Question

For a relation R(A,B,C,D,E), we propose a decomposition R into the three relation schemas: R1(A,B,C), R2(B,C,D), and R3(A,C,E). For each of the following sets of FD’s that hold on R, use the technique discussed in the class to (i) prove/disprove the proposed decomposition is lossless, and (ii) determine whether or not, the proposed decomposition is dependency-preserving.

(a). A→D,D→E,andB→D. (b). CD→E,E→D,andA→D.

Answer 1

To prove the proposed decomposition is lossless, check if the union of the decomposed relations is equal to the original relation. To determine if it is dependency-preserving, check if all functional dependencies are preserved. For case (a), the proposed decomposition is lossless and dependency-preserving. For case (b), it is not lossless and not dependency-preserving.

Step-by-step explanation:

To prove the proposed decomposition is lossless, we need to check if the union of the three decomposed relations (R1, R2, and R3) is equal to the original relation (R). If the union is equal, then the decomposition is lossless. If not, it is not lossless.

To determine if the proposed decomposition is dependency-preserving, we need to check if all the functional dependencies in the original relation (R) are preserved in the decomposed relations (R1, R2, and R3). If all dependencies are preserved, then the decomposition is dependency-preserving. If any dependency is lost, then it is not dependency-preserving.

(a) For the given set of functional dependencies A→D, D→E, and B→D, we can prove that the proposed decomposition is lossless and dependency-preserving.

(b) For the given set of functional dependencies CD→E, E→D, and A→D, we can disprove that the proposed decomposition is lossless and dependency-preserving.