Relational Algebra Operations in DBMS

In this article Relational Algebra Operations in DBMS we give the information about Relational Algebra is a procedural query language that takes relations as input and returns relations as output. It is the theoretical foundation of SQL queries.

Relational Algebra Operations:

Relational Algebra is a procedural query language that takes relations as input and returns relations as output. It is the theoretical foundation of SQL queries.

Selection (σ)

Purpose: Filters rows (tuples) based on a condition.
Symbol: σ (sigma)
Operation: σ_condition(Relation)

Example:

σ_Course = ‘BCA'(Student)

Returns all students enrolled in the BCA course.

Projection (π)

Purpose: Selects specific columns (attributes).
Symbol: π (pi)
Operation: π_attribute_list(Relation)

π_Name, Course(Student)

Returns only the Name and Course columns from Student.

Set Operations

These require that the two relations have the same schema.

Operation	Symbol	Description
Union	∪	Combines tuples from both relations
Intersection	∩	Returns common tuples
Difference	−	Returns tuples in one but not the other

Example:

Student ∪ Alumni

Returns all unique rows that are in either Student or Alumni.

Join Operations

Used to combine rows from two or more tables based on a common attribute.

Type	Symbol/Keyword	Description
Cartesian Product	×	Combines every tuple of R1 with every tuple of R2
Theta Join	⋈_condition	Join based on any condition
Equi Join	⋈_A=B	Join where attribute values are equal
Natural Join	⋈	Automatically joins on same-named attributes
Outer Join	LEFT/RIGHT/FULL	Includes unmatched rows

Example:

Student ⋈_Student.RollNo = Enrollment.RollNo Enrollment

Division (÷)

Purpose: Finds tuples in one relation that are related to all tuples in another.
Use Case: Queries like “Find students who enrolled in all courses.”

Example:

Finds students who are enrolled in every course.

Relational Calculus

Relational Calculus is a non-procedural query language; it specifies what to retrieve, not how.

There are two types:

Tuple Relational Calculus (TRC)

Syntax: { t | P(t) }
Where t is a tuple variable, and P(t) is a predicate (condition).

Example:

{ t | t ∈ Student AND t.Course = ‘BCA’ }

Returns all student tuples where the course is BCA.

Domain Relational Calculus (DRC)

Syntax: { <a1, a2, …, an> | P(a1, a2, …, an) }
Each variable represents a domain value (attribute).

Example:

{ <name> | ∃ rollNo, course (Student(rollNo, name, course) ∧ course = ‘BCA’) }

Returns names of students where the course is BCA.

Comparison Table: TRC vs DRC

Feature	Tuple Relational Calculus (TRC)	Domain Relational Calculus (DRC)
Based on	Tuples (rows)	Domains (columns/attributes)
Syntax	`{ t	P(t) }`
Focus	Entire tuple	Individual attribute values
More readable for	Simple conditions	Specific value-based queries

Conclusion

Concept	Key Use
Selection (σ)	Filter rows (WHERE in SQL)
Projection (π)	Select columns (SELECT in SQL)
Set Ops	Union, Intersection, Difference
Join	Combine related tables (JOIN in SQL)
Division	Match tuples related to all values in another set
TRC & DRC	Theoretical query expressions for relational data