3. For the universal relation R(A, B, C, D), consider the decomposition D consisting of R1(A, B) and R2(B, C, D), and the set F of functional dependencies { A -> D ; B -> A,C ; D -> A,B }.

a. Compute the projection of F on R1.

b. Compute the projection of F on R2.

c. Does the decomposition D preserve the set of dependencies F? Give a detailed explanation why or why not. (That is, dont just repeat back the definition of the dependency preservation property, but rather show why the decomposition D either has or does not have this property.)

4. Perform the chase test for the lossless join property for the universal relation R(A1, A2, A3, A4), and the decomposition D and set of functional dependencies F given below: D = { R1(A1, A2) , R2(A2, A3, A4) , R3(A1, A4) } F = { A1,A3 -> A2 , A2 -> A1 , A4 -> A3 }

a. Show the initial state of the matrix S, and show each step of your work as you modify the matrix using the functional dependencies. (That is, dont just show me the final state of the matrix each time you change it, show the change, and show which functional dependency you applied to make the change. Answers that show only the final matrix without showing the steps taken to obtain it will receive no credit.)

b. Does the decomposition D have the lossless join property? Explain why or why not in terms of the final state of the matrix S.