Exercise 15.6 Briefly answer the following questions: Explain the role of relational algeb...

Exercise 15.6 Briefly answer the following questions:

1. Explain the role of relational algebra equivalences in the System R optimizer.
2. Consider a relational algebra expression of the form σ_c(π_l(R × S)). Suppose that the equivalent expression with selections and projections pushed as much as possible, taking into account only relational algebra equivalences, is in one of the following forms. In each case give an illustrative example of the selection conditions and the projection lists (c, l, c1, l1, etc.).
   1. Equivalent maximally pushed form: π_l1(σ_c1(R) × S).
   2. Equivalent maximally pushed form: π_l1(σ_c1(R) × σ_c2(S)).
   3. Equivalent maximally pushed form: σ_c(π_l1(π_l2(R) × S)).
   4. Equivalent maximally pushed form: σ_c1(π_l1( σ_c2(π_l2(R)) × S)).
   5. Equivalent maximally pushed form: σ_c1(π_l1(π_l2(σ_c2(R)) × S)).
   6. Equivalent maximally pushed form: π_l(σ_c1(π_l1(π_l2(σ_c2(R)) × S))).