Questions Concerning the Chase Test

> 2. [5 Marks] Let R(A,B,C,D,E) be decomposed into relations with the following three set of attributes {A,B,C}, {B,C,D}, and {A,C,E}. For each of the following sets of FD's, use the chase test to tell whether the decomposition of R is lossless. For those that are not lossless, give an example of an instance of R that returns more than R when projected onto the decomposed relation and rejoined. > > a. A→D, CD→E and E→D. > > b. A→D, D→E and C→D. For the question above, my work for each of the questions is below the concerns. Here are my main concerns: 1. Does the order in which you use the relations matter? 2. Can you end up with less tuples with the chase test? 3. Is my approach correct? -----------------------Part A below---------------------------------- InitialTableau = T₁ ⋈ T₂ ⋈ T₂

+----+----+----+----+----+
| A  | B  | C  | D₁ | E₁ |
+----+----+----+----+----+
| A₂ | B  | C  | D  | E₂ |
+----+----+----+----+----+
| A  | B₂ | C  | D₂ | E  |
+----+----+----+----+----+

-----------------------ANSWER TO QUESTIONS START NOW----------------------- a) After changing the initial tableau in a way that ensures that the FD's given in the question are satisfied, we get the following tableau.

+----+----+----+----+----+
| A  | B  | C  | D₁ | E  |
+----+----+----+----+----+
| A₂ | B  | C  | D  | E₂ |
+----+----+----+----+----+
| A  | B₂ | C  | D₁ | E  |
+----+----+----+----+----+

- Since we do not have an unsubscribed row, this relation is lossy/not lossless. Example of an instance R (Were going to use the final tableau): R₁(A,B,C)

+----+----+----+
| A  | B  | C  |
+----+----+----+
| A₂ | B  | C  |
+----+----+----+
| A  | B₂ | C  |
+----+----+----+

R₂(B,C,D)

+----+----+----+
| B  | C  | D₁ |
+----+----+----+
| B  | C  | D  |
+----+----+----+
| B₂ | C  | D₁ |
+----+----+----+

R₃(A,C,E)

+----+----+----+
| A  | C  | E₂ |
+----+----+----+
| A₂ | C  | E₂ |
+----+----+----+
| A  | C  | E  |
+----+----+----+

After Joining the above relations, we get:

+----+----+----+----+----+
| A  | B  | C  | D₁ | E  |
+----+----+----+----+----+
| A  | B  | C  | D  | E  |
+----+----+----+----+----+
| A₂ | B  | C  | D₁ | E₂ |
+----+----+----+----+----+
| A₂ | B  | C  | D  | E₂ |
+----+----+----+----+----+
| A  | B₂ | C  | D  | E  |
+----+----+----+----+----+

- **since we have 2 more rows than the original tableau, this decomposition is not lossless.** b) After changing the initial tableau in a way that ensures that the FD's given in the question are satisfied, we get the following tableau.

+----+----+----+----+----+
| A  | B  | C  | D  | E  |
+----+----+----+----+----+
| A₂ | B  | C  | D  | E  |
+----+----+----+----+----+
| A  | B₂ | C  | D  | E  |
+----+----+----+----+----+

- Since we have an unsubscribed row, this decomposition is lossless.