Relational Calculus

Question 1
A
I and II
B
I and III
C
II and IV
D
III and IV
       Database Management System        Relational Calculus       2009
Question 1 Explanation: 
R={a,b,c}
S={c}

It looks like Division operator. Since a division operator in E(A,B)/ P(B) will be equal to

Now replacing A=R-S P=r
B=S
E=r
we will get,

equivalent to I
∴ It is equivalent to division operator.
⇒ r(R-S,S)/r(S)

This logical statement means that
① Select t(R-S) from r such that
② for all tuples U in S,
③ there exists a tuple V in r, such that
④ U=V[S] & t=V[R-S]
A(x,y) & B(y)
A/B = {(x) | ∃(x,y)∈A(y)∈B}
which means that A/B contains all x tuples, such that for every tuple in B, there is an xy tuple in A.
So, this is just equivalent to I.

This logical statement means that
① Select t(R-S) from r such that
② for all tuples V in r,
③ there exists a tuple U in r, such that
④ U=V[S] & t=V[R-S]
⇒ Select (R-S) values from r, where the S value is in (r/r), which will be true only if S in r is a foreign key referring to S is r.

This selects (a,b) from all tuples from r which has an equivalent value in S.
Question 2
A
I only
B
II only
C
III only
D
III and IV only
       Database Management System        Relational Calculus       Gate-2008
Question 2 Explanation: 
Demorgan law:
∀xP(x)≡∼∃x(∼P(x))
∼∀x(∼P(x))≡∃x(P(x))
Given: ∀t ∈ r(P(t))------ (1)
As per Demorgan law
(1) ⇒ ∼∃t ∈ r(∼P(t))
which is option (III).
Question 3
A
The empty set
B
schools with more than 35% of its students enrolled in some exam or the other
C
schools with a pass percentage above 35% over all exams taken together
D
schools with a pass percentage above 35% over each exam
       Database Management System        Relational Calculus       Gate 2008-IT
Question 3 Explanation: 
Query having the division with
{ x | x ∈ Enrolment ∧ x . school-id = t } | * 100 > 35 }
This is school with enrollment % is 35 or above.
Question 4
 
A
Names of employees with a male supervisor.
B
Names of employees with no immediate male subordinates.
C
Names of employees with no immediate female subordinates.
D
Names of employees with a female supervisor.
       Database Management System        Relational Calculus       Gate-2007
Question 4 Explanation: 
The given Tuple Relational calculus produce names of employees with no immediate female subordinates.
Question 5
Which of the following relational calculus expressions is not safe?
A
{t|∃u ∈ R1 (t[A] = u[A])∧ ¬∃s ∈ R2 (t[A] = s[A])}
B
{t|∀u ∈ R1 (u[A]= "x" ⇒ ∃s ∈ R2 (t[A] = s[A] ∧ s[A] = u[A]))}
C
{t|¬(t ∈ R1)}
D
{t|∃u ∈ R1 (t[A] = u[A])∧ ∃s ∈ R2 (t[A] = s[A])}
       Database Management System        Relational Calculus       Gate-2001
There are 5 questions to complete.