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Relational Algebra, Tuple Calculus and SQL PYQs for GATE CS

Solve 11+ Relational Algebra, Tuple Calculus and SQL previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

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Question 1

2026 Slot Set1 PYQ

Consider a relational database schema with two relations R (P, Q) and S (X, Y) . Let E = {⟨ u ⟩ ∣ \exists v \exists w ⟨ u, v ⟩ \in R \land ⟨ v, w ⟩ \in S} be a tuple relational calculus expression. Which one of the following relational algebraic expressions is equivalent to E ?

Question 2

2025 Slot Set2 PYQ

Consider the following relational schema: Students (rollno: integer, name: string, age: integer, cgpa: real) Courses (courseno: integer, cname: string, credits: integer) Enrolled (rollno: integer, courseno: integer, grade: string) Which of the following options is/are correct SQL query/queries to retrieve the names of the students enrolled in course number (i.e., courseno) 1470?

Question 3

2025 Slot Set1 PYQ

Consider two relations describing t e am s and pl a y er s in a sports league: • t e am s (t i d, t nam e) : t i d, t nam e are team-id and team-name, respectively • pl a y er s (p i d, p nam e, t i d) : p i d, p nam e, and t i d denote player-id, player-name and the team-id of the player, respectively Which ONE of the following tuple relational calculus queries returns the name of the players who play for the team having t nam e as ‘ M I ’?

Question 4

2025 Slot Set1 PYQ

Consider the following database tables of a sports league. player(pid,pname,age) team(tid,tname,city,cid) coach(cid,cname) members(pid,tid) An instance of the table and an SQL query are given. player coach team members pid pname age 1 Jasprit 312 Atharva 243 Ishan 264 Axar 30 cid cname 101 Ricky 102 Mark 103 Trevor tid tname city cid 10 MI Mumbai 10220 DC Delhi 10130 PK Mohali 103 pid tid 110230310420 SELECT MIN(P.age) FROM player P WHERE P.pid IN (SELECT M.pid FROM team T, coach C, members M WHERE C.cname = 'Mark' AND T.cid = C.cid AND M.tid = T.tid) The value returned by the given SQL query is ______. (Answer in integer)

Question 5

2024 Slot Set2 PYQ

The relation schema, Person(pid,city), describes the city of residence for every person uniquely identified by pid. The following relational algebra operators are available: selection, projection, cross product, and rename. To find the list of cities where at least 3 persons reside, using the above operators, the minimum number of cross product operations that must be used is

Question 6

2024 Slot Set1 PYQ

Consider the following two relations, R (A, B) and S (A, C) : R A B 10202030304030505095 S A C 109030454080 The total number of tuples obtained by evaluating the following expression σ_{B < C} (R ⋈_{R . A = S . A} S) is __________

Question 7

2022 PYQ

Consider the relational database with the following four schemas and their respective instances. Student(sNo, sName, dNo) Dept(dNo, dName) Course(cNo, cName, dNo) Register(sNo, cNo) Student sNo sName dNo S01 James D01 S02 Rocky D01 S03 Jackson D02 S04 Jane D01 S05 Milli D02 Dept dNo dName D01 CSE D02 EEE Course cNo cName dNo C11 DS D01 C12 OS D01 C21 DE D02 C22 PT D02 C23 CV D03 Register sNo cNo S01 C11 S01 C12 S02 C11 S03 C21 S03 C22 S03 C23 S04 C11 S04 C12 S05 C11 S05 C21 SQL Query: SELECT * FROM Student AS S WHERE NOT EXIST (SELECT cNo FROM Course WHERE dNo = “D01” EXCEPT SELECT cNo FROM Register WHERE sNo = S.sNo) The number of rows returned by the above SQL query is___________.

Question 8

2022 PYQ

Consider the following three relations in a relational database. Employee (\underline{eId}, Name), Brand (\underline{bId}, bName), Own (\underline{eId}, \underline{bId}) Which of the following relational algebra expressions return the set of e I d s who own all the brands?

