🧮 RELATIONAL ALGEBRA

🧮 RELATIONAL ALGEBRA

Complete Interactive Visual Learning Module

🌟 What is Relational Algebra?

Relational Algebra is a formal system of operations used to retrieve and manipulate information stored in relational database tables.

A relation can be thought of as a table containing rows (tuples) and columns (attributes).

For example: STUDENT may contain:

Student_ID, Name, Age, Course_ID

Relational Algebra allows us to ask questions such as:

👉 Which students are older than 18?
👉 Show only student names.
👉 Combine Student and Course information.
👉 Find students present in two different relations.
👉 Find students who completed ALL courses.

📚 Example Database

STUDENT

Student_ID	Name	Age	Course_ID
101	Rahul	20	10
102	Anita	19	20
103	Arjun	22	30
104	Riya	17	40

COURSE

Course_ID	Course_Name
10	BCA
20	BBA
30	B.Sc
50	MCA

All major Relational Algebra operations are available as interactive buttons above.

Click any button to learn its definition, formula and step-by-step example.

σ Selection

Selection chooses specific ROWS from a relation according to a condition.

Think: "Which rows do I want?"

σ_{Age > 18}(STUDENT)

1

Read the first student: Rahul, Age = 20.

2

Check 20 > 18 → TRUE → Keep Rahul.

3

Anita: 19 > 18 → TRUE → Keep Anita.

4

Arjun: 22 > 18 → TRUE → Keep Arjun.

5

Riya: 17 > 18 → FALSE → Remove Riya.

Result: Rahul, Anita and Arjun.

🧠 σ = ROWS

π Projection

Projection selects specific COLUMNS from a relation.

π_{Name, Age}(STUDENT)

1

Start with the STUDENT relation.

2

Select only Name and Age columns.

3

Ignore Student_ID and Course_ID.

Result:

Name	Age
Rahul	20
Anita	19
Arjun	22
Riya	17

🧠 π = COLUMNS

∪ Union

Union combines tuples from two union-compatible relations. Duplicate tuples are removed.

A ∪ B

A = {Rahul, Anita, Arjun}

B = {Anita, Riya}

A ∪ B

= {Rahul, Anita, Arjun, Riya}

🎯 Union means: ALL records from A and B without duplicates.

− Set Difference

Difference returns tuples present in the first relation but not in the second relation.

A − B

A = {Rahul, Anita, Arjun}
B = {Anita, Arjun}

A − B

= {Rahul}

🎯 Rahul exists in A but not in B.

∩ Intersection

Intersection returns only tuples common to both relations.

A ∩ B

A = {Rahul, Anita, Arjun}
B = {Anita, Arjun, Riya}

A ∩ B

= {Anita, Arjun}

🎯 Intersection = COMMON records

× Cartesian Product

Cartesian Product combines every row of one relation with every row of another relation.

STUDENT × COURSE

1

Suppose STUDENT has 2 rows.

2

Suppose COURSE has 3 rows.

3

Every student is combined with every course.

Number of rows = 2 × 3 = 6

🎯 Cartesian Product = EVERY POSSIBLE COMBINATION

ρ Rename

Rename changes the name of a relation or attributes.

ρ_S(STUDENT)

1

Original relation name = STUDENT.

2

Apply ρ operator.

3

New relation name = S.

STUDENT → S

The actual data does not change.

θ Theta Join

Theta Join uses a comparison condition between two relations.

R ⋈_θ S

Possible comparison operators:

=    ≠    <    >    ≤    ≥

Example:

STUDENT.Course_ID < COURSE.Course_ID

This is a Theta Join.

= Equi Join

Equi Join is a join whose condition uses the equality operator (=).

STUDENT ⋈ _{STUDENT.Course_ID = COURSE.Course_ID} COURSE

1

Compare Student Course_ID with Course Course_ID.

2

10 = 10 → Match.

3

20 = 20 → Match.

4

30 = 30 → Match.

Equi Join = Join using equality (=)

🌿 Natural Join

Natural Join automatically uses common attributes with compatible domains. Duplicate join attributes are represented only once.

STUDENT ⋈ COURSE

1

Find common attribute names.

2

Course_ID exists in both tables.

3

Match equal Course_ID values.

4

Combine matching rows.

Natural Join automatically identifies the common attribute.

🔗 Inner Join

Returns only rows that have matching values in both tables.

STUDENT ⋈ COURSE

Name	Course_ID	Course_Name
Rahul	10	BCA
Anita	20	BBA
Arjun	30	B.Sc

Only matching Course_ID values are returned.

⬅ Left Outer Join

Returns all rows from the LEFT relation and matching rows from the RIGHT relation. Unmatched right-side attributes become NULL.

