Relational Algebra in DBMS
What is Relational Algebra in DBMS?
On this page, we will learn about Relational Algebra in DBMS.
Relational Algebra in DBMS is a query language which is procedural in nature, both of its input and output are relations. The theoretical foundations of relational databases and SQL is provided by Relational Algebra.
Relational Algebra in DBMS
Following operations
can be applied via relational algebra –
 Select
 Project
 Union
 Set Different
 Cartesian product
 Rename
Select Operation (σ)
Use – Fetching rows (tuples) from a table, which satisfied a given condition.
Notation – σ_{p}(r)
Breakdown –
σ
represents select predicater
for relationp
for proposition logics like –− =, ≠, ≥, < , >, ≤
. with us of connectors likeOR
,AND
, orNOT
Like we used in the earlier example in the image above, of selecting rows for people who had age > 25 and give results in Fname format.
Example 1
 Query –
σ_{age} > 25 (Student)
σ
(select predicate)r
(relation) – Employeep
(proposition logic) – age >25 Result – returning the list of students with age greater than 25.
 In SQL – select * from Student where age > 25
Example 2
 Query
σ_{age > 25 and Fname = 'Arya'} (Student)
σ
(select predicate)r
(relation) – Employeep
(proposition logic) – age >25 and Fname = ‘Arya’ Result – returning the list of students with age greater than 25 and Fname equals to arya
 In SQL – select * from Student where age > 25 & Fname = ‘Arya’
 Query
Example 3
 Query
σ_{Lname = 'Stark'} (Student)
σ
(select predicate)r
(relation) – Employeep
(proposition logic) – Lname = ‘Stark’ Result – returning the list of students Lname = ‘Stark’
 In SQL – select * from Student where Lname = ‘Stark’
Project Operation (∏)
Use – Fetching on specific columns from a table
Notation – ∏ _{A1,A2,An}(r)
Breakdown –
∏
represents Project predicater
for relationA1, A2, A3
for selection from columns for projection
Example 1
 Query
σ_{Lname = 'Stark'} (Student)
σ
(select predicate)r
(relation) – Employeep
(proposition logic) – Lname = ‘Stark’ Result – returning the list of students Lname = ‘Stark’
 In SQL – select * from Student where Lname = ‘Stark’
ID  Fname  Lname  Age 

1  Jon  Stark  25 
2  Arya  Stark  28 
3  Bran  Stark  26 
4  Sansa  Stark  27 
Union Operation (∪)
Union Operation performs as expected, it essentially finds the union of the tables included in the union i.e finds only the unique rows/tuples from multiple tables, removing the duplications.
Use – Fetching union rows (tuples), i.e unique rows (tuples) from multiple tables removing the duplications
Notation – A(∪)B
Breakdown –
(∪)
represents Union Operation A and B are the tables
 Query
A(∪)B
(∪)
(Union Operation)A and B
Table In SQL – SELECT * FROM A UNION SELECT * FROM B;
Resultant
1  Jon  Stark  25 
Resultant
Table A
ID  FNAME  LNAME  AGE 

1  John  Stark  25 
2  Arya  Stark  28 
3  Bran  Stark  26 
4  Sansa  Stark  27 
5  Cersie  Lannister  40 
6  Jamie  Lannister  40 
7  Tywin  Lannister  65 
Intersection Operation (∩)
Intersection operation works simply by helping to find the rows (tuples) that are common i.e. exists in both (all) the tables involved in the intersection operation.
Use – Fetching union rows (tuples), i.e common rows (tuples) from multiple tables and only the rows that exist in both (all) tables involved in the operation.
Notation – A(∩)B
Breakdown –
(∩)
represents Intersection Operation A and B are the tables
Table A
ID  Fname  Lname  Age 

1  Jon  Stark  25 
2  Arya  Stark  28 
3  Bran  Stark  26 
4  Sansa  Stark  27 
Table B
ID  Fname  Lname  Age 

1  Jon  Stark  25 
5  Cersie  Lannister  40 
6  Jamie  Lannister  40 
7  Tywin  Lannister  65 
 Query
A(∩)B

A and B
Tables(∩)
(Intersection Operation) In SQL – SELECT * FROM A INTERSECT SELECT * FROM B;
Resultant
ID  Fname  Lname  Age 
1  Jon  Stark  25 
∏ _{player} (Cricket) ∪ ∏ _{player} (Football)The following gives the result for selecting the people who either play only cricket or only play football or play both of them.
Set Difference (−)
Use – Fetching rows from which are present in relation but not the other one. Example players who play cricket but don’t play football.
Notation – A  B
Breakdown –

represents Set Difference Operation A and B are the tables (relations)
Table Plays Cricket
ID  Fname 

1  Jon 
2  Arya 
3  Bran 
4  Sansa 
Table Football
ID  Fname 

1  Jon 
2  Arya 
5  Cersi 
6  Jamie 
 Query
A  B

A and B
Tables A: Plays Cricket B: Plays Football
(Intersection Operation) In SQL – SELECT * FROM A INTERSECT SELECT * FROM B;
Resultant A  B
ID  Fname 

3  Bran 
4  Sansa 
Resultant B  A
ID  Fname 

5  Cersie 
6  Jamie 
Cartesian Product (Χ)
Use – Merging columns from two different relations, i.e. combining them together. It is mostly not a suitable and meaningful data representation if we just calculate the cartesian product alone. To get meaningful data we must follow the same by subsequent operations. We will understand more about this in detail.
Notation – A X B
Breakdown –
X
represents Cartesian Product A and B are the tables (relations)
ID  Fname  Country 

3  Bran  UK 
4  Sansa  UK 
5  Jon  India 
ID  Fname  Country 

5  Cersie  India 
6  Jamie  Australia 
5  Samwell  India 
Imaging that we have list of all Facebook and Twitter users and we want to find out the unique list of people from both of them who live in India.
σ_{Fname = 'Country'}(Facebook Χ Twitter)
Result
Fname 

Jon 
Samwell 
Rename Operation (ρ)
Use – When we find the result of a some relational operation they are just displayed !. However, we may want to store them in a new relation with a name that we can further use. For the same we use rename operation.
Don’t get confused with the name, the name might give idea that it is done to rename an existing table. However, this is used to store results of relation in a new named table.
Notation – ρ_{x} (E)
Breakdown –
ρ
i.e. rho represents rename operation.x
is the new name of the resultant table, E is the expression used for result.
Use – ρ _{users} (E =
A (∩) B)
The results of expression will be stored in new table names as users.
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