GATE CS 2022 Q15 — Relational Model: Relational Algebra

To find the set of employees ($eId$) who own all the brands, we can analyze each option:

• Option A: $$\Pi_{eId} \left(\Pi_{eId,bId} (Own) / \Pi_{bId} (Brand) \right)$$ The division operator ($/$) is specifically designed for "all" queries. Dividing the active ownership pairs by the set of all brands yields the $eId$s associated with every $bId$ in $\text{Brand}$. Thus, Option A is correct.
• Option B: $$\Pi_{eId} (Own) – \Pi_{eId} \left( ( \Pi_{eId} (Own) \times \Pi_{bId} (Brand)) – \Pi_{eId,bId} (Own)\right)$$ This is the standard double-negation formulation of division:
  1. $(\Pi_{eId} (Own) \times \Pi_{bId} (Brand))$ represents all possible combinations of existing owners and all brands.
  2. Subtracting $\Pi_{eId,bId} (Own)$ gives the pairs of $(eId, bId)$ that are missing (i.e., brands that an owner does not own).
  3. Projecting this on $eId$ gives the set of employees who are missing at least one brand.
  4. Subtracting this set from all active owners $\Pi_{eId}(Own)$ leaves only those employees who do not miss any brand (i.e., they own all brands).
  Thus, Option B is correct.
• Option C: $$\Pi_{eId} \left(\Pi_{eId,bId} (Own) / \Pi_{bId} (Own) \right)$$ This divides by $\Pi_{bId}(Own)$ (brands owned by at least one person) instead of $\Pi_{bId}(Brand)$ (all existing brands). If some brand is not owned by anyone, this expression fails to check for it. Thus, Option C is incorrect.
• Option D: $$\Pi_{eId} \left( (\Pi_{eId}(Own) \times \Pi_{bId} (Own)) / \Pi_{bId} (Brand) \right)$$ The cartesian product creates artificial pairs of every owner with every owned brand, losing the actual ownership information from the $Own$ relation. Thus, Option D is incorrect.