Question 9

2021 Slot Set2 PYQ

The relation scheme given below is used to store information about the employees of a company, where empId is the key and deptId indicates the department to which the employee is assigned. Each employee is assigned to exactly one department. emp(empId, name, gender, salary, deptId) Consider the following SQL query: select deptId, count(*) from emp where gender = "female" and salary > (select avg(salary) from emp) group by deptId; The above query gives, for each department in the company, the number of female employees whose salary is greater than the average salary of

Question 10

2021 Slot Set1 PYQ

The following relation records the age of 500 employees of a company, where empNo (indicating the employee number) is the key: empAge (\underline{empNo}, age) Consider the following relational algebra expression: Π_{empNo} (empAge ⋈_{(age > age1)} ρ_{empNo1, age1} (empAge)) What does the above expression generate?

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Relational Algebra, Tuple Calculus and SQL PYQs for GATE CS

Solve 11+ Relational Algebra, Tuple Calculus and SQL previous year questions for GATE CS with answers and detailed solutions. Free sample questions below.

Chapter Roadmap: Relational Algebra, Tuple Calculus and SQL

Chapter Roadmap

Relational Algebra, Tuple Calculus and SQL

By the end of this chapter, you will master the translation between declarative query languages and procedural operations, write complex SQL aggregations, and estimate query execution costs.

1. Tuple Relational Calculus and RA Equivalence 2 PYQs

Declarative querying, quantifiers, and translating TRC to Relational Algebra.

2. SQL Queries, Joins and Aggregation 4 PYQs

Writing complex nested queries, group by, having, and various join types.

3. Relational Algebra Operations, Division and Self-Joins 4 PYQs

Procedural query formulation, division operator, and self-join patterns.

4. Selection Cardinality and Query Result Estimation 1 PYQ

Estimating the number of tuples returned by selections and joins.

Tuple Relational Calculus and RA Equivalence

Bridging the gap between declarative "what" and procedural "how" in database queries.

Context: Relational Algebra, Tuple Calculus and SQL > Tuple Relational Calculus and Relational Algebra Equivalence

What you will learn here:

The declarative syntax of Tuple Relational Calculus (TRC).
How existential and universal quantifiers map to relational algebra operations.
Step-by-step methods for translating TRC expressions into Relational Algebra.
Common traps involving free and bound variables.

Relational Algebra, Tuple Calculus and SQL: Solved Questions with Step-by-Step Explanations (10 Problems)

Question 1 · Databases · 2026_Set1 MCQ

Consider a relational database schema with two relations

R (P, Q)

and

S (X, Y)

.

Let

E = {⟨ u ⟩ ∣ \exists v \exists w ⟨ u, v ⟩ \in R \land ⟨ v, w ⟩ \in S}

be a tuple relational calculus expression.

Which one of the following relational algebraic expressions is equivalent to

E

A.
$Π_{P} (R ⋈_{R . P = S . X} S)$
B.
$Π_{P} (S ⋈_{S . X = R . Q} R)$
C.
$Π_{P} (R ⋈_{R . P = S . Y} S)$
D.
$Π_{P} (S ⋈_{S . Y = R . Q} R)$

Question 2 · Databases · 2025_Set2 MSQ

Consider the following relational schema:

Students (rollno: integer, name: string, age: integer, cgpa: real)

Courses (courseno: integer, cname: string, credits: integer)

Enrolled (rollno: integer, courseno: integer, grade: string)

Which of the following options is/are correct SQL query/queries to retrieve the names of the students enrolled in course number (i.e., courseno) 1470?

