The Relational Algebra

1 Relational Algebra Operations From Set Theory * UNION * INTERSECTION * MINUS * CARTESIAN OPERATION

2 Binary Relational Operations *JOIN *DIVISION

CONTENTS

UNION Operation

The result of this operation, denoted by R S, is a relation that includes all

tuples that are either in R or in S or in both R and S. Duplicate tuples are

eliminated.

Union operation is a binary which means that it always operates on two

relations.

The notation for the union of relations(table) R &S is R S Is Q

We have two relations R&S , then the union of R &S as shown in fig.

Relational Algebra Operations FromSet Theory

ID NAME

501 AMIT

503 ANAND

504 SONAL

506 MADHU

507 RAJNI

510 KAPIL

512 RANJIT

ID NAME

503 ANAND

504 SONAL

506 MADHU

510 KAPIL

ID NAME

501 AMIT

503 ANAND

504 SONAL

507 RAJNI

510 KAPIL

512 RANJIT

R

SR S

Properties of Union Operation

1.The input relations must be union compatible.

2.Commutativity -It means that result of R S is same as

that of S R.

3. Associativity- It means that

P ( Q S)= (P Q) S Where P,Q and S

are relations.

BACK TO INDEX

INTERSECTION OPERATION

The result of this operation, denoted by R S, is a relation that includes all tuples that are in both R and S. The two relations used in intersection operation must be “union compatible’’

The intersection operation is a binary operation.

The notation for the Intersection of relations(table) R &S is R S Is Q

We have two relations R&S , then the intersection of R &S as shown in fig.

ID NAME

503 ANAND

504 SONAL

510 KAPIL

ID NAME

503 ANAND

504 SONAL

506 MADHU

510 KAPIL

ID NAME

501 AMIT

503 ANAND

504 SONAL

507 RAJNI

510 KAPIL

512 RANJIT

R

S

Q=R S

Properties of Intersection Operation

1.The input relations must be union compatible.

2.Commutativity -It means that result of R S is same as

that of S R.

3. Associativity- It means that

P ( Q S)= (P Q) S Where P,Q and S

are relations.

BACK TO INDEX

Set Difference (or MINUS) Operation

The result of this operation, denoted by R - S, is a relation that includes all tuples that are in R but not in S. The two relations used in difference operation must be “union compatible’’

The Difference operation is a binary operation, which means that it always operates on two relations

The notation for the Difference of relations(table) R &S is R - S Is Q

ID NAME

501 AMIT

507 RAJNI

512 RANJIT

ID NAME

503 ANAND

504 SONAL

506 MADHU

510 KAPIL

ID NAME

501 AMIT

503 ANAND

504 SONAL

507 RAJNI

510 KAPIL

512 RANJIT

R

S

Q=R - S

Properties of Difference Operation

1.The input relations must be union compatible.

2. They are not Commutative-It means that result of R - S is same as that

of S - R.

3. They are not Associative- It means that

P - ( Q - S) ≠ (P - Q) - S Where P,Q and S

are relations.

BACK TO INDEX

CARTESIAN (or cross product) Operation

The Cartesian Product of two relations result in new relation that include concatenation of every tuple of first relation with every tuple of second relation.In other words , the new relation is created consisting of all possible combinations of the tuple.It is also known as the Cross Product or Cross Join.

The Cartesian operation is a binary operation. Which means that it always operates on two relations.

The notation for the Cartesian Product of relations(table) R &S is RXS Is Q

R.ID R.NAME

S.ID

S.NAME

501 AMIT 503 ANAND

501 AMIT 506 MADHU

503 ANAND 503 ANAND

503 ANAND 506 MADHU

504 SONAL 503 ANAND

504 SONAL 506 MADHU

ID NAME

503 ANAND

506 MADHU

ID NAME

501 AMIT

503 ANAND

504 SONAL

R Q=RX S

S

Properties of Cartesian Product Operation1.The relations on which Cartesian product operation is applied need not necessary to be union compatible.2.Commutativity-It means that result of R X S is same as that of S X R. 3.Associativity- It means that P X ( Q XS) = (P X Q) XS Where P,Q and S are relations.4.The resultant relation(table) may hold duplicate attributes if some attributes of two relations are defined on common domains.5.The Degree of the resultant operation is equal to the sum of the degree of all the relations i.e. Degree of R= Degree of P + Degree of Q6. The total number of rows in the resultant operation is equal to the product of the number of rows of the first and the second relation i.e. Total number of Tuples of R = Total number of Tuples of P X Total number of Tuples of Q.

