1) Let R and S be two relations with the following schema
R (P,Q,R1,R2,R3)
S (P,Q,S1,S2)
Where {P, Q} is the key for both schemas. Which of the following queries are equivalent?

Question

1) Let R and S be two relations with the following schema R (P,Q,R1,R2,R3) S (P,Q,S1,S2) Where {P, Q} is the key for both schemas. Which of the following queries are equivalent?

Accepted Answer

Correct answer: D — Answer: Queries I, III and IV are equivalent; II is not. Reasoning: Key fact: {P,Q} is a key for both R and S, so each (P,Q) pair appears at most once in each relation. I: Π_P(R ⋈ S) returns the set of P values for which there exists a Q such that the pair (P,Q) appears in both R and S (i.e., common (P,Q) pairs). III: Π_P(Π_{P,Q}(R) ∩ Π_{P,Q}(S)) computes the intersection of (P,Q) pairs present in both relations and then projects P. This yields exactly the same set as I. IV: Let A = Π_{P,Q}(R) and B = Π_{P,Q}(S). Then A − (A − B) = A ∩ B, so Π_P(A − (A − B)) = Π_P(A ∩ B), which is the same as III and hence I. II: Π_P(R) ⋈ Π_P(S) joins only on P, so it returns P values that appear in R and in S possibly with different Q values. For example, if R contains (p,q1) and S contains (p,q2) with q1 ≠ q2, II includes p while I, III and IV do not. Therefore II is not equivalent to the others. Conclusion: I, III and IV are equivalent; II is different because it ignores matching on Q.

Answer

A. Only I and II

Answer

B. Only I and III

Answer

C. Only I, II and III

Answer

D. Only I, III and IV

1) Let R and S be two relations with the following schema R (P,Q,R1,R2,R3) S…