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Extended Operators in Relational Algebra

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  • Difficulty Level : Easy
  • Last Updated : 18 Nov, 2022
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Basic idea about  relational model and basic operators in Relational Algebra: Relational Model Basic Operators 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_NONAMEADDRESSPHONEAGE
1RAMDELHI945512345118
2RAMESHGURGAON965243154318
3SUJITROHTAK915625313120
4SURESHDELHI915676897118

Table 1

  STUDENT_SPORTS         

ROLL_NOSPORTS
1Badminton
2Cricket
2Badminton
4Badminton

Table 2

 ALL_SPORTS

SPORTS
Badminton
Cricket 

Table 3

EMPLOYEE                                                               

EMP_NONAMEADDRESSPHONEAGE
1RAMDELHI945512345118
5NARESHHISAR978291819222
6SWETARANCHI985261762121
4SURESHDELHI915676897118

 Table 4

Intersection (∩): Intersection on two relations R1 and R2 can only be computed if R1 and R2 are union compatible (These two relation should have same number of attributes and corresponding attributes in two relations have same domain). Intersection operator when applied on two relations as R1R2 will give a relation with tuples which are in R1 as well as R2. Syntax:

 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:

ROLL_NONAMEADDRESSPHONEAGE
1RAMDELHI945512345118
4SURESHDELHI915676897118

  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

STUDENTc 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:

ROLL_NONAMEADDRESSPHONEAGEEMP_NONAMEADDRESSPHONEAGE
2RAMESHGURGAON9652431543181RAMDELHI945512345118
3SUJITROHTAK9156253131201RAMDELHI945512345118
4SURESHDELHI9156768971181RAMDELHI945512345118

Equijoin(⋈): Equijoin is a special case of conditional join where only equality condition holds between a pair of attributes. As values of two attributes will be equal in result of equijoin, only one attribute will be appeared in result. 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:

ROLL_NONAMEADDRESSPHONEAGENAMEADDRESSPHONEAGE
1RAMDELHI945512345118RAMDELHI945512345118
4SURESHDELHI915676897118SURESHDELHI915676897118

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:

ROLL_NONAMEADDRESSPHONEAGESPORTS
1RAMDELHI945512345118Badminton
2RAMESHGURGAON965243154318Cricket
2RAMESHGURGAON965243154318Badminton
4SURESHDELHI915676897118Badminton

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

STUDENTSTUDENT.ROLL_NO>EMPLOYEE.EMP_NOEMPLOYEE

RESULT

ROLL_NONAMEADDRESSPHONEAGEEMP_NONAMEADDRESSPHONEAGE
2RAMESHGURGAON9652431543181RAMDELHI945512345118
3SUJITROHTAK9156253131201RAMDELHI945512345118
4SURESHDELHI9156768971181RAMDELHI945512345118
1RAMDELHI945512345118NULLNULLNULLNULLNULL

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:

ROLL_NONAMEADDRESSPHONEAGEEMP_NONAMEADDRESSPHONEAGE
2RAMESHGURGAON9652431543181RAMDELHI945512345118
3SUJITROHTAK9156253131201RAMDELHI945512345118
4SURESHDELHI9156768971181RAMDELHI945512345118
NULLNULLNULLNULLNULL5NARESHHISAR978291819222
NULLNULLNULLNULLNULL6SWETARANCHI985261762121
NULLNULLNULLNULLNULL4SURESHDELHI915676897118

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:

ROLL_NONAMEADDRESSPHONEAGEEMP_NONAMEADDRESSPHONEAGE
2RAMESHGURGAON9652431543181RAMDELHI945512345118
3SUJITROHTAK9156253131201RAMDELHI945512345118
4SURESHDELHI9156768971181RAMDELHI945512345118
NULLNULLNULLNULLNULL5NARESHHISAR978291819222
NULLNULLNULLNULLNULL6SWETARANCHI985261762121
NULLNULLNULLNULLNULL4SURESHDELHI915676897118
1RAMDELHI945512345118NULLNULLNULLNULLNULL

Division Operator (÷): Division operator A÷B or A/B can be applied if and only if:

  • 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.

                                                                                                  A

xy
a1
b2
a2
d4

                                                                                                    ÷∏

                                                                                                  B

y
1
2

 The resultant of A/B is  

                                                                                                A ÷ B

x
a

 Division can be expressed in terms of Cross Product , Set Difference and Projection.

 In the above example , for A/B , compute all  x values that are not disqualified by some y in B.

 x value is disqualified if attaching y value from B, we obtain xy tuple that is not in A.

 Disqualified x values:     ∏x(( ∏x(A) × B ) – A)

 So     A/B  = ∏x( A )  − all disqualified tuples

          A/B   = ∏x( A )  −  ∏x(( ∏x(A) × B ) – A)

 In the above example , disqualified tuples are 

b2
d4

 So, the resultant is 

x
a

Overview of Relational Algebra Operators Previous Year Gate Questions https://www.geeksforgeeks.org/gate-gate-cs-2012-question-50/ https://www.geeksforgeeks.org/gate-gate-cs-2012-question-43/ Article contributed by Sonal Tuteja. Please write comments if you find anything incorrect, or you want to share more information about the topic discussed above.


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