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

Consider the relations r₁(P, Q, R) and r₂(R, S, T) with primary keys P and R respectively. The relation r₁ contains 2000 tuples and r₂ contains 2500 tuples. The maximum size of the join $${r_1} \Join {r_2}$$ is