GATE CS 1998 Q2.19 — Tuple Calculus and SQL

To determine which transformation is incorrect, let us analyze each option:

• Option A: $\sigma_{C_1} \left(\sigma_{C_2}\left(R_1\right)\right) \to \sigma_{C_2} \left(\sigma_{C_1}\left(R_1\right)\right)$
  Selection operations commute. The order in which we apply independent selection conditions does not affect the final set of tuples. Thus, this transformation is correct.
• Option B: $\sigma_{C_1} \left(\pi_{A_1}\left(R_1\right)\right) \to \pi_{A_1} \left(\sigma_{C_1}\left(R_1\right)\right)$
  For the left-hand side (LHS) expression to be valid, the selection condition $C_1$ must only involve attributes that are present in the projection list $A_1$. Since $C_1$ only uses attributes from $A_1$, applying the selection before or after the projection yields the same result. Thus, this transformation is correct.
• Option C: $\sigma_{C_1} \left(R_1 \cup R_2\right) \to \sigma_{C_1}\left(R_1\right) \cup \sigma_{C_1} \left(R_2\right)$
  Selection distributes over set union. Filtering the union of two relations is equivalent to filtering each relation individually and then taking their union. Thus, this transformation is correct.
• Option D: $\pi_{A_1} \left(\sigma_{C_1}\left(R_1\right)\right) \to \sigma_{C_1} \left(\pi_{A_1}\left(R_1\right)\right)$
  On the LHS, the selection condition $C_1$ can involve any attributes of $R_1$, including those not present in $A_1$. Once the selection is applied, we project only the attributes in $A_1$.
  However, on the RHS, the projection $\pi_{A_1}(R_1)$ is performed first, which discards all attributes not in $A_1$. If $C_1$ references any of these discarded attributes, the subsequent selection $\sigma_{C_1}$ cannot be evaluated, making the RHS expression invalid. Thus, this transformation is incorrect.