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**Relational Algebra**

Relational Algebra is procedural query language, which takes Relation as input and generate relation as output. Relational algebra mainly provides theoretical foundation for relational databases and SQL.

**Operators in Relational Algebra**

**Projection (π)**

Projection is used to project required column data from a relation.

**Example :**

**Note: **By Default projection removes duplicate data.

**Selection (σ)**

Selection is used to select required tuples of the relations.

for the above relation

σ (c>3)R

will select the tuples which have c more than 3.

**Note: **selection operator only selects the required tuples but does not display them. For displaying, data projection operator is used.

For the above selected tuples, to display we need to use projection also.

**Union (U)**

Union operation in relational algebra is same as union operation in set theory, only constraint is for union of two relation both relation must have same set of Attributes.

**Set Difference (-)**

Set Difference in relational algebra is same set difference operation as in set theory with the constraint that both relation should have same set of attributes.

**Rename (ρ)**

Rename is a unary operation used for renaming attributes of a relation.

ρ (a/b)R will rename the attribute ‘b’ of relation by ‘a’.

**Cross Product (X)**

Cross product between two relations let say A and B, so cross product between A X B will results all the attributes of A followed by each attribute of B. Each record of A will pairs with every record of B.

below is the example

**Note:** if A has ‘n’ tuples and B has ‘m’ tuples then A X B will have ‘n*m’ tuples.

**Natural Join (⋈)**

Natural join is a binary operator. Natural join between two or more relations will result set of all combination of tuples where they have equal common attribute.

Let us see below example

**Conditional Join**

Conditional join works similar to natural join. In natural join, by default condition is equal between common attribute while in conditional join we can specify the any condition such as greater than, less than, not equal

Let us see below example

Relational Algebra is a procedural query language which takes relations as an input and returns relation as an output. There are some basic operators which can be applied on relations to produce required results which we will discuss one by one. We will use STUDENT_SPORTS, EMPLOYEE and STUDENT relations as given in Table 1, Table 2 and Table 3 respectively to understand the various operators.

**Table 1 : STUDENT_SPORTS**

ROLL_NO | SPORTS |

1 | Badminton |

2 | Cricket |

3 | Badminton |

4 | Badminton |

**Table 2 : EMPLOYEE**

EMP_NO | NAME | ADDRESS | PHONE | AGE |

1 | RAM | DELHI | 9455123451 | 18 |

5 | NARESH | HISAR | 9782918192 | 22 |

6 | SWETA | RANCHI | 9852617621 | 21 |

4 | SURESH | DELHI | 9156768971 | 18 |

**Table 3 : STUDENT**

ROLL_NO | NAME | ADDRESS | PHONE | AGE |

1 | RAM | DELHI | 9455123451 | 18 |

5 | NARESH | GURGAON | 9652431543 | 18 |

6 | SUJIT | ROHTAK | 9156253131 | 20 |

4 | SURESH | DELHI | 9156768971 | 18 |

* Selection operator (σ): *Selection operator is used to select tuples from a relation based on some condition. Syntax:

σ (Cond)(Relation Name)

Extract students whose age is greater than 18 from STUDENT relation given in Table 1

σ (AGE>18)(STUDENT)

**RESULT :**

ROLL_NO | NAME | ADDRESS | PHONE | AGE |

3 | SUJIT | ROHTAK | 9156253131 | 20 |

* Projection Operator (∏):* Projection operator is used to project particular columns from a relation. Syntax:

∏(Column 1,Column 2….Column n)(Relation Name)

Extract ROLL_NO and NAME from STUDENT relation given in Table 1

∏(ROLL_NO,NAME)(STUDENT)

**RESULT :**

ROLL_NO | NAME |

1 | RAM |

2 | RAMESH |

3 | SUJIT |

4 | SURESH |

**Note: **If resultant relation after projection has duplicate rows, it will be removed. For Example: ∏_{(ADDRESS)}(STUDENT) will remove one duplicate row with value DELHI and return three rows.

* Cross Product(X):* Cross product is used to join two relations. For every row of Relation1, each row of Relation2 is concatenated. If Relation1 has m tuples and and Relation2 has n tuples, cross product of Relation1 and Relation2 will have m X n tuples. Syntax:

Relation1 X Relation2

To apply Cross Product on STUDENT relation given in Table 1 and STUDENT_SPORTS relation given in Table 2,

