relational algebra

1. 5th Chapter
Relational Algebra and
Calculus

2. Relational algebra is a procedural language , which takes
instances of relations as input and yields instances of relations
as output.
• It uses operators to perform queries .
• An operator can be either unary or binary .
• They accept relations as their input and yields relations as
their output.
Relational Algebra Operations:
Union
Intersection
Difference
Division
Cartesian Product
Selection
Projection
Join

3. Union: It is a binary operation , which combines the tuples of
two relations.
The two must contains the same columns.
Each column of the first set must be either the same datatype as
the corresponding column of the second test.
It is denoted by ‘U’ symbol.
R U S
R and S are the relations and U is the operator.
Intersection : This operation is a binary operation . It results in
relation with tuples that are in both the relations.
It is denoted by ‘⋂’ symbol.
R ⋂ S

4. Difference: This is a binary operator . This operator creates a
new relation with tuples that are in one relation but not in other
relation.
It is denoted by ‘-’ Symbol.
R - S
R and S are the relations .
Cartesian Product :
This is a binary relation operation .It combines the tuples
of two relations into one relation .
It is denoted by ‘×’ Symbol.
R × S
R and S are two relations and × is the operator.
Division: This operation is used to find the tuples with
phrase ‘for all’.
It is denoted by ‘÷’ Symbol.
R ÷ S

5. Selection Operation : This is a unary relational operation . This
operation pulls the horizontal subset of the relations that satisfies the
condition.
This can use operators like <,>,<=,>=,=and != to filter the data from
the relation.
It can also use logical AND,OR and NOT operators to combine the
various filtering conditions .
Denoted by: σp(r) ,Where σ stands for selection predicate
and r stands for relation. p is prepositional logic formula which may
use connectors like and, or, and not.
Projection Operators: This is a unary relation operation .
It creates the subset of relation based on the conditions specified
.Here , it selects only selected columns /attributes from the relation-
vertical subset of relations .
The selection operation above creates subset of relation but
For all the attributes in the relation.
Denoted by: ∏A1, A2, An (r)
∏ is the operator for projection, r is the relation and a1,a2,a3 are the
attributes of the relations.

6. Join operations: A join means combining fields from two tables
by using values common to each.
The different types of join operations are :
Natural join
Inner join
Self join
Left Outer join
Right Outer join
Full Outer join
Cartesian join

7. Natural join : Natural join does not use any comparison
operator.
We can perform a Natural join only if there is at least one
common attribute that exists between two relations.
In addition the attributes must have the same name and
domain.
Natural join acts on those matching attributes where the values
of attributes in both the relations are same .
R ⋈ S Here , R and S are the relations
Syntax: Select table1.column1,table2.column2…from table1
Natural join table2 where table1.common=table2.common.

8. Inner join: The inner join selects all rows from both participating
tables as long as there is a match between the columns.
Inner join is same as join clause, combining rows from two or
more tables.
Syntax: select table1.column,table2.column2..FROM table1
INNER JOIN table2 ON table1.common=table2.common;

9. Left Outer join: In this operation , all the tuples in the left hand
side relation is retained .
All matching attribute in the right hand relation is displayed with
values and the ones which does not have values are shown as
Null value .
Syntax: Select table1.column1,table2.column2….from table1 left
outer join table2 on table1.common=table2.common;

10. Right Outer Join: In this operation , all the attributes of right
hand side is retained and it matching attribute in left hand
relation is found and displayed . If no matching is found then
null is displayed.
Syntax: Select table1.column1,table2.column2….from table1
Right outer join table2 on table1.common=table2.common;

11. Full Outer Join:
In this operation combination of both left and right outer join .
It displays all the attributes from both the relation .
If the matching attribute exists in other relation ,then that will be
displayed ,else as those attributes are shown as null.
Syntax: Select table1.column1,table2.column2…from table1 full
outer join on table1.common_field=table2.common_field;

12. Self join: The SQL self join is used to join a table to itself as if the
were two tables , temporarily remaining at least one table in the
SQL statement.
Syntax: Select a. column_name,b.column_name ……
From table a, table b
Where a .common=b.common;

13. Cartesian Join: A Cartesian join or Cross join returns the
Cartesian product of the sets of records from the two or more
joined tables. Thus , it equates to an inner join where the join-
condition always evaluates to true or where the join-condition is
absent from the statement.

