Question 25

To find all eId who own all brands in relation Brand:

1. Expression (A) computes $\Pi_{\text{eId}, \text{bId}}(\text{Own}) / \Pi_{\text{bId}}(\text{Brand})$. Relational division $R(A, B) / S(B)$ returns the set of all $A$-values that are associated in $R$ with every $B$-value present in $S$. Projecting eId yields precisely the employees who own all brands. Thus (A) is correct.
2. Expression (B) uses the fundamental definition of division using projection, cartesian product, and difference:
   • $\Pi_{\text{eId}}(\text{Own}) \times \Pi_{\text{bId}}(\text{Brand})$ forms all possible pairs of (eId, brand) for each employee who owns at least one brand.
   • Subtracting $\Pi_{\text{eId}, \text{bId}}(\text{Own})$ yields the pairs (eId, brand) representing brands that an employee does NOT own.
   • Projecting on eId gives all employees who are missing at least one brand.
   • Subtracting this set from $\Pi_{\text{eId}}(\text{Own})$ leaves only those employees who own all brands. Thus (B) is correct.
3. Expressions (C) and (D) divide by $\Pi_{\text{bId}}(\text{Own})$, which only considers brands currently owned by someone rather than all brands defined in the Brand relation.