Unit-3 (3.3) Relational Algebra

Relational model stores data in the form of tables. This concept purposed by Dr. E.F. Codd, a researcher of IBM in the year 1960s. The relational model consists of three major components:

1. The set of relations and set of domains that defines the way data can be represented (data structure).

2. Integrity rules that define the procedure to protect the data (data integrity).

3. The operations that can be performed on data (data manipulation).

A rational model database is defined as a database that allows to group its data items into one or more independent tables that can be related to one another by using fields common to each related table.

Relational model can represent as a table with columns and rows. Each row is known as a tuple. Each table of the column has a name or attribute.

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Relational Model Concepts

1. Attribute: Each column in a Table. Attributes are the properties which define a relation. e.g., Student_Rollno, NAME,etc.
2. Tables – In the Relational model the, relations are saved in the table format. It is stored along with its entities. A table has two properties rows and columns. Rows represent records and columns represent attributes.
3. Tuple – It is nothing but a single row of a table, which contains a single record.
4. Relation Schema: A relation schema represents the name of the relation with its attributes.
5. Degree: The total number of attributes which in the relation is called the degree of the relation.
6. Cardinality: Total number of rows present in the Table.
7. Column: The column represents the set of values for a specific attribute.
8. Relation instance – Relation instance is a finite set of tuples in the RDBMS system. Relation instances never have duplicate tuples.
9. Relation key - Every row has one, two or multiple attributes, which is called relation key.
10. Attribute domain – Every attribute has some pre-defined value and scope which is known as attribute domain

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Domain: It contains a set of atomic values that an attribute can take.

Tables − In relational data model, relations are saved in the format of Tables. This format stores the relation among entities. A table has rows and columns, where rows represents records and columns represent the attributes.

Tuple − A single row of a table, which contains a single record for that relation is called a tuple.

Relation instance − A finite set of tuples in the relational database system represents relation instance. Relation instances do not have duplicate tuples. Or In the relational database system, the relational instance is represented by a finite set of tuples. Relation instances do not have duplicate tuples.

Relation schema − A relation schema describes the relation name (table name), attributes, and their names. Or  A relational schema contains the name of the relation and name of all columns or attributes.

Relation key − Each row has one or more attributes, known as relation key, which can identify the row in the relation (table) uniquely. Or In the relational key, each row has one or more attributes. It can identify the row in the relation uniquely

Attribute domain − Every attribute has some pre-defined value scope, known as attribute domain. It contains the name of a column in a particular table. Each attribute Ai must have a domain, dom(Ai)

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Properties of Relations

Ø  Name of the relation is distinct from all other relations.

Ø  Each relation cell contains exactly one atomic (single) value

Ø  Each attribute contains a distinct name

Ø  Attribute domain has no significance

Ø  tuple has no duplicate value

Ø  Order of tuple can have a different sequence

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Ø  A table has a name that is distinct from all other tables in the database.

Ø  There are no duplicate rows; each row is distinct.

Ø  Entries in columns are atomic. The table does not contain repeating groups or multivalued attributes.

Ø  Entries from columns are from the same domain based on their data type including:

o   number (numeric, integer, float, smallint,…)

o   character (string)

o   date

o   logical (true or false)

Ø  Operations combining different data types are disallowed.

Ø  Each attribute has a distinct name.

Ø  The sequence of columns is insignificant.

Ø  The sequence of rows is insignificant.

Relational Algebra

Relational algebra is a procedural query language. It gives a step by step process to obtain the result of the query. It uses operators to perform queries.

We have learned the 6 fundamental operations of relational algebra:

Rename _ (Rho) ρ

Selection _ σ Sigma

Projection _ uppercase of (pi) ∏

Set union - ∪

Set difference : -

Cartesian product _ X

Types of Relational operation

1. Select Operation:

• The select operation selects tuples that satisfy a given predicate.
• It is denoted by sigma (σ).

Notation:  σ p(r)

Where:

σ is used for selection prediction
r is used for relation
p is used as a propositional logic formula which may use connectors like: AND OR and NOT. These relational can use as relational operators like =, ≠, ≥, <, >, ≤.

For example: LOAN Relation

BRANCH_NAME	LOAN_NO	AMOUNT
Downtown	L-17	1000
Redwood	L-23	2000
Perryride	L-15	1500
Downtown	L-14	1500
Mianus	L-13	500
Roundhill	L-11	900
Perryride	L-16	1300

Input:

σ BRANCH_NAME="perryride" (LOAN)  

Output:

BRANCH_NAME	LOAN_NO	AMOUNT
Perryride	L-15	1500
Perryride	L-16	1300

2. Project Operation:

• This operation shows the list of those attributes that we wish to appear in the result. Rest of the attributes are eliminated from the table.
• It is denoted by ∏.

