Consider the following three relations in a relational database. $Employee(\underline{eId}, Name)$, $Brand(\underline{bId}, bName)$, $Own(\underline{eId}, \underline{bId})$ Which of the following relational algebra expressions return the set of $eIds$ who own all the brands?

To find the set of $eId$s who own all the brands listed in the $Brand$ relation, we use the relational division operation:

• Option A: $\Pi_{eId} (\Pi_{eId, bId}(Own) / \Pi_{bId}(Brand))$ correctly divides the pairs $(eId, bId)$ in $Own$ by all $bId$s in $Brand$. The resulting schema already consists solely of $eId$, so the outer projection $\Pi_{eId}$ retains the correct result. Hence, A is correct.

• Option B: Relational division $R(A, B) / S(B)$ expressed in terms of fundamental relational algebra operations is:

  $$\Pi_A(R) - \Pi_A((\Pi_A(R) \times S) - R)$$

  Substituting $R = \Pi_{eId, bId}(Own)$ and $S = \Pi_{bId}(Brand)$ gives:

  $$\Pi_{eId}(Own) - \Pi_{eId}((\Pi_{eId}(Own) \times \Pi_{bId}(Brand)) - \Pi_{eId, bId}(Own))$$

  This finds all $eId$s and removes those who are missing at least one brand from $Brand$. Hence, B is correct.

• Option C: Dividing by $\Pi_{bId}(Own)$ only finds employees who own all brands that are owned by *at least one* employee, which may not be the complete set of brands in $Brand$. Hence, C is incorrect.

• Option D: The Cartesian product $\Pi_{eId}(Own) \times \Pi_{bId}(Own)$ generates artificial pairings of every employee with every owned brand, losing the actual ownership information from $Own$. Hence, D is incorrect.