DBMS : Relational Algebra

DBMS – Lecture 2 : Relational Algebra
Lecture Plan
• Relational Algebra (operators)
• How it allows data independence
For more visit: www.technotz.info

Recapitulate: Database Abstraction
• We have seen relational model as an abstraction.
• Q: How do we operate on the relations/tables? How do
we access data?
Ans: Use a language such as:
- relational algebra
- SQL

Relational Algebra
• Queries can be expressed in relational algebra
- To retrieve data from tables
• Five operators
- Project: Cutting a table vertically
- Restrict or Select: Cutting horizontally
- Cartesian-product: Putting two tables together
- Union: Adding rows
- Difference: Deleting rows

Project Operation
Cuts a table vertically.
• Retrieves some (or all) columns from a table R
 Example: Retrieve rollno and address of each student
PROJECT (rollno, name, student)
or
student [rollno, name]
rollno name
123
45
Ram
Mohan

From our familiar relation: student
rollno* name addr city birth date
123
45
Ram
Mohan
30 MG Marg
10 Laxmi Bai
Hyd
Dlh
83-06-21
83-09-26

Project Operation: Definition
• Definition
PROJECT (A1, ..., Ak, R)
or
R [A1, ..., Ak]
- A1 to Ak are attribute names,
- R is a relation/table.
* Result: A table with all the rows of R but only k
attributes (A1 to Ak)

Restrict or Select Operation
Cutting a table horizontally
• Retrieves some (or all) of the rows from a table R.
- Those that satisfy a given condition.
 Ex. Retrieve information about the student with rollno 45
RESTRICT ((rollno = 45), student)
rollno* name addr city birth date
45 Mohan 10 Laxmi Bai Dlh 83-09-26

 Ex. Retrieve students from Hyderabad
RESTRICT (city = 'Hyd', student)
roll no* name addr city birth date
123 Ram 30 MG
Marg
Hyd 83-06-21

RESTRICT (F, R)
• F is a condition, and R a table
• Condition F is:
* Equality or inequality involving attributes and
values
 Ex. (A2 < 16)
 Ex. (A2 = A3)
* logical-and, logical-or or logical-not of
conditions.
 Ex. ( (A2 < 16) AND (A2 = A3) )
* Result: A table with all the attributes of R, but only
those rows that satisfy condition.

Composition of Operations
• Possible to combine the operations
 Ex. Names of students from Hyderabad
PROJECT (sname, RESTRICT (city = 'Hyd', student))
 Ex. Names of students from Hyderabad with roll no. greater
than 50
PROJECT (sname,
RESTRICT ((city = 'Hyd') AND
(rollno > 50), student) )
name
Ram
- Above condition has logical-and

Cross Product (*)
• A "multiplication" of two relations (R, S):
R * S
- Let R have m attributes (A1 to Am) and r rows,
- Let S have n attributes (B1 to Bn) and s rows,
- Result of cross product (R * S) is a relation with
 (m+n) attributes: A1 ... Am, B1 ... Bn
 r * s rows
* Every row of R is pasted with every row of S.

Cross Product Example
S = PROJECT (rollno, name, student)
S
roll no name
123 Ram
45 Mohan
enroll
cno* rollno* grade
IT365 123
IT355 45
IT365 45
IT340 123 For more visit: www.technotz.info

Cross-Product: S * register
rollno name cno* rollno* grade
123
Ram
IT365
123
123
Ram
IT355
45
123
Ram
IT365
45
123
Ram
IT340
123
45
Mohan
IT365
123
45
Mohan
IT355
45
45
Mohan
IT365
45
45
Mohan
IT340
123

Join Operation: Example
Join relates rows of two tables on some columns.
 Ex. Find rollno. and name of students enrolled in
courses.
JOIN ((S.rollno = enroll.rollno), S, enroll) (where S is a
projection of student as above)
rollno name cno grade
123
123
45
45
Ram
Ram
Mohan
Mohan
IT365
IT340
IT355
IT365
* Take the cross-product of: S and enroll
* Keep only those rows where rollno is the same.
* Do not keep two columns for rollno
- Both have same value in each row
 Same as: PROJECT For more (rollno, visit: www.technotz.name, info
cno, grade,
(continue...)

...continued
RESTRICT [ (S.rollno = enroll.rollno), S * enroll ] )
rollno name cno grade
123
123
45
45
Ram
Ram
Mohan
Mohan
IT365
IT340
IT355
IT365

Join Operation: Definition
• Join of R1, R2 on some logic formula F:
JOIN (F, R1, R2)
• Join: is a combination of CROSS-PROD and
RESTRICT.
The above is same as:
RESTRICT (F, R1 CROSS-PROD R2)
or
RESTRICT (F, R1 * R2)

Equi-join and Natural Join
• Equi-join: A join operation in which F contains equality
only.
• Natural join: A join operation in which equi-join is
performed over
the same column names across two relations.
 Example:
JOIN ((student.rollno = enroll.rollno), student,
enroll)
same as:
NATURAL-JOIN (student, enroll)

Union (U) Operation
• Takes two tables R and S (with same columns) and pastes
them
horizontally
(rows add up): R U S
 Ex. Students doing AI (IT365), and students doing
DBMS (IT355)
courses
PROJECT (rollno, RESTRICT (cno='IT365', enroll))
U
PROJECT (rollno, RESTRICT (cno='IT355', enroll))
rollno rollno
U rollno =
123
123
45 45
45

Difference (-) Operation
• Takes two tables R and S (with same columns) and removes rows
of S from
rows of R: R - S
 Ex. Students doing AI (IT365) but not DBMS (IT355).
( PROJECT (rollno, RESTRICT (cno='IT365', enroll))
PROJECT (rollno, RESTRICT (cno='IT355', enroll)) )
rollno rollno rollno
123
- =
45 123
45

Rename Operation
• This is used to give new names to columns.
RENAME (A1 as B1, ... An as Bn, R)
- Attributes A's are renamed to B's.

Conclusions
• Relational algebra is a query language
- Allows us to access or retrieve data from tables
* Query is in terms of tables, attributes, etc.
- Is independent of physical representation
* Does not depend on what organization is used:
 What kinds of files are used ?
 Whether sequential, b-tree, or hashed ?
• As a result:
- When underlying file organization changed,
- queries need not change

DBMS : Relational Algebra

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