STUDENT ⟕ COURSE

Name	Course_ID	Course_Name
Rahul	10	BCA
Anita	20	BBA
Arjun	30	B.Sc
Riya	40	NULL

⬅ LEFT JOIN = ALL LEFT RECORDS

➡ Right Outer Join

Returns all rows from the RIGHT relation and matching rows from the LEFT relation.

STUDENT ⟖ COURSE

Name	Course_ID	Course_Name
Rahul	10	BCA
Anita	20	BBA
Arjun	30	B.Sc
NULL	50	MCA

➡ RIGHT JOIN = ALL RIGHT RECORDS

↔ Full Outer Join

Returns all rows from both relations. Matching records are combined. Unmatched records receive NULL values.

STUDENT ⟗ COURSE

Name	Course_ID	Course_Name
Rahul	10	BCA
Anita	20	BBA
Arjun	30	B.Sc
Riya	40	NULL
NULL	50	MCA

↔ FULL JOIN = ALL RECORDS FROM BOTH TABLES

👤 Self Join

Self Join joins a relation with itself. It is useful when rows in the same table are related to one another.

EMPLOYEE

Emp_ID	Employee	Manager_ID
101	Rahul	103
102	Anita	103
103	Arjun	NULL

E1 ⋈_{E1.Manager_ID = E2.Emp_ID} E2

Result:

Rahul → Arjun
Anita → Arjun

Here Arjun is the manager of Rahul and Anita.

✖ Cross Join

Cross Join creates the Cartesian Product of two relations.

STUDENT × COURSE

Suppose:

STUDENT = 3 rows
COURSE = 4 rows

Number of result rows:

3 × 4 = 12

Every student is paired with every course.

÷ Division

Division is used when a query contains the idea of "ALL".

STUDENT_COURSE ÷ REQUIRED_COURSE

Required Courses

Course_ID
BCA101
BCA102
BCA103

Rahul completed:
BCA101 ✓
BCA102 ✓
BCA103 ✓

Anita completed:
BCA101 ✓
BCA102 ✓
BCA103 ✗

Division result:

Rahul

Because Rahul completed ALL required courses.

❓ Extra BIT Questions & Answers

Click any question to reveal its answer.

Q1. What is Relational Algebra?

Relational Algebra is a formal procedural language used to perform operations on relational database relations.

Q2. Which operator selects rows?

Selection (σ) selects rows satisfying a condition.

Q3. Which operator selects columns?

Projection (π) selects columns from a relation.

Q4. What is the difference between Selection and Projection?

Selection works mainly on rows, while Projection works mainly on columns.

σ → Rows
π → Columns

Q5. What is Union?

Union combines tuples from two union-compatible relations and removes duplicate tuples.

Q6. What is Cartesian Product?

Cartesian Product combines every tuple of one relation with every tuple of another relation.

Q7. If A has 5 rows and B has 4 rows, how many rows can A × B produce?

5 × 4 = 20 rows.

Q8. What is Natural Join?

Natural Join automatically joins relations using common attributes with compatible domains.

Q9. What is an Equi Join?

Equi Join uses equality (=) as the join condition.

Q10. What is a Theta Join?

Theta Join uses comparison operators such as =, ≠, <, >, ≤ and ≥.

Q11. What does Left Outer Join return?

It returns all rows from the left relation and matching rows from the right relation.

Q12. What does Right Outer Join return?

It returns all rows from the right relation and matching rows from the left relation.

Q13. What does Full Outer Join return?

It returns all rows from both relations, including unmatched rows.

Q14. What is Self Join?

Self Join joins a relation with itself, usually using different aliases.

Q15. What is Division used for?

Division is useful for queries involving the concept of "ALL".

Q16. Which operator renames a relation?

ρ (Rename).

Q17. What is the result of A ∩ B?

The result contains tuples common to both A and B.

Q18. What does A − B mean?

It returns tuples present in A but not present in B.

Q19. Which operation produces every possible combination?

Cartesian Product / Cross Join (×).

Q20. Which JOIN keeps all records from both tables?

Full Outer Join.

🎯 COMPLETE RELATIONAL ALGEBRA REVISION

σ Selection π Projection ∪ Union − Difference ∩ Intersection × Cartesian Product ρ Rename θ Theta Join Equi Join Natural Join Inner Join Left Join Right Join Full Join Self Join Cross Join ÷ Division

σ = ROWS   |   π = COLUMNS   |   × = COMBINATIONS

B.K. PAUL

Bhairab Ganguly College

Database Management System • Relational Algebra

© 2026 B.K. PAUL • Bhairab Ganguly College