Chapter 19, Problem 15E

Explanation of Solution

To prove the decomposition is not a dependency preserving decomposition:

The schema R= (A,B,C,D,E) is being decomposed into (A,B,C) and (D,E,F) along with functional dependencies that are defined and holds true.

The decomposition does not preserve the functional dependencies $\text{B}\to \text{D}$

Proof:

Considering the restriction of the functional dependency F to the decomposition (A,B,C) also called as F_1:

$\begin{array}{l}{\text{F}}_1=\text{A}\to \text{ABC},\text{A}\to \text{A}\to \text{B},\text{A}\to \text{AC},\text{A}\to \text{BC},\text{A}\to \text{B},\text{A}\to \text{C},\\ \text{A}\to \text{A,B}\to \text{B},\text{C}\to \text{C},\text{AB}\to \text{C},\text{AB}\to \text{ABC},\text{AB}\to \text{BC},\text{AB}\to \text{AB},\\ \text{AB}\to \text{A},\text{AB}\to \text{B}\end{array}$