Relational algebra is the formal system used to manipulate data in a relational database. It consists of operators like select, project, join, union, and difference that are applied to relations to retrieve or modify data. Relational calculus provides an alternative, non-procedural way to query databases using logic and variables. Both are based on relational and algebraic concepts and underpin the functionality of relational databases and SQL.

1. Relational Algebra
Relational Algebra is :
• The formal description of how a relational database operates
• An interface to the data stored in the database itself
• The mathematics which underpin SQL operations
Operators in relational algebra are not necessarily the same as SQL operators, even
if they have the same name.
For example, the SELECT statement exists in SQL, and also exists in relational
algebra. These two uses of SELECT are not the same. The DBMS must take whatever
SQL statements the user types in and translate them into relational algebra
operations before applying them to the database.

2. Important Terminology
• Relation - a set of tuples.
• Tuple - a collection of attributes which describe some real
world entity.
• Attribute - a real world role played by a named domain.
• Domain - a set of atomic values.
• Set - a mathematical definition for a collection of objects
which contains no duplicates.

3. Operators -Write
• INSERT - provides a list of attribute values for a new tuple in a relation. This
operator is the same as SQL.
• DELETE - provides a condition on the attributes of a relation to determine which
tuple(s) to remove from the relation. This operator is the same as SQL.
• MODIFY - changes the values of one or more attributes in one or more tuples of a
relation, as identified by a condition operating on the attributes of the relation.
This is equivalent to SQL UPDATE.
Operators –Retrieval
There are two groups of operations:
• Mathematical set theory based relations:
UNION, INTERSECTION, DIFFERENCE, and CARTESIAN PRODUCT.
• Special database operations:
SELECT (not the same as SQL SELECT), PROJECT, and JOIN.

4. Relational SELECT
• SELECT is used to obtain a subset of the tuples of a relation that satisfy a select
condition.
For example, find all employees born after 1st Jan 1950:
SELECTdob '01/JAN/1950'(employee)
Relational PROJECT
• The PROJECT operation is used to select a subset of the attributes of a relation
by specifying the names of the required attributes.
For example, to get a list of all employees surnames and employee numbers:
PROJECTsurname,empno(employee)

5. • SELECT and PROJECT
SELECT and PROJECT can be combined together.
For example, to get a list of employee numbers for employees in
department number 1:

6. JOIN Operator
• JOIN is used to combine related tuples from two relations:
• In its simplest form the JOIN operator is just the cross product of
the two relations.
• As the join becomes more complex, tuples are removed within
the cross product to make the result of the join more meaningful.
• JOIN allows you to evaluate a join condition between the
attributes of the relations on which the join is undertaken.
The notation used is
R JOINjoin condition S

7. The relational model is based on the principle of relational algebra.
Relational algebra is a collection of operators that operate on
relations. Each operator takes one or two relations as its input and
produces a new relation as its output.
RESTRICT:
The Restrict operator extracts specified tuples or rows from a given relation, based on a
condition.
For example, a student table has the attributes ROLL NO[ roll number], STUDNAME [ student
name], AGE and GENDER. The condition is to extract the tuples of only those students whose
age is more than 25. The resultant relation appears, as follows:

8. PROJECT :
• The Project operator extracts specified attributes or columns
from a given relation
For example, if only the name and age of the students is to be
extracted, the resultant relation appears, as follows [assume
that the student table has only six tuples]

9. PRODUCT :
The product operator builds a relation from two specified
relations. It consists of an possible combinations of tuples, one
from each of the two relations.
To be product compatible, the two tables must have common attributes.
The Product operator creates Cartesian product between the two tables.

10. UNION :
• The Union operator builds a relation from tuples appearing in either or both
of the specified relations.
• To be union compatible, the two tables should have the same types of
attributes [sets of columns having the same data types]
• For example, Consider two tables A and B.
• A contains the roll numbers and names of all students whose major
discipline is computer science. B contains the roll numbers and names of all
students whose major discipline is mathematics. These tables are union
compatible because they have the same types of attributes.

11. Relational Calculus
Relational algebra is a procedural language, in which user has to write the steps or
procedure to obtain the required results but in general a user should not have to be
concerned with the details of how to obtain information. In relational calculus user
is not concerned with the procedure to obtain the results, he/she just tell his/her
requirements and the output is available without knowing the method about its
retrieval
In relational calculus, a query is expressed as a formula consisting of a number of
variables and an expression involving these variables. It is up to the DBMS to
transform these nonprocedural queries into equivalent, efficient, procedural
queries.
The relational calculus is used to measure the selective power of relational
languages. A language that can be used to produce any relation that can be derived
using the relational calculus is said to be relationally complete.

12. Relational Calculus
Relation calculus, which in effect means calculating with relations, is based on
predicate calculus, which is calculating with predicates. It is a formal language used
to symbolize logical arguments in mathematics. Propositions specifying a property
consist of an expression that names an individual object, and another expression,
called the predicate, that stands for the property that the individual object
possesses.
If for instance, p and q are propositions, we can build other propositions "not p", "p
or q", "p and q" and so on. In predicate calculus, propositions may be built not only
out of other propositions but also out of elements that are not themselves
propositions. In this manner we can build a proposition that specifies a certain
property or characteristic of an object.

13. Relational Calculus can be classified in two categories:
• Tuple Oriented relational Calculus
• Domain Oriented relational calculus
In relational tuple calculus, the variables represent the tuples
from specified relations; in relational domain calculus, the
variables represent values drawn from specified domains.

14. Tuple Oriented Relational Calculus
The tuple relational calculus is based on specifying a number of tuple variables. Each
such tuple variable normally ranges over a particular database relation. This means
that the variable may take any individual tuple from that relation as its value.
A simple tuple relational calculus query is of the form { t I COND(t)·}, where '1' is a
tuple variable and COND(t) is a conditional expression involving '1'. The result of such
a query is a relation that contains all the tuples (rows) that satisfy COND(t)
For example,
The relational calculus query {t I BOOK(t) and t.PRICE>lOO} will get you all the books
whose price is greater than 100. In the above example, the condition 'BOOK(t)'
specifies that the range relation of the tuple variable '1' is BOOK. Each BOOK tuple 't'
that satisfies the condition 't.PRICE> 100' will be retrieved. Note that 't.PRICE'
references the attribute PRICE of the tuple variable '1'.

15. Domain Oriented Relational Calculus
The domain calculus differs from the tuple calculus in the type of variables used
in formulas. In domain calculus the variables range over single values from
domains of attributes rather than ranging over tuples. To form a relation of
degree 'n' for a query result, we must have 'n' of these domain variables-one for
each attribute.
An expression of the domain calculus is of the following form:
• {Xl, X2, ... , Xn I COND(XI, X2, .. ·, Xn, Xn+b Xn+2, , Xn+m)}
In the above expression Xl, X2, ... , Xn, Xn+b Xn+2, , Xn+m are domain variables
that range over domains of attributes and COND is a condition or formula of the
domain relational calculus.

16. Expression of the domain calculus are constructed from the following
elements:
Domain variables Xl, X2, ... , Xn, Xn+b Xn+2, ... , Xn+m each domain variable is
to range over some specified domain .
• Conditions, which can take two forms:
• Simple comparisons of the form x * y, as for the tuple calculus, except that x and yare now domain variables.
• Membership conditions, of the form R (term,
term ...).
Here, R is a relation, and each "term" is a pair AV, where A in turn is an attribute
Of R and V is either a domain variable or a constant.
For example EMP (empno: 100, ename: 'Ajay') is a membership condition (which evaluates to
true if and only if there exists an EMP tuple having empno=100 and ename = 'Ajay') .