Consider the following relations P(X,Y,Z), Q(X,Y,T) and R(Y,V).

How many tuples will be returned by the following relational algebra query?

          ∏x(σ(P.Y=R.Y ∧ R.V=V2)(P × R)) - ∏x(σ(Q.Y=R.Y ∧ Q.T>2)(Q × R))

Consider the relations r(A, B) and s(B, C), where s.B is a primary key and r.B is a foreign key referencing s.B. Consider the query

Q: r⋈(σ_B<5(s))

Let LOJ denote the natural left outer-join operation. Assume that r and s contain no null values.

Which one of the following is NOT equivalent to Q?

Consider a database that has the relation schemas EMP(EmpId, EmpName, DeptId), and DEPT(DeptName, DeptId). Note that the DeptId can be permitted to a NULL in the relation EMP. Consider the following queries on the database expressed in tuple relational calculus.

(I) {t│∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∀v ∈ DEPT(t[DeptId] ≠ v[DeptId]))}
(II) {t│∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∃v ∈ DEPT(t[DeptId] ≠ v[DeptId]))}
(III) {t│∃u ∈ EMP(t[EmpName] = u[EmpName] ∧ ∃v ∈ DEPT(t[DeptId] = v[DeptId]))}

Which of the above queries are safe?

Consider a database that has the relation schema CR(StudentName, CourseName). An instance of the schema CR is as given below.

The following query is made on the database.

         T1 ← πCourseName(σStudentName='SA'(CR))
         T2 ← CR ÷ T1

The number of rows in T2 is ____________.

Consider two relations R₁(A,B) with the tuples (1,5), (3,7) and R₂(A,C) = (1,7), (4,9). Assume that R(A,B,C) is the full natural outer join of R₁ and R₂. Consider the following tuples of the form (A,B,C): a = (1.5,null), b = (1,null,7), c = (3,null,9), d = (4,7,null), e = (1,5,7), f = (3,7,null), g = (4,null,9). Which one of the following statements is correct?  

Suppose R₁(A, B) and R₂(C, D) are two relation schemas. Let r₁ and r₂ be the corresponding relation instances. B is a foreign key that refers to C in R₂. If data in r₁ and r₂ satisfy referential integrity constraints, which of the following is ALWAYS TRUE?

Consider the following relations A, B, C.

How many tuples does the result of the following relational algebra expression contain? Assume that the schema of AUB is the same as that of A.

(AUB)⋈A.Id>40∨C.Id<15 C

Which of the following functional dependencies hold for relations R(A, B, C) and S(B, D, E):

B -> A
A -> C

The relation R contains 200 tuples and the rel ation S contains 100 tuples. What is the maximum number of tuples possible in the natural join of R and S (R natural join S)

Let R and S be two relations with the following schema

R(P,Q,R1,R2,R3)
S(P,Q,S1,S2)

Where {P, Q} is the key for both schemas. Which of the following queries are equivalent?

I. Π_P(R ⨝ S)
II. Π_P(R) ⨝ Π_P(S)
III. Π_P(Π_P,Q(R) ∩ Π_P,Q(S))
IV. Π_P(Π_P,Q(R) - (Π_P,Q(R) - Π_P,Q(S)))

Information about a collection of students is given by the relation studinfo(studId, name, sex). The relation enroll(studId, courseId) gives which student has enrolled for (or taken) that course(s). Assume that every course is taken by at least one male and at least one female student. What does the following relational algebra expression represent?

Let r be a relation instance with schema R = (A, B, C, D). We define r₁ = Π_A,B,C(R) and r₂ = Π_A,D(R). Let s = r₁*r₂ where * denotes natural join. Given that the decomposition of r into r₁ and r₂ is lossy, which one of the following is TRUE?

Let R₁(A,B,C) and R₂(D,E) be two relation schema, where the primary keys are shown underlined, and let C be a foreign key in R₁ referring to R₂. Suppose there is no violation of the above referential integrity constraint in the corresponding relation instances r₁ and r₂. Which one of the following relational algebra expressions would necessarily produce an empty relation?

Consider the following relation schema pertaining to a students database:

   Student (rollno, name, address)
   Enroll (rollno, courseno, coursename) 

Where the primary keys are shown underlined. The number of tuples in the Student and Enroll tables are 120 and 8 respectively. What are the maximum and minimum number of tuples that can be present in (Student * Enroll), where '*' denotes natural join ?

Consider the relation Student (name, sex, marks), where the primary key is shown underlined, pertaining to students in a class that has at least one boy and one girl. What does the following relational algebra expression produce? (Note: p is the rename operator).

The condition in join is "(sex = female ^ x = male ^ marks ≤ m)"

With regard to the expressive power of the formal relational query languages, which of the following statements is true?

Suppose the adjacency relation of vertices in a graph is represented in a table Adj(X,Y). Which of the following queries cannot be expressed by a relational algebra expression of constant length?

Let r and s be two relations over the relation schemes R and S respectively, and let A be an attribute in R. Then the relational algebra expression σ_A=a(r⋈s) is always equal to

Given the relations

   employee (name, salary, deptno) and
   department (deptno, deptname, address)  

Which of the following queries cannot be expressed using the basic relational algebra operations (σ, π, ×, ⋈, ∪, ∩, -)?

Consider the join of a relation R with a relation S. If R has m tuples and S has n tuples then the maximum and minimum sizes of the join respectively are

The relational algebra expression equivalent to the following tuple calculus expression:

Given two union compatible relations R₁(A,B) and R₂(C,D), what is the result of the operation R₁A = CAB = DR₂?

Which of the following query transformations (i.e. replacing the l.h.s. expression by the r.h.s. expression) is incorrect? R₁ and R₂ are relations, C₁, C₂ are selection conditions and A₁, A₂ are attributes of R₁?

Give a relational algebra expression using only the minimum number of operators from (∪, −) which is equivalent to R ∩ S.

Consider a selection of the form σ_A≤100(r), where r is a relation with 1000 tuples. Assume that the attribute values for A among the tuples are uniformly distributed in the interval [0, 500]. Which one of the following options is the best estimate of the number of tuples returned by the given selection query ?

Consider the following relation schemas:
b-Schema = (b-name, b-city, assets)
a-Schema = (a-num, b-name, bal)
d-Schema = (c-name, a-number)

Let branch, account and depositor be respectively instances of the above schemas. Assume that account and depositor relations are much bigger than the branch relation.

Consider the following query:

 Пc-name (σb-city = "Agra" ⋀ bal < 0 (branch ⋈ (account ⋈ depositor)  

Which one of the following queries is the most efficient version of the above query?

Consider the relations r₁(P, Q, R) and r₂(R, S, T) with primary keys P and R respectively. The relation r₁ contains 2000 tuples and r₂ contains 2500 tuples. The maximum size of the join r₁⋈ r₂ is :

Consider the schema R = (A, B, C, D, E, F) on which the following functional dependencies hold :

A ➝ B
B, C ➝ D
E ➝ C
D ➝ A

What are the candidate keys of R ?

Relations and fields can be renamed in relational algebra using the renaming operator

The ____ defines a set of operations on relations, paralleling the usual algebraic operations such as addition, subtraction or multiplication, which operates on numbers.

Which of the following is a unary operation?

The number of tuples in the result of a left outer join operation will always be