_{1}and T

_{2}and four schedules S

_{1}, S

_{2}, S

_{3}, S

_{4}of T

_{1}and T

_{2}as given below:

T_{1}: R_{1}[ x ] W_{1}[ x ] W_{1}[ y ]

T_{2}: R_{2}[ x ] R_{2}[ y ] W_{2}[ y ]

S_{1}: R_{1}[ x ] R_{2}[ x ] R_{2}[ y ] W_{1}[ x ] W_{1}[ y ] W_{2}[ y ]

S_{2}: R_{1}[ x ] R_{2}[ x ] R_{2}[ y ] W_{1}[ x ] W_{2}[ y ] W_{1}[ y ]

S_{3}: R_{1}[ x ] W_{1}[ x ] R_{2}[ x ] W_{1}[ y ] R_{2}[ y ] W_{2}[ y ]

S_{4}: R_{2}[ x ] R_{2}[ y ] R_{1}[ x ] W_{1}[ x ] W_{1}[ y ] W_{2}[ y ]

I. $$\pi_{R-S}(r) - \pi_{R-S} \left (\pi_{R-S} (r) \times s - \pi_{R-S,S}(r)\right )$$

II. $$\left\{t \mid t \in \pi_{R-S} (r) \wedge \forall u \in s \left(\exists v \in r \left(u = v[S] \wedge t = v\left[R-S\right]\right )\right )\right\}$$

III.$$\left\{t \mid t \in \pi_{R-S} (r) \wedge \forall v \in r \left(\exists u \in s \left(u = v[S] \wedge t = v\left[R-S\right]\right )\right ) \right\}$$

```
IV. Select R.a, R.b
From R, S
Where R.c = S.c
```

Which of the above queries are equivalent?_{1}, T

_{2}and T

_{3}.

[ 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?