A clustering index is defined on the fields which are of type

Consider the following three schedules of transactions T₁, T₂ and T₃.
[ Notation: In the following NYO represents the action Y (R for read, W for write) performed by transac­tion N on object O. ]

(S1) 2RA 2WA 3RC 2WB 3WA 3WC 1RA 1RB 1WA 1WB

(S2) 3RC 2RA 2WA 2WB 3WA 1RA 1RB 1WA 1WB 3WC

(S3) 2RZ 3RC 3WA 2WA 2WB 3WC 1RA 1RB 1WA 1WB

Which of the following statements is TRUE?

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. $$\exists t \notin r \left(\neg P\left(t\right)\right)$$