Consider two transactions T₁ and T₂ and four schedules S₁, S₂, S₃, S₄ of T₁ and T₂ as given below:

T₁: R₁[ x ] W₁[ x ] W₁[ y ]
T₂: R₂[ x ] R₂[ y ] W₂[ y ]
S₁: R₁[ x ] R₂[ x ] R₂[ y ] W₁[ x ] W₁[ y ] W₂[ y ]
S₂: R₁[ x ] R₂[ x ] R₂[ y ] W₁[ x ] W₂[ y ] W₁[ y ]
S₃: R₁[ x ] W₁[ x ] R₂[ x ] W₁[ y ] R₂[ y ] W₂[ y ]
S₄: R₂[ x ] R₂[ y ] R₁[ x ] W₁[ x ] W₁[ y ] W₂[ y ]

Which of the above schedules are conflict-serializable?

Let R and S be relational schemes such that R = { a, b, c } and S = { c }. Now consider the following queries on the database:

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?

The following key values are inserted into a B⁺-tree in which order of the internal nodes is 3, and that of the leaf nodes is 2, in the sequence given below. The order of internal nodes is the maximum number of tree pointers in each node, and the order of leaf nodes is the maximum number of data items that can be stored in it. The B⁺- tree is initially empty.

10, 3, 6, 8, 4, 2, 1

The maximum number of times leaf nodes would get split up as a result of these insertions is

$${\left( {1217} \right)_8}$$ is equivalent to