A.
<pre>SELECT S.name

FROM Students S

WHERE EXISTS (SELECT * FROM Enrolled E

WHERE E.courseno = 1470

AND E.rollno = S.rollno);</pre>
B.
<pre>SELECT S.name

FROM Students S

WHERE SIZEOF (SELECT * FROM Enrolled E

WHERE E.courseno = 1470

AND E.rollno = S.rollno) > 0;</pre>
C.
<pre>SELECT S.name

FROM Students S

WHERE 0 < (SELECT COUNT(*)

FROM Enrolled E

WHERE E.courseno = 1470

AND E.rollno = S.rollno);</pre>
D.
<pre>SELECT S.name

FROM Students S NATURAL JOIN Enrolled E

WHERE E.courseno = 1470;</pre>

Question 3 · Databases · 2025_Set1 MCQ

Consider two relations describing

t e am s

and

pl a y er s

in a sports league:

•

t e am s (t i d, t nam e)

t i d, t nam e

are team-id and team-name, respectively
•

pl a y er s (p i d, p nam e, t i d)

p i d, p nam e

, and

t i d

denote player-id, player-name and the team-id of the player, respectively

Which ONE of the following tuple relational calculus queries returns the name of the players who play for the team having

t nam e

as ‘

M I

’?

A.
${p . p nam e ∣ p \in pl a y er s \land \exists t (t \in t e am s \land p . t i d = t . t i d \land t . t nam e = ‘MI’)}$
B.
${p . p nam e ∣ p \in t e am s \land \exists t (t \in pl a y er s \land p . t i d = t . t i d \land t . t nam e = ‘MI’)}$
C.
${p . p nam e ∣ p \in pl a y er s \land \exists t (t \in t e am s \land t . t nam e = ‘MI’)}$
D.
${p . p nam e ∣ p \in t e am s \land \exists t (t \in pl a y er s \land t . t nam e = ‘MI’)}$

Question 4 · Databases · 2025_Set1 NAT

Consider the following database tables of a sports league.

player(pid,pname,age)                    team(tid,tname,city,cid)
coach(cid,cname)                           members(pid,tid)

An instance of the table and an SQL query are given.

SELECT MIN(P.age)
FROM player P
WHERE P.pid IN (
    SELECT M.pid
    FROM team T, coach C, members M
    WHERE C.cname = 'Mark'
          AND T.cid = C.cid
          AND M.tid = T.tid
)

The value returned by the given SQL query is ______. (Answer in integer)

Question 5 · Databases · 2024_Set2 MCQ

The relation schema, Person(pid,city), describes the city of residence for every person uniquely identified by pid. The following relational algebra operators are available: selection, projection, cross product, and rename.

To find the list of cities where at least 3 persons reside, using the above operators, the minimum number of cross product operations that must be used is

A.
1
B.
2
C.
3
D.
4

Question 6 · Databases · 2024_Set1 NAT

Consider the following two relations,

R (A, B)

and

S (A, C)

:

The total number of tuples obtained by evaluating the following expression

σ_{B < C} (R ⋈_{R . A = S . A} S)

is __________

Question 7 · Databases · 2022 NAT

Consider the relational database with the following four schemas and their respective instances.

Student(sNo, sName, dNo)  Dept(dNo, dName)

Course(cNo, cName, dNo)  Register(sNo, cNo)

Student
sNo	sName	dNo
S01	James	D01
S02	Rocky	D01
S03	Jackson	D02
S04	Jane	D01
S05	Milli	D02

Dept
dNo	dName
D01	CSE
D02	EEE

Course
cNo	cName	dNo
C11	DS	D01
C12	OS	D01
C21	DE	D02
C22	PT	D02
C23	CV	D03

Register
sNo	cNo
S01	C11
S01	C12
S02	C11
S03	C21
S03	C22
S03	C23
S04	C11
S04	C12
S05	C11
S05	C21

SQL Query:

SELECT * FROM Student AS S WHERE NOT EXIST
  (SELECT cNo FROM Course WHERE dNo = “D01”
        EXCEPT
   SELECT cNo FROM Register WHERE sNo = S.sNo)

The number of rows returned by the above SQL query is___________.