STUDENT X STUDENT_SPORTS

**RESULT :**

ROLL_NO | NAME | ADDRESS | PHONE | AGE | ROLL_NO | SPORTS |

1 | RAM | DELHI | 9455123451 | 18 | 1 | Badminton |

1 | RAM | DELHI | 9455123451 | 18 | 2 | Cricket |

1 | RAM | DELHI | 9455123451 | 18 | 2 | Badminton |

1 | RAM | DELHI | 9455123451 | 18 | 4 | Badminton |

2 | RAMESH | GURGAON | 9652431543 | 18 | 1 | Badminton |

2 | RAMESH | GURGAON | 9652431543 | 18 | 2 | Cricket |

2 | RAMESH | GURGAON | 9652431543 | 18 | 2 | Badminton |

2 | RAMESH | GURGAON | 9652431543 | 18 | 4 | Badminton |

3 | SUJIT | ROHTAK | 9156253131 | 20 | 1 | Badminton |

3 | SUJIT | ROHTAK | 9156253131 | 20 | 2 | Cricket |

3 | SUJIT | ROHTAK | 9156253131 | 20 | 2 | Badminton |

3 | SUJIT | ROHTAK | 9156253131 | 20 | 4 | Badminton |

4 | SURESH | DELHI | 9156768971 | 18 | 1 | Badminton |

4 | SURESH | DELHI | 9156768971 | 18 | 2 | Cricket |

4 | SURESH | DELHI | 9156768971 | 18 | 2 | Badminton |

4 | SURESH | DELHI | 9156768971 | 18 | 4 | Badminton |

* Union (U): *Union on two relations R1 and R2 can only be computed if R1 and R2 are

Relation1 U Relation2

Find person who are either student or employee, we can use Union operator like:

STUDENT U EMPLOYEE

**RESULT :**

ROLL_NO | NAME | ADDRESS | PHONE | AGE |

1 | RAM | DELHI | 9455123451 | 18 |

2 | RAMESH | GURGAON | 9652431543 | 18 |

3 | SUJIT | ROHTAK | 9156253131 | 20 |

4 | SURESH | DELHI | 9156768971 | 18 |

5 | NARESH | HISAR | 9782918192 | 22 |

6 | SWETA | RANCHI | 9852617621 | 21 |

* Minus (-):* Minus on two relations R1 and R2 can only be computed if R1 and R2 are

Relation1 - Relation2

Find person who are student but not employee, we can use minus operator like:

STUDENT - EMPLOYEE

**RESULT :**

ROLL_NO | NAME | ADDRESS | PHONE | AGE |

2 | RAMESH | GURGAON | 9652431543 | 18 |

3 | SUJIT | ROHTAK | 9156253131 | 20 |

* Rename(ρ): *Rename operator is used to give another name to a relation. Syntax:

ρ(Relation2, Relation1)

To rename STUDENT relation to STUDENT1, we can use rename operator like:

ρ(STUDENT1, STUDENT)

** **If you want to create a relation STUDENT_NAMES with ROLL_NO and NAME from STUDENT, it can be done using rename operator as:

ρ(STUDENT_NAMES, ∏(ROLL_NO, NAME)(STUDENT))

Basic idea about relational model and basic operators in Relational Algebra:

Relational model

Basicoperators in Relational Algebra

Extended operators are those operators which can be derived from basic operators.There are mainly three types of extended operators in Relational Algebra:

**Join****Intersection****Divide**

The relations used to understand extended operators are STUDENT, STUDENT_SPORTS, ALL_SPORTS and EMPLOYEE which are shown in Table 1, Table 2, Table 3 and Table 4 respectively.

**STUDENT**

ROLL_NO | NAME | ADDRESS | PHONE | AGE |

1 | RAM | DELHI | 9455123451 | 18 |

2 | RAMESH | GURGAON | 9652431543 | 18 |

3 | SUJIT | ROHTAK | 9156253131 | 20 |

4 | SURESH | DELHI | 9156768971 | 18 |

**Table 1**

**STUDENT_SPORTS**

ROLL_NO | SPORTS |

1 | Badminton |

2 | Cricket |

2 | Badminton |

4 | Badminton |

**Table 2**

**ALL_SPORTS**

**Table 3**

**EMPLOYEE**

EMP_NO | NAME | ADDRESS | PHONE | AGE |

1 | RAM | DELHI | 9455123451 | 18 |

5 | NARESH | HISAR | 9782918192 | 22 |

6 | SWETA | RANCHI | 9852617621 | 21 |

4 | SURESH | DELHI | 9156768971 | 18 |

**Table 4**

* Intersection (∩):* Intersection on two relations R1 and R2 can only be computed if R1 and R2 are

Relation1 ∩ Relation2

Example: Find a person who is student as well as employee- STUDENT ∩ EMPLOYEE

In terms of basic operators (union and minus) :

STUDENT ∩ EMPLOYEE = STUDENT + EMPLOYEE - (STUDENT U EMPLOYEE)

**RESULT :**

* Conditional Join(⋈_{c}):* Conditional Join is used when you want to join two or more relation based on some conditions. Example: Select students whose ROLL_NO is greater than EMP_NO of employees

