What is the optimized version of the relation algebra expression $\pi_{A1}(\pi_{A2}(\sigma_{F1}(\sigma_{F2}(r))))$, where $A1, A2$ are sets of attributes in $r$ with $A1 \subset A2$ and $F1, F2$ are Boolean expressions based on the attributes in $r$?

We use standard relational algebra equivalence rules:

1. Cascade of Selections:

   $$\sigma_{F1}(\sigma_{F2}(r)) \equiv \sigma_{F1 \land F2}(r)$$

   Applying consecutive selection conditions is equivalent to applying a single selection with the logical AND of the conditions.

2. Cascade of Projections: If $A1 \subseteq A2$, then:

   $$\pi_{A1}(\pi_{A2}(E)) \equiv \pi_{A1}(E)$$

   Because $A1$ is a subset of $A2$, projecting onto $A2$ first and then onto $A1$ preserves the exact same set of attributes as directly projecting onto $A1$.

Combining these two optimizations gives:

Thus, option (A) is the correct optimized expression.