Question 8 · Databases · 2022 MSQ

Consider the following three relations in a relational database.

Employee (\underline{eId}, Name), Brand (\underline{bId}, bName), Own (\underline{eId}, \underline{bId})

Which of the following relational algebra expressions return the set of

e I d s

who own all the brands?

A.
$Π_{e I d} (Π_{e I d, b I d} (O w n) / Π_{b I d} (B r an d))$
B.
$Π_{e I d} (O w n) - Π_{e I d} ((Π_{e I d} (O w n) \times Π_{b I d} (B r an d)) - Π_{e I d, b I d} (O w n))$
C.
$Π_{e I d} (Π_{e I d, b I d} (O w n) / Π_{b I d} (O w n))$
D.
$Π_{e I d} ((Π_{e I d} (O w n) \times Π_{b I d} (O w n)) / Π_{b I d} (B r an d))$

Question 9 · Databases · 2021_Set2 MCQ

The relation scheme given below is used to store information about the employees of a company, where empId is the key and deptId indicates the department to which the employee is assigned. Each employee is assigned to exactly one department.

emp(empId, name, gender, salary, deptId)

Consider the following SQL query:

select deptId, count(*)
from emp
where gender = "female" and salary > (select avg(salary) from emp)
group by deptId;

The above query gives, for each department in the company, the number of female employees whose salary is greater than the average salary of

A.
employees in the department.
B.
employees in the company.
C.
female employees in the department.
D.
female employees in the company.

Question 10 · Databases · 2021_Set1 MCQ

The following relation records the age of 500 employees of a company, where empNo (indicating the employee number) is the key:

empAge (\underline{empNo}, age)

Consider the following relational algebra expression:

Π_{empNo} (empAge ⋈_{(age > age1)} ρ_{empNo1, age1} (empAge))

What does the above expression generate?

A.
Employee numbers of only those employees whose age is the maximum.
B.
Employee numbers of only those employees whose age is more than the age of exactly one other employee.
C.
Employee numbers of all employees whose age is not the minimum.
D.
Employee numbers of all employees whose age is the minimum.

Register
sNo	cNo
S01	C11
S01	C12
S02	C11
S03	C21
S03	C22
S03	C23
S04	C11
S04	C12
S05	C11
S05	C21

Register
sNo	cNo
S01	C11
S01	C12
S02	C11
S03	C21
S03	C22
S03	C23
S04	C11
S04	C12
S05	C11
S05	C21

Register
sNo	cNo
S01	C11
S01	C12
S02	C11
S03	C21
S03	C22
S03	C23
S04	C11
S04	C12
S05	C11
S05	C21

Register
sNo	cNo
S01	C11
S01	C12
S02	C11
S03	C21
S03	C22
S03	C23
S04	C11
S04	C12
S05	C11
S05	C21

Try a question

Login to view the complete previous-year questions and solutions

Personalised first. High quality throughout.

Built around you, not around a syllabus PDF

Revision that hits your weak spots

Questions calibrated to the real exam

Notes written for recall, not for volume

One place for everything

Honest progress

Unlock the whole course

Relational Algebra, Tuple Calculus and SQL PYQs for GATE CS

Chapter Roadmap: Relational Algebra, Tuple Calculus and SQL

Chapter Roadmap

Tuple Relational Calculus and RA Equivalence

Tuple Relational Calculus and RA Equivalence

What you will learn here:

Relational Algebra, Tuple Calculus and SQL: Solved Questions with Step-by-Step Explanations (10 Problems)

More previous year questions (pyqs) in this unit

Register
sNo	cNo
S01	C11
S01	C12
S02	C11
S03	C21
S03	C22
S03	C23
S04	C11
S04	C12
S05	C11
S05	C21

Register
sNo	cNo
S01	C11
S01	C12
S02	C11
S03	C21
S03	C22
S03	C23
S04	C11
S04	C12
S05	C11
S05	C21