Relational Calculus
Question 1 
I and II
 
I and III
 
II and IV  
III and IV 
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=RS P=r
B=S
E=r
we will get,
equivalent to I
∴ It is equivalent to division operator.
⇒ r(RS,S)/r(S)
This logical statement means that
① Select t(RS) 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[RS]
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(RS) 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[RS]
⇒ Select (RS) 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.
S={c}
It looks like Division operator. Since a division operator in E(A,B)/ P(B) will be equal to
Now replacing A=RS P=r
B=S
E=r
we will get,
equivalent to I
∴ It is equivalent to division operator.
⇒ r(RS,S)/r(S)
This logical statement means that
① Select t(RS) 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[RS]
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(RS) 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[RS]
⇒ Select (RS) 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 
I only  
II only  
III only  
III and IV only 
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).
∀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 
The empty set  
schools with more than 35% of its students enrolled in some exam or the other  
schools with a pass percentage above 35% over all exams taken together  
schools with a pass percentage above 35% over each exam 
Question 3 Explanation:
Query having the division with
{ x  x ∈ Enrolment ∧ x . schoolid = t }  * 100 > 35 }
This is school with enrollment % is 35 or above.
{ x  x ∈ Enrolment ∧ x . schoolid = t }  * 100 > 35 }
This is school with enrollment % is 35 or above.
Question 4 
Names of employees with a male supervisor.
 
Names of employees with no immediate male subordinates.  
Names of employees with no immediate female subordinates.  
Names of employees with a female supervisor.

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?
{t∃u ∈ R_{1} (t[A] = u[A])∧ ¬∃s ∈ R_{2} (t[A] = s[A])}  
{t∀u ∈ R_{1} (u[A]= "x" ⇒ ∃s ∈ R_{2} (t[A] = s[A] ∧ s[A] = u[A]))}  
{t¬(t ∈ R_{1})}  
{t∃u ∈ R_{1} (t[A] = u[A])∧ ∃s ∈ R_{2} (t[A] = s[A])} 
There are 5 questions to complete.