Relational Calculus in DBMS

1. Introduction to Relational Calculus

Relational Calculus is a non-procedural query language used in relational databases. Unlike Relational Algebra, which focuses on how to retrieve data, Relational Calculus focuses on what data to retrieve.

🔹 Key Idea: Instead of specifying step-by-step operations, it defines conditions that must be met to retrieve data.
🔹 Used in: Query Optimization & Database Query Languages like SQL.

Comparison: Relational Algebra vs. Relational Calculus

Feature	Relational Algebra	Relational Calculus
Type	Procedural	Non-Procedural
Focus	How to retrieve data	What data to retrieve
Approach	Uses operations like selection, projection, join	Uses predicate logic
Output	New relation (table)	Desired tuples satisfying conditions
Basis of	Query Execution	Query Definition

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2. Types of Relational Calculus

Relational Calculus is of two types:

1️⃣ Tuple Relational Calculus (TRC) → Uses tuples (rows) as variables.
2️⃣ Domain Relational Calculus (DRC) → Uses columns (attributes) as variables.

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3. Tuple Relational Calculus (TRC)

Tuple Relational Calculus uses tuples (rows) as variables and applies conditions to select tuples.

Syntax of TRC

{ t | Condition (t) }  

🔹 { t } → Represents the tuples (rows) to be selected.
🔹 Condition(t) → Specifies the condition that must be satisfied.

Example 1: Retrieve Students Enrolled in ‘CS’ Course

{ t | t ∈ STUDENT ∧ t.Course = 'CS' }  

✅ Explanation:

• t ∈ STUDENT → t is a tuple from the STUDENT table.
• t.Course = 'CS' → Select only those students who are in the CS course.

📌 Equivalent SQL Query:

SELECT * FROM STUDENT WHERE Course = 'CS';

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Example 2: Retrieve Names of Students Aged Over 21

{ t.Name | t ∈ STUDENT ∧ t.Age > 21 }  

✅ Explanation:

• t ∈ STUDENT → t is a tuple from STUDENT.
• t.Age > 21 → Select students with Age > 21.
• { t.Name } → Only return the Name attribute.

📌 Equivalent SQL Query:

SELECT Name FROM STUDENT WHERE Age > 21;

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TRC Using Logical Connectors

Symbol	Meaning
∧ (AND)	Both conditions must be true
∨ (OR)	At least one condition must be true
¬ (NOT)	Negation of a condition
∃ (Exists)	There exists at least one tuple satisfying the condition
∀ (For All)	The condition must hold for all tuples

🔹 Example: Retrieve students who are not enrolled in CS.

{ t | t ∈ STUDENT ∧ ¬(t.Course = 'CS') }  

📌 Equivalent SQL Query:

SELECT * FROM STUDENT WHERE Course <> 'CS';

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4. Domain Relational Calculus (DRC)

Domain Relational Calculus uses column values (attributes) as variables instead of tuples.

Syntax of DRC

{ <X1, X2, ..., Xn> | Condition (X1, X2, ..., Xn) }  

🔹 <X1, X2, ..., Xn> → Specifies the attributes (columns) to be retrieved.
🔹 Condition (X1, X2, ..., Xn) → Specifies the conditions that must be met.

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Example 1: Retrieve Names of Students in CS Course

{ <Name> | ∃ Age, Course (STUDENT(Name, Age, Course) ∧ Course = 'CS') }  

✅ Explanation:

• ∃ Age, Course → These attributes exist in the STUDENT table.
• STUDENT(Name, Age, Course) → Name, Age, and Course are columns.
• Course = 'CS' → Select only CS students.

📌 Equivalent SQL Query:

SELECT Name FROM STUDENT WHERE Course = 'CS';

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Example 2: Retrieve Students Aged Above 21

{ <Name> | ∃ Age, Course (STUDENT(Name, Age, Course) ∧ Age > 21) }  

📌 Equivalent SQL Query:

SELECT Name FROM STUDENT WHERE Age > 21;

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5. Differences Between Tuple and Domain Relational Calculus

Feature	Tuple Relational Calculus (TRC)	Domain Relational Calculus (DRC)
Based On	Tuples (rows)	Attributes (columns)
Variables	Represent rows in a relation	Represent values of attributes
Syntax	`{ t	Condition(t) }`
Example	`{ t	t ∈ STUDENT ∧ t.Age > 21 }`

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6. Advantages of Relational Calculus

✅ Declarative Approach: Focuses on what data is needed, not how to retrieve it.
✅ Logical and Flexible: Uses logical expressions to define queries.
✅ Foundation of SQL: Helps in query formulation and optimization.

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7. Disadvantages of Relational Calculus

❌ Complex Queries: Writing queries with logical expressions can be difficult.
❌ Less Efficient Execution: Since it lacks step-by-step operations, DBMS must convert it into Relational Algebra for execution.

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8. Conclusion

Relational Calculus is a powerful non-procedural query language that defines conditions for retrieving data logically rather than specifying steps. It plays a crucial role in query formulation, optimization, and database theory.