RelationalCalculus

Relational Calculus

Relational calculus is a declarative query language based on mathematical logic that describes “What to retrieve” rather than “How to retrieve it”. It uses predicate logic to specify the desired properties of the result without specifying the procedure to obtain it.

Tuple Relational Calculus (TRC)

TRC uses tuple variables that range over relations. The general form is { t | P(t) } where t is a tuple variable and P(t) is a predicate that must be true.

SAMPLE DATA FOR EXAMPLES

Employee

EmpID	Name	DeptID	Salary
1	Alice	10	60000
2	Bob	20	55000
3	Carol	10	70000
4	David	30	40000
5	Eve	20	75000

Department

DeptID	DeptName	Location
10	HR	Delhi
20	IT	Mumbai
30	Finance	Bangalore

Project

ProjID	ProjName	DeptID
101	Alpha	10
102	Beta	20
103	Gamma	10

WorksOn

EmpID	ProjID	Hours
1	101	20
1	103	15
2	102	30
3	101	25
5	102	40

Operations

Selection (Filtering Rows)

Selection retrieves tuples that satisfy a condition using predicates on tuple attributes. Find all employees with salary greater than 60000 $t\mid t\in Employee\wedge t.Salary>60000$

EmpID	Name	DeptID	Salary
3	Carol	10	70000
5	Eve	20	75000

Projection (Selecting Columns)

Projection extracts specific attributes from tuples using existential quantifiers. Get names and salaries of all employees $t\mid \exists e\in Employee(t.Name=e.Name\wedge t.Salary=e.Salary)$

Name	Salary
Alice	60000
Bob	55000
Carol	70000
David	40000
Eve	75000

Combined Selection and Projection

Combines filtering and attribute extraction in a single query. Find names of employees in DeptID 10 $t\mid \exists e\in Employee(e.DeptID=10\wedge t.Name=e.Name)$

Name
Alice
Carol

Existential Quantifier (∃)

Existential quantifier checks if there exists at least one tuple satisfying the condition. Find employees who work on at least one project $t\mid t\in Employee\wedge \exists w\in WorksOn(w.EmpID=t.EmpID)$

EmpID	Name	DeptID	Salary
1	Alice	10	60000
2	Bob	20	55000
3	Carol	10	70000
5	Eve	20	75000

Universal Quantifier (∀)

Universal quantifier checks if a condition holds for all tuples in a relation. Find employees who work on ALL projects in DeptID 10 $t\mid t\in Employee\wedge \forall p\in Project(p.DeptID\ne 10\vee \exists w\in WorksOn(w.EmpID=t.EmpID\wedge w.ProjID=p.ProjID))$

EmpID	Name	DeptID	Salary
1	Alice	10	60000

Negation (¬)

Negation filters tuples that do not satisfy a given condition. Find employees who do NOT work on any project $t\mid t\in Employee\wedge \neg \exists w\in WorksOn(w.EmpID=t.EmpID)$

EmpID	Name	DeptID	Salary
4	David	30	40000

Natural Join

Natural join combines tuples from two relations based on common attributes. Find employee names with their department names $t\mid \exists e\in Employee,\exists d\in Department(e.DeptID=d.DeptID\wedge t.Name=e.Name\wedge t.DeptName=d.DeptName)$

Name	DeptName
Alice	HR
Bob	IT
Carol	HR
David	Finance
Eve	IT

Multiple Conditions

Multiple conditions combine several predicates using logical operators. Find names of employees in IT department with salary > 60000 $t\mid \exists e\in Employee,\exists d\in Department(e.DeptID=d.DeptID\wedge d.DeptName{=}^{\prime }I{T}^{\prime }\wedge e.Salary>60000\wedge t.Name=e.Name)$

Name
Eve

Division

Division finds tuples that are related to all tuples in another relation. Find employees who work on ALL projects $t\mid t\in Employee\wedge \forall p\in Project(\exists w\in WorksOn(w.EmpID=t.EmpID\wedge w.ProjID=p.ProjID))$

EmpID	Name	DeptID	Salary

Domain Relational Calculus (DRC)

DRC uses domain variables that range over attribute values. The general form is { ⟨x₁, x₂, ...⟩ | P(x₁, x₂, ...) } where variables represent individual attribute values.

Operations

Selection

Selection filters domain values based on predicates. Find EmpID and Name of employees with salary > 60000 $⟨i,n⟩\mid \exists d,\exists s(Employee(i,n,d,s)\wedge s>60000)$

EmpID	Name
3	Carol
5	Eve

Projection

Projection extracts specific domain values from relations. Get all employee names $⟨n⟩\mid \exists i,\exists d,\exists s(Employee(i,n,d,s))$

Name
Alice
Bob
Carol
David
Eve

Join

Join combines domain values from multiple relations based on common attributes. Find employee names and their department locations $⟨n,l⟩\mid \exists i,\exists d1,\exists s,\exists d2,\exists dn(Employee(i,n,d1,s)\wedge Department(d2,dn,l)\wedge d1=d2)$

Name	Location
Alice	Delhi
Bob	Mumbai
Carol	Delhi
David	Bangalore
Eve	Mumbai

Existential (∃)

Existential quantifier checks for existence of domain values satisfying conditions. Find names of employees who work on project 101 $⟨n⟩\mid \exists i,\exists d,\exists s,\exists p,\exists h(Employee(i,n,d,s)\wedge WorksOn(i,p,h)\wedge p=101)$

Name
Alice
Carol

Universal (∀)

Universal quantifier checks if conditions hold for all domain values. Find employees who work on projects with hours ≥ 20 for ALL their assigned projects $⟨i,n⟩\mid \exists d,\exists s(Employee(i,n,d,s)\wedge \forall p,\forall h(WorksOn(i,p,h)\to h\ge 20))$

EmpID	Name
2	Bob
3	Carol
4	David
5	Eve