, Salary ) (  FNAME,LNAME,SALARY (EMPLOYEE)) or
Emp(First_name, Last_name, Salary)   FNAME,LNAME,SALARY (EMPLOYEE)

18. Set Operators
A relation is a set of tuples; therefore set operations are applicable:
Intersection: 
Union: 
Set difference (Minus): 
Set operators are binary operators
Operator: Relation  Relation  Relation
Result of set operation is a relation that has the same schema as the combining relations

19. Examples: Set OperationsY A B x1 x2 x3 x4 A B x1 x2 x5 x6
X  Y X  Y X – Y A B x1 x2 A B x1 x2 x3 x4 x5 x6 A B x3 x4

20. Example STUDENT  INSTRUCTOR STUDENT – INSTRUCTOR STUDENT  INSTRUCTOR

21. Union Compatible Relations
Set operations are limited to union compatible relations
Two relations are union compatible if:
they have the same number of attributes, and
the domains of corresponding attributes are type compatible (i.e. dom(Ai)=dom(Bi) for i=1, 2, ..., n).
The resulting relation has the same attribute names as the first operand relation

22. Example Tables: Student (SSN, Name, Address, Status)
Professor (Id, Name, Office, Phone)
are not union compatible.
But
 Name (Student) and  Name (Professor)
are union compatible

23. Exercise Relations: Find the SSN of student employees
Employee (SSN, Name, Address)
Professor (SSN, Office, Phone)
Student (SSN, Status)
Find the SSN of student employees
 SSN (Employee)   SSN (Student)
Find the SSN of employees who are not professors
 SSN (Employee) –  SSN (Professor)
Find the SSN of employees who are neither professors nor students
 SSN (Employee) – ( SSN (Student)   SSN (Professor))

24. Cartesian Product A B C D A B C D x1 x2 y1 y2 x1 x2 y1 y2
R  S is expensive to compute:
Number of columns = degree(R) + degree(S)
Number of rows = number of rows (R) × number of rows (S)
A B C D A B C D
x1 x y1 y x1 x2 y1 y2
x3 x y3 y x1 x2 y3 y4
x3 x4 y1 y2
R S x3 x4 y3 y4
R S

25. Example List all the possible Broker-Client pairs
Broker (BrokerId, BrokerName)
Client (ClientId, ClientName)
List all the possible Broker-Client pairs
Broker  Client
BrokerID
BrokerName 1
Merrill Lynch 2
Morgan Stanley 3
Salomon Smith Barney
ClientId
ClientName
A11
Bill Gates
B11
Steve Jobs
BrokerId
BrokerName
ClientId
ClientName 1
Merrill Lynch
A11
Bill Gates
B11
Steve Jobs 2
Morgan Stanley 3
Salomon Smith Barney

26. More Complex Examples
Find the names of employees who are not supervisors

27. Example
Find the names of employees who earn more than their supervisors

28. Summary (Relational Algebra Operators)
Unary operators
SELECT condition(R)
PROJECT Attribute-List(R)
RENAME S(A1,A2,…Ak)(R)
Set operators
R  S
R  S
R – S or S – R
Cartesian product
R  S