7.29

Show that the following decomposition of the schema $R$ of Exercise 7.1 is not a lossless decomposition:

$$\begin{gathered} (A,B,C) \\ (C,D,E) \end{gathered}$$

Hint: Give an example of a relation $r(R)$ such that ${\mathrm{\Pi }}_{A,B,C}(r)\bowtie {\mathrm{\Pi }}_{C,D,E}(r)\ne r$

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Take the following instance of $r(R)$:-

A	B	C	D	E
1	6	5	7	3
2	8	5	9	4

Then ${\mathrm{\Pi }}_{A,B,C}(r)$ is:-

A	B	C
1	6	5
2	8	5

${\mathrm{\Pi }}_{C,D,E}(r)$ is:-

C	D	E
5	7	3
5	9	4

And their natural join ${\mathrm{\Pi }}_{A,B,C}(r)\bowtie {\mathrm{\Pi }}_{C,D,E}(r)$ 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.