14. Relational Calculus: Relational Calculus is a non-procedural query
language .
The relational calculus tells , the system what data to be retrieved but
does not tell how to retrieve it.
Relational Calculus exists in two forms:
Tuple Relational Calculus (TRC)
Domain Relational Calculus(DRC)
Tuple Relational Calculus : Tuple Relational Calculus is a non
procedural query language which specifies to select the tuples in a
relation .
It can select the tuples with range of values or tuples for certain
attribute values.
The resulting relation can have one or more tuples.
{t| P(t)} or {t | condition (t)}
t=> resulting tuple
P(t) => condition used to fetch
Example: {t | Employee (t) and t. salary>10000}
It selects the tuples from employee relation such that resulting
employee tuples will have salary greater than 10000.

15. Domain Relational Calculus: Domain relational calculus uses list
of attributes to be selected from the relation based on the
condition .
It is same as Tuple relational calculus , but differs by selecting
attributes rather than selecting whole tuples.
Denoted by, { <a1,a2,a3,…..an>| P(a1,a2,a3,….an)}
a1,a2,a3……are the attributes .
p=> condition
Example: {<EMPID, ENAME> | <EMPID,ENAME>? EMPLOYEE ^
DEPTID = 10}
selects empid empname of employees who work for
department.
Constraints: Every relation has some conditions that must hold
for it to be a valid relation .
These conditions are called Relational Integrity Constraints.

16. There are 3 types of Integrity Constraints:
Key Constraints
Domain Constraints
Referential Constraints
Key Constraints: There must be at least one minimal subset of
attributes in the relation , which can identify a tuple uniquely.
This minimal subset of attributes is called key for that relation.
Key constraints are also referred to as Entity Constraints.
Key constraints force that :
In a relation with a key attribute , no two tuples can have
identical values for key attributes.
A key attribute cannot have NULL values.

17. Domain Constraints: Attributes have specific values in real-
world scenario.
Example: Age can only be a positive integer.
Referential integrity Constraints: Referential integrity
Constraints Work on the concept of Foreign keys.
A Foreign key is a key attribute of a relation that can be referred
in other relation .
Referential integrity constraint states that if a relation refers to a
Key attribute of a different or same relation , then that key
element must exist.
Integrity Rules: Integrity rules may sound very technical but
they are simple and straightforward rules that each table must
follow.
These are very important in database design, when tables break
any of the integrity rules our database will contain errors when
retrieving information.
Hence the name integrity which describes reliability and
consistency of values.

18. There are two types of integrity rules:
Entity integrity rule
Referential integrity rule
Entity integrity rule: The entity integrity rule refers to the rules
the primary key must follow.
The primary key value cannot be null.
The primary key value must be unique.
Referential Integrity Rule:
The referential integrity rule refers to the foreign key.
The foreign key may be null and may have the same value.
The foreign key value must match a record in the table it is
referring to.

19. Normalization of Database:
 It is a technique of organizing data in a database .
 Normalization is a systematic approach of decomposing
tables to eliminate data redundancy and undesirable
characteristics
 Like Insertion , Update and Deletion Anomalies.
 It is a multi-step process that puts data into tabular form by
removing duplicated data from the relational tables.
Normalization is used for mainly two purpose(goals).
 Eliminating redundant data.
 Ensuring data dependencies make sense i.e data is logically
stored .

20. Problem without Normalization:
Empid Ename Address Department
1001 Surya Bangalore Biology
1002 Chandra Mysore Maths
1003 Rama Kolar Maths
1004 Surya Bangalore Physics

21. Updation Anomaly: To upadate the address of a employee who
occurs twice or more than twice in a table ,we will have to
update the Address column in all the rows , else data will
become inconsistent.
Insertion Anomaly: Suppose for a new employee , we have
empid, ename and address of a employee but if employee has
not opted for any department yet then we have to insert Null
there ,leading to insertion anomaly.
Deletion Anomaly: If empid 1001 has only one department and
temporarily he drops it , when we delete that row , entire
employee record will be deleted along with it.
Types of Normal form:
 First Normal Form
 Second Normal Form
 Third Normal Form

22. First Normal Form (1NF) : As per the rule of first normal form ,
an attribute of a table cannot hold multiple values .
It should hold only atomic values.
Table without Normalization:

23. Table with 1NF:

24. Second Normal Form(2NF)
A table is said to be in 2NF if both the following conditions hold:
Table in 1NF
No non-prime attribute is dependent on the proper subset of
any candidate key of table.