1.            Notation: ∏ A1, A2, An (r)   

Where

A1, A2, A3 is used as an attribute name of relation r.

Example: CUSTOMER RELATION

NAME	STREET	CITY
Jones	Main	Harrison
Smith	North	Rye
Hays	Main	Harrison
Curry	North	Rye
Johnson	Alma	Brooklyn
Brooks	Senator	Brooklyn

Input:

1.               ∏ NAME, CITY (CUSTOMER)  

Output:

NAME	CITY
Jones	Harrison
Smith	Rye
Hays	Harrison
Curry	Rye
Johnson	Brooklyn
Brooks	Brooklyn

3. Union Operation:

• Suppose there are two tuples R and S. The union operation contains all the tuples that are either in R or S or both in R & S.
• It eliminates the duplicate tuples. It is denoted by ∪.

1.   Notation: R ∪ S   

A union operation must hold the following condition:

• R and S must have the attribute of the same number.
• Duplicate tuples are eliminated automatically.

Example:

DEPOSITOR RELATION

CUSTOMER_NAME	ACCOUNT_NO
Johnson	A-101
Smith	A-121
Mayes	A-321
Turner	A-176
Johnson	A-273
Jones	A-472
Lindsay	A-284

BORROW RELATION

CUSTOMER_NAME	LOAN_NO
Jones	L-17
Smith	L-23
Hayes	L-15
Jackson	L-14
Curry	L-93
Smith	L-11
Williams	L-17

Input:

1.                  ∏ CUSTOMER_NAME (BORROW) ∪ ∏ CUSTOMER_NAME (DEPOSITOR)  

Output:

CUSTOMER_NAME

Johnson

Smith

Hayes

Turner

Jones

Lindsay

Jackson

Curry

Williams

Mayes

4. Set Intersection:

• Suppose there are two tuples R and S. The set intersection operation contains all tuples that are in both R & S.
• It is denoted by intersection ∩.

1.               Notation: R ∩ S   

Example: Using the above DEPOSITOR table and BORROW table

Input:

1.                     ∏ CUSTOMER_NAME (BORROW) ∩ ∏ CUSTOMER_NAME (DEPOSITOR)  

Output:

CUSTOMER_NAME

Smith

Jones

5. Set Difference:

• Suppose there are two tuples R and S. The set intersection operation contains all tuples that are in R but not in S.
• It is denoted by intersection minus (-).

1.                  Notation: R - S  

Example: Using the above DEPOSITOR table and BORROW table

Input:

1.                        ∏ CUSTOMER_NAME (BORROW) - ∏ CUSTOMER_NAME (DEPOSITOR)  

Output:

CUSTOMER_NAME

Jackson

Hayes

Willians

Curry

6. Cartesian product

• The Cartesian product is used to combine each row in one table with each row in the other table. It is also known as a cross product.
• It is denoted by X.

1.      Notation: E X D  

Example:

EMPLOYEE

EMP_ID	EMP_NAME	EMP_DEPT
1	Smith	A
2	Harry	C
3	John	B

DEPARTMENT

DEPT_NO	DEPT_NAME
A	Marketing
B	Sales
C	Legal

Input:

1.                        EMPLOYEE X DEPARTMENT  

Output:

EMP_ID	EMP_NAME	EMP_DEPT	DEPT_NO	DEPT_NAME
1	Smith	A	A	Marketing
1	Smith	A	B	Sales
1	Smith	A	C	Legal
2	Harry	C	A	Marketing
2	Harry	C	B	Sales
2	Harry	C	C	Legal
3	John	B	A	Marketing
3	John	B	B	Sales
3	John	B	C	Legal

7. Rename Operation:

The rename operation is used to rename the output relation. It is denoted by rho (ρ).

Example: We can use the rename operator to rename STUDENT relation to STUDENT1.

1.                        ρ(STUDENT1, STUDENT)  

Join Operations:

A Join operation combines related tuples from different relations, if and only if a given join condition is satisfied. It is denoted by ⋈.

Example:

EMPLOYEE

EMP_CODE	EMP_NAME
101	Stephan
102	Jack
103	Harry

SALARY

EMP_CODE	SALARY
101	50000
102	30000
103	25000

1.      Operation: (EMPLOYEE ⋈ SALARY)

Result:

EMP_CODE	EMP_NAME	SALARY
101	Stephan	50000
102	Jack	30000
103	Harry	25000

Types of Join operations:

1. Natural Join:

• A natural join is the set of tuples of all combinations in R and S that are equal on their common attribute names.
• It is denoted by ⋈.

Example: Let's use the above EMPLOYEE table and SALARY table:

Input:

1.      ∏EMP_NAME, SALARY (EMPLOYEE ⋈ SALARY)

Output:

EMP_NAME	SALARY
Stephan	50000
Jack	30000
Harry	25000

2. Outer Join:

The outer join operation is an extension of the join operation. It is used to deal with missing information.

