Question 31

Given the relational algebra expression:

where $A1, A2$ are sets of attributes with $A1 \subset A2$, and $F1, F2$ are selection conditions on attributes of $r$.

1. Cascade of Selections:
   Successive selection operations commute and combine via logical conjunction (AND):

   $$\sigma_{F1}(\sigma_{F2}(r)) = \sigma_{F1 \land F2}(r).$$

2. Cascade of Projections:
   For any attribute sets $X$ and $Y$ where $X \subseteq Y$:

   $$\pi_X(\pi_Y(R)) = \pi_X(R).$$

   Since $A1 \subset A2$, performing projection onto $A2$ first and then onto $A1$ simply yields the projection onto $A1$ directly:

   $$\pi_{A1}(\pi_{A2}(\dots)) = \pi_{A1}(\dots).$$

3. Combining both rules:

   $$E = \pi_{A1}(\sigma_{F1 \land F2}(r)).$$

This matches option A.