Relational algebra consists of operators that allow deriving new relations from old ones. The five main operators are: 1) Union combines relations, 2) Intersection returns common tuples between relations, 3) Difference returns tuples in one relation not in another, 4) Selection returns tuples satisfying a condition, and 5) Projection eliminates columns and removes duplicates. Other operators include Cartesian Product, which pairs all tuples between relations, and Rename, which changes the schema but not instances.

1. Relational Algebra

2. Relational Algebra
• The relational algebra consists of a set of operators
that (speaking very loosely) allow us to derive
“new” relations from “old” ones.
• Each such operator takes one or more relations as
input and produces another relation as output.

3. Five operators:
– Union: 
– Intersection: 
– Difference: -
– Selection: s
– Projection: P
– Cartesian Product: 
– Rename : r

5. 1. Union

7. 2. Intersection
• It is used to derive common record
• R1  R2

8. 3. Difference

10. 3. Selection
• Returns all tuples which satisfy a condition
• Notation: sc(R)
• Examples
– sSalary > 40000 (Employee)
– sname = “Smith” (Employee)
• The condition c can be =, <, , >, , <>

11. sSalary > 400000 (Employee)
SSN Name Salary
1234545 John 200000
5423341 Smith 600000
4352342 Fred 500000
SSN Name Salary
5423341 Smith 600000
4352342 Fred 500000

13. Another selection

14. 4. Projection
• Eliminates columns, then removes duplicates
• Notation: P A1,…,An (R)
• Example: project social-security number and
names:
– P SSN, Name (Employee)
– Output schema: Answer(SSN, Name)

15. P Name,Salary (Employee)
SSN Name Salary
1234545 John 200000
5423341 John 600000
4352342 John 200000
Name Salary
John 20000
John 60000

16. Example of Projection

17. Projection removes duplicates

18. 5. Cartesian Product
• Each tuple in R1 with each tuple in R2
• Notation: R1  R2
• Example:
– Employee  Dependents
• Very rare in practice; mainly used to express
joins

19. Cartesian Product Example
Employee
Name SSN
John 999999999
Tony 777777777
Dependents
EmployeeSSN Dname
999999999 Emily
777777777 Joe
Employee x Dependents
Name SSN EmployeeSSN Dname
John 999999999 999999999 Emily
John 999999999 777777777 Joe
Tony 777777777 999999999 Emily
Tony 777777777 777777777 Joe

20. Renaming
• Changes the schema, not the instance
• Notation: r B1,…,Bn (R)
• Example:
– rLastName, SocSocNo (Employee)
– Output schema:
Answer(LastName, SocSocNo)

21. Renaming Example
Employee
Name SSN
John 999999999
Tony 777777777
LastName SocSocNo
John 999999999
Tony 777777777
rLastName, SocSocNo (Employee)