Example:

EMPLOYEE

EMP_NAME	STREET	CITY
Ram	Civil line	Mumbai
Shyam	Park street	Kolkata
Ravi	M.G. Street	Delhi
Hari	Nehru nagar	Hyderabad

FACT_WORKERS

EMP_NAME	BRANCH	SALARY
Ram	Infosys	10000
Shyam	Wipro	20000
Kuber	HCL	30000
Hari	TCS	50000

Input:

1.      (EMPLOYEE ⋈ FACT_WORKERS)

Output:

EMP_NAME	STREET	CITY	BRANCH	SALARY
Ram	Civil line	Mumbai	Infosys	10000
Shyam	Park street	Kolkata	Wipro	20000
Hari	Nehru nagar	Hyderabad	TCS	50000

An outer join is basically of three types:

1. Left outer join
2. Right outer join
3. Full outer join

a. Left outer join:

• Left outer join contains the set of tuples of all combinations in R and S that are equal on their common attribute names.
• In the left outer join, tuples in R have no matching tuples in S.
• It is denoted by ⟕.

Example: Using the above EMPLOYEE table and FACT_WORKERS table

Input:

1.      EMPLOYEE ⟕ FACT_WORKERS

EMP_NAME	STREET	CITY	BRANCH	SALARY
Ram	Civil line	Mumbai	Infosys	10000
Shyam	Park street	Kolkata	Wipro	20000
Hari	Nehru street	Hyderabad	TCS	50000
Ravi	M.G. Street	Delhi	NULL	NULL

b. Right outer join:

• Right outer join contains the set of tuples of all combinations in R and S that are equal on their common attribute names.
• In right outer join, tuples in S have no matching tuples in R.
• It is denoted by ⟖.

Example: Using the above EMPLOYEE table and FACT_WORKERS Relation

Input:

1.      EMPLOYEE ⟖ FACT_WORKERS

Output:

EMP_NAME	BRANCH	SALARY	STREET	CITY
Ram	Infosys	10000	Civil line	Mumbai
Shyam	Wipro	20000	Park street	Kolkata
Hari	TCS	50000	Nehru street	Hyderabad
Kuber	HCL	30000	NULL	NULL

c. Full outer join:

• Full outer join is like a left or right join except that it contains all rows from both tables.
• In full outer join, tuples in R that have no matching tuples in S and tuples in S that have no matching tuples in R in their common attribute name.
• It is denoted by ⟗.

Example: Using the above EMPLOYEE table and FACT_WORKERS table

Input:

1.      EMPLOYEE ⟗ FACT_WORKERS

Output:

EMP_NAME	STREET	CITY	BRANCH	SALARY
Ram	Civil line	Mumbai	Infosys	10000
Shyam	Park street	Kolkata	Wipro	20000
Hari	Nehru street	Hyderabad	TCS	50000
Ravi	M.G. Street	Delhi	NULL	NULL
Kuber	NULL	NULL	HCL	30000

3. Equi join:

It is also known as an inner join. It is the most common join. It is based on matched data as per the equality condition. The equi join uses the comparison operator(=).

Example:

CUSTOMER RELATION

CLASS_ID	NAME
1	John
2	Harry
3	Jackson

PRODUCT

PRODUCT_ID	CITY
1	Delhi
2	Mumbai
3	Noida

Input:

1.                  CUSTOMER ⋈ PRODUCT    

Output:

CLASS_ID	NAME	PRODUCT_ID	CITY
1	John	1	Delhi
2	Harry	2	Mumbai
3	Harry	3	Noida

Division Operator (÷): 

Division operator 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.

Consider the relation STUDENT_SPORTS and ALL_SPORTS given in Table 1 and Table 2 above.

Table 1

STUDENT_SPORTS

ROLL_NO	SPORTS
1	Badminton
2	Cricket
2	Badminton
4	Badminton

Table 2

ALL_SPORTS

SPORTS

Badminton

Cricket

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:

ROLL_NO

2

Assignment

Assignments are often used to increase clarity by cutting a long query into multiple steps, each of which can be described by a short line.

Denoted by:  T   ß[expression]

where [expression] is a relational algebra expression, and T is a table variable.

The assignment stores in T the table output by [expression].

Reference:

https://www.guru99.com/relational-data-model-dbms.html

http://ecomputernotes.com/fundamental/what-is-a-database/relational-model

https://www.javatpoint.com/dbms-relational-model-concept

https://www.javatpoint.com/dbms-relational-algebra

https://www.javatpoint.com/dbms-join-operation

https://www.geeksforgeeks.org/extended-operators-in-relational-algebra/

https://www.assignmenthelp.net/assignment_help/relational-algebra-assignment-help