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. $$\Pi_P \left(R \bowtie S\right)$$
II. $$\Pi_P \left(R\right) \bowtie \Pi_P\left(S\right)$$
III. $$\Pi_P \left(\Pi_{P, Q} \left(R\right) \cap \Pi_{P,Q} \left(S\right) \right)$$
IV. $$\Pi_P \left(\Pi_{P, Q} \left(R\right) - \left(\Pi_{P,Q} \left(R\right) - \Pi_{P,Q} \left(S\right)\right)\right)$$

Which of the following tuple relational calculus expression(s) is/are equivalent to $$\forall t \in r \left(P\left(t\right)\right)$$?

I. $$\neg \exists t \in r \left(P\left(t\right)\right)$$
II. $$\exists t \notin r \left(P\left(t\right)\right)$$
III. $$\neg \exists t \in r \left(\neg P\left(t\right)\right)$$
IV. $$\exists t \notin r \left(\neg P\left(t\right)\right)$$

Consider The Following Relational Scheme

Student (school-id, sch-roll-no, sname, saddress)
School (school-id, sch-name, sch-address, sch-phone)
Enrolment (school-id, sch-roll-no, erollno, examname)
ExamResult (Erollno, examname, marks)

What does the following SQL query output?

SELECT sch-name, COUNT (*) 
FROM School C, Enrolment E, 
ExamResult R 
WHERE E.school-id = C.school-id 
AND E.examname = R.examname 
AND E.erollno = R.erollno
AND R.marks = 100 AND S.school-id IN 
(SELECT school-id 
 FROM student 
 GROUP BY school-id 
 HAVING COUNT (*) > 200) 
GROUP BY school-id;

Consider The Following Relational Scheme

Student (school-id, sch-roll-no, sname, saddress)
School (school-id, sch-name, sch-address, sch-phone)
Enrolment (school-id, sch-roll-no, erollno, examname)
ExamResult (Erollno, examname, marks)

Consider the following tuple relational calculus query

{ t | ∃E ∈ Enrolment t = E.school-id ∧ 
| { x | x ∈ ExamResult B.school-id = 
t ∧ ( ∃B ∈ ExamResult B.erollno = 
x.erollno ∧ B.examname = x.examname ∧ 
B.marks > 35 } | ÷ | 
{ x | x ∈ Enrolment ∧ x.school-id = t } 
| * 100 > 35 }

If a student needs to score more than 35 marks to pass an exam what does the query return?