BACK TO INDEX

JOIN Operation◦ The sequence of Cartesian product followed by select is used quite

commonly to identify and select related tuples from two relations, a special operation, called JOIN. It is denoted by ∞

◦ This operation is very important for any relational database with more than a single relation, because it allows us to process relationships among relations.

◦ The general form of a join operation on two relations R(A1, A2, . . ., An) and S(B1, B2, . . ., Bm) is:

R∞<join condition>S

where R and S can be any relations that result from general relational algebra expressions.

Consider the example of EMP & DEPT table

Binary Relational Operations

EMP ID ENAME SALARY DEPT ID

101 ANU 12000 01

103 PRIYA 15000 01

106 ANKUR 13000 02

DEPT ID

DNAME

01 FINANCE

01 MARKETING

02 PACKINGEMP ID ENAME SALARY DEPT

IDDEPT ID

DNAME

101 ANU 12000 01 01 FINANCE

103 PRIYA 15000 01 01 MARKETING

106 ANKUR 13000 02 02 PACKING

EMP∞dept_id=dept_no DEPT

EMP DEPT

In the above example, the DEPT ID attribute of EMP table is matched with the DEPTNO attribute of DEPT table and if both are equal then the resultant tuples from both the tables are fetched as a single row

The common attribute in both the tables should not have the same name so as to preserve the uniqueness of the column names in the resultant relation.

If they have the same name then they should be renamed to preserve uniqueness.

There is another join known as Natural Join in which there is no need of explicitly naming the columns. The join is Performed by joining all those col from the first table to any col in the second table with the same name

The result of Natural Join in the above example is as shown

EMP ID ENAME SALARY DEPT ID

DNAME

101 ANU 12000 01 FINANCE

103 PRIYA 15000 01 MARKETING

106 ANKUR 13000 02 PACKING

Natural Join on EMP & DEPT table

The natural Join is also referred as Inner Join

The Another type of join in which relation is joined to it self by comparing the values with a col of a single relation is known as Self Join

EMP ID

ENAME MANG ID

101 REKHA -------

103 KAPIL 101

104 AMAR 106

106 ATUL -------

107 PUNEET 1-1

ENAME

ENAME

KAPIL REKHA

AMAR ATUL

PUNEET

REKHA

EMPTHE RESULTANT RELATION

THE RESULT OF SELF JOIN

Properties of Join Operation1.Commutativity-It means that result of R ⋈ S is same as that of S ⋈ R. 2.Associativity- Using this the same rows appear in the final resuIt regardless of the order in which relations are joined. P ⋈ ( Q ⋈ S) = (P ⋈ Q) ⋈ S Where P,Q and S are relations.3.We can combine a selection with a cross product to form a join. R ⋈ S= (R X S)4.We can commute a selection with a join . If the selection condition only involves attributes of one of the arguments to the join. (R ⋈ S )=(R) ⋈ S . The attributes must appear only in R and not in S.5. Rows whose join attributes are null,don’t appear in the resultants relation(table).6.More than two tables can also be joined but it increases the complexity.7.Joins are most commonly used when there exists a relationship between tables i.e. where a join condition is based on a Primary key and Foreign key columns.

BACK TO INDEX

DIVISION Operation The division operation results in a new relation such that every tuple

appearing in the resultant relation must exist in the dividend relation in

combination with the every tuple in divisor relation .

It is binary operation which operates in two relations.

The division is denoted by a symbol “÷”

The notation for the division of two relations P & Q is P÷Q. where P is the dividend relation and Q is the divisor relation.

Consider an example, in which we take two tables P and Q

N

n1

n2

N

n1

M N

m1 n1

m1 n2

m2 n1

m3 n3

m4 n1

m4 n2

m1 n3

N

n1

n2

n3

Q

-

P Q Q Q Q

i ii iii iv

M

m1

m4

M

m1

m2

m3

M

m1

M

m1

m2

m3

m4

R R R R

The resultant relation R obtained by P÷Q in each case when tuples of Q are different

Properties of Divide Operation

1. It is a binary operation as it operates on two relations(tables).

2.In the Operation P÷Q , the relation P is known as dividend relation and

relation Q is known as a divisor relation . If the dividend relation has

degree m+n and the divisor relation has a degree of n , then the resultant

relation will have a degree m.

3.The division operation is suited to queries that include the phrase “for

all”.

4.It is very rarely used in database applications.

BACK TO INDEX

THANKS

BACK TO INDEX