Consider the following two relations, R(A, B) and S(A, C):

R
A	B
10	20
20	30
30	40
30	50
50	95

S
A	C
10	90
30	45
40	80

The total number of tuples obtained by evaluating the following expression

$$ \sigma_{B < C}(R \bowtie_{R.A = S.A} S) $$

is _________

Consider the following three relations in a relational database.

Employee ( $$\underline {eld} $$ , Name), Brand ( $$\underline {bld} $$ , bName), Own ( $$\underline {eld} $$ , $$\underline {bld} $$)

Which of the following relational algebra expressions return the set of elds who own all the brands?

What is the optimized version of the relation algebra expression $$\pi_{A1}(\pi_{A2}(\sigma_{F1}(\sigma_{F2}(r))))$$, where $$A1, A2$$ are sets of attributes in r with $$A1 \subset A2$$ and $$F1,F2$$ are Boolean expressions based on the attributes in r?

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