25. Third Normal Form (3NF)
A table design is said to be in 3NF if both the following conditions
hold:
Table must be in 2NF
Transitive functional dependency of non-prime attribute on any
super key should be removed.
An attribute that is not part of an any candidate key is known as
prime attribute.

26. Table complies with 3NF:
The advantage of removing transitive dependency is ,
• Amount of data duplication is reduced.
• Data integrity achieved.

27. Properties of Normalized Relation:
No data value should be duplicated in different rows,
unnecessarily.
A value must be specified for every attribute in a row.
Each relation should be self-contained . In other words ,if a
row from a relation is deleted , important information should
not be accidently lost.
When a row is added to a relation , other relations in the
database should not be affected.
A value of an attribute in a tuple may be changed
independent of other tuples in the relation and other
relations.

28. Advantages of Normalization:
 More efficient data structure.
 Avoid redundant fields or columns.
 More flexible data structure i.e we should be able to add new
rows and data values easily.
 Better understanding of data.
 Ensures that distinct tables exist when necessary.
 Easier to maintain data structure i.e it is easy to perform
operations and complex queries can be easily handled.
 Minimizes data duplication .
 Close modeling of real word entities ,process and their
relationships.

29. Disadvantages:
 On Normalizing the relations to higher normal forms i.e
4NF,5NF the performance degrades.
 It is very time consuming and difficult process in normalizing
relations of higher degree.
 Careless decompositions may leads to bad design
Functional Dependency: Functional dependency (FD) is a set of
constraints between two attributes in a relation .
 Functional dependency says that if two tuples have same
values for attributes A1,A2,A3 …….An, then those two tuples
must have to have same values for the attributes B1,B2,…..Bn.
 Functional Dependency is represented by a arrow sign (--->)
that is XY , where X functionally determines Y.
 The left hand side attributes determine the values of
attributes on the right hand side .

30. An important property of functional dependency is Armstrong ‘s
axiom, which is used in database normalization.
In a relation R , with three attributes (X,Y,Z) Armstrong’s axiom holds
strong if the following conditions are satisfied.
Transitivity: if X Y and YZ, then XZ
Reflexivity (Subset Property): if Y is a subset of X , then XY
Augmentation: if X Y, then XZYZ

32. Queries: A query is a request for data or information from a
database table or combinations of tables.
This data may be generated as results returned by structured Query
language (SQL).
Query languages generate different in data types according to
function.
SQL returns data in neat rows and columns .
Other query language generate data as graphs or other complex
data manipulation
Example: Select * from student.
Sub Queries: A sub query or a inner query is a query within another
SQL query and embedded within the WHERE clause.
A sub query is used to return the data that will be used in the main
query as a condition to further restrict the data to be retrieved.
Sub Queries can be used along with the SELECT ,INSERT, UPDATE,
and DELETE statements along with the operations like
=,<,>,>=,<=,IN, BETWEEN etc.

33. Rules for Sub Queries:
Sub Queries must be enclosed within parentheses.
A sub query can have only one column in the SELECT clause, unless
multiple columns in a are in the main query to compare its selected
columns.
An ORDER BY cannot be used in a sub query, although the main query can
be use an ORDER BY .
The group by can be used to perform the same function as the ORDER BY
in a sub query.
Sub Queries that return more than one row can only be used with
multiple value operators, such as the IN operators.
A sub Query cannot be immediately enclosed in a set function.
The BEWEEN operator cannot be used with a sub query ; However , the
BEWEEN operator can be used within the sub query.
Syntax: SELECT column_name [,column_name]
FROM table1, [,table2]
WHERE column_name OPERATOR
(SELECT column_name[,column_name]
FROM table1 [,table2]
[WHERE])

34. Example: select count (*) accident from participated where regno
in (select regno from car where model like ‘ALTO’);
Nested Sub Queries: A sub Query can be nested inside other sub
queries.
SQL has an ability to nest queries within one another.
A sub query is a select statement that is nested within another
Select statement and which return intermediate results.
Syntax: Select column1,column2…..from table_name[where
condition ](select column1,column2,..from table_name where
condition );