STUDENT⋈c STUDENT.ROLL_NO>EMPLOYEE.EMP_NOEMPLOYEE

In terms of basic operators (cross product and selection) :

σ (STUDENT.ROLL_NO>EMPLOYEE.EMP_NO)(STUDENT×EMPLOYEE)

**RESULT :**

* Equijoin:* Equijoin is a

Example:Select students whose ROLL_NO is equal to EMP_NO of employees

STUDENT⋈STUDENT.ROLL_NO=EMPLOYEE.EMP_NOEMPLOYEE

In terms of basic operators (cross product, selection and projection) :

∏(STUDENT.ROLL_NO, STUDENT.NAME, STUDENT.ADDRESS, STUDENT.PHONE, STUDENT.AGE EMPLOYEE.NAME, EMPLOYEE.ADDRESS, EMPLOYEE.PHONE, EMPLOYEE>AGE)(σ (STUDENT.ROLL_NO=EMPLOYEE.EMP_NO) (STUDENT×EMPLOYEE))

**RESULT :**

* Natural Join(⋈): *It is a special case of equijoin in which equality condition hold on all attributes which have same name in relations R and S (relations on which join operation is applied). While applying natural join on two relations, there is no need to write equality condition explicitly. Natural Join will also return the similar attributes only once as their value will be same in resulting relation.

Example: Select students whose ROLL_NO is equal to ROLL_NO of STUDENT_SPORTS as:

STUDENT⋈STUDENT_SPORTS

In terms of basic operators (cross product, selection and projection) :

∏(STUDENT.ROLL_NO, STUDENT.NAME, STUDENT.ADDRESS, STUDENT.PHONE, STUDENT.AGE STUDENT_SPORTS.SPORTS)(σ (STUDENT.ROLL_NO=STUDENT_SPORTS.ROLL_NO) (STUDENT×STUDENT_SPORTS))

**RESULT :**

Natural Join is by default inner join because the tuples which does not satisfy the conditions of join does not appear in result set. e.g.; The tuple having ROLL_NO 3 in STUDENT does not match with any tuple in STUDENT_SPORTS, so it has not been a part of result set.

**Left Outer Join (****)** When applying join on two relations R and S, some tuples of R or S does not appear in result set which does not satisfy the join conditions. But Left Outer Joins gives all tuples of R in the result set. The tuples of R which do not satisfy join condition will have values as NULL for attributes of S.

**Example:**Select students whose ROLL_NO is greater than EMP_NO of employees and details of other students as well

STUDENT(⟕)STUDENT.ROLL_NO>EMPLOYEE.EMP_NOEMPLOYEE

**RESULT**

**Right Outer Join (****)** When applying join on two relations R and S, some tuples of R or S does not appear in result set which does not satisfy the join conditions. But Right Outer Joins gives all tuples of S in the result set. The tuples of S which do not satisfy join condition will have values as NULL for attributes of R.

**Example:** Select students whose ROLL_NO is greater than EMP_NO of employees and details of other Employees as well

STUDENT(⟖)STUDENT.ROLL_NO>EMPLOYEE.EMP_NOEMPLOYEE

**RESULT :**

*Full Outer Join ***(****) **

When applying join on two relations R and S, some tuples of R or S does not appear in result set which does not satisfy the join conditions. But Full Outer Joins gives all tuples of S and all tuples of R in the result set. The tuples of S which do not satisfy join condition will have values as NULL for attributes of R and vice versa.

**Example: **Select students whose ROLL_NO is greater than EMP_NO of employees and details of other Employees as well and other Students as well

STUDENT(⟗)STUDENT.ROLL_NO>EMPLOYEE.EMP_NOEMPLOYEE

**RESULT :**

* Division Operator (÷): *Division operator A

- Attributes of B is proper subset of Attributes of A.
- The relation returned by division operator will have attributes = (All attributes of A – All Attributes of B)
- The relation returned by division operator will return those tuples from relation A which are associated to every B’s tuple.

Consider the relation STUDENT_SPORTS and ALL_SPORTS given in Table 2 and Table 3 above.

To apply division operator as

STUDENT_SPORTS÷ ALL_SPORTS

- The operation is valid as attributes in ALL_SPORTS is a proper subset of attributes in STUDENT_SPORTS.
- The attributes in resulting relation will have attributes {ROLL_NO,SPORTS}-{SPORTS}=ROLL_NO
- The tuples in resulting relation will have those ROLL_NO which are associated with all B’s tuple {Badminton, Cricket}. ROLL_NO 1 and 4 are associated to Badminton only. ROLL_NO 2 is associated to all tuples of B. So the resulting relation will be:

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