7.29
Show that the following decomposition of the schema \(R\) of Exercise 7.1 is not a lossless decomposition:
\[ (A,B,C) \\ (C,D,E) \\ \]
Hint: Give an example of a relation \(r(R)\) such that \(\Pi_{A,B,C}(r) \bowtie \Pi_{C,D,E}(r) \not = r\)
Take the following instance of \(r(R)\):-
| A | B | C | D | E |
|---|---|---|---|---|
| 1 | 6 | 5 | 7 | 3 |
| 2 | 8 | 5 | 9 | 4 |
Then \(\Pi_{A,B,C}(r)\) is:-
| A | B | C |
|---|---|---|
| 1 | 6 | 5 |
| 2 | 8 | 5 |
\(\Pi_{C,D,E}(r)\) is:-
| C | D | E |
|---|---|---|
| 5 | 7 | 3 |
| 5 | 9 | 4 |
And their natural join \(\Pi_{A,B,C}(r) \bowtie \Pi_{C,D,E}(r)\) is:-
| A | B | C | D | E |
|---|---|---|---|---|
| 1 | 6 | 5 | 7 | 3 |
| 1 | 6 | 5 | 9 | 4 |
| 2 | 8 | 5 | 7 | 3 |
| 2 | 8 | 5 | 9 | 4 |
Thus, the decomposition is a lossy decomposition.