The following functional dependencies hold for relations R(A, B, C) and S(B, D, E): $$$\eqalign{ & B \to A \cr & A \to C \cr} $$$

The relation R contains 200 tuples and the relation S contains 100 tuples. What is the maximum number of tuples possible in the natural join R $$\Join$$ S?

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. $$\Pi_P \left(\Pi_{P, Q} \left(R\right) - \left(\Pi_{P,Q} \left(R\right) - \Pi_{P,Q} \left(S\right)\right)\right)$$

Information about a collection of students is given by the relation studInfo(studId, name, sex). The relation enroll(studId, courseId) gives which student has enrolled for (or taken) what course(s). Assume that every course is taken by at least one male and at least one female student. What does the following relational algebra expression represent?

$$\eqalign{ & \prod\nolimits_{courseId} {((\prod\nolimits_{studId} {({\sigma _{sex = 'female'}}} } \cr & (studInfo)) \times \prod\nolimits_{courseId} {(enroll)) - enroll)} \cr} $$

Consider the relation employee(name, sex, supervisorName) with name as the key, supervisorName gives the name of the supervisor of the employee under consideration. What does the following Tuple Relational Calculus query produce?

$$\eqalign{ & \{ e.name\,|\,employee(e) \wedge \cr & (\forall x)[\neg employee(x) \vee \cr & x.\sup ervisorName \ne e.name\, \vee \cr & x.sex = 'male']\} \cr} $$