Chapter 19, Problem 14E

Explanation of Solution

Proof:

The schema U has been decomposed into relation R₁, R₂,….R_n.

Consider a record t that is present in the relation $\text{u}\left(\text{U}\right)$

Here,

${\text{r}}_{\text{i}}\text{=}{\displaystyle {\prod }_{\text{Ri}}\left(\text{u}\right)}$  and it is $\text{t}\left[{\text{R}}_{\text{i}}\right]\in {\text{r}}_{\text{i}}\text{,1}\le \text{i}\le \text{n}$

So,

$\begin{array}{l}\text{t}\left[{\text{R}}_1\right]⊳⊲\text{t}\left[{\text{R}}_2\right]⊳⊲\mathrm{.....}⊳⊲\text{t}\left[{\text{R}}_n\right]\\ \in {\text{r}}_{\text{1}}⊳⊲{\text{r}}_{\text{2}}⊳⊲\mathrm{....}⊳⊲{\text{r}}_{\text{n}}\end{array}$

Using the definition of the natural join the below equation is obtained,

$\begin{array}{l}\text{t}\left[{\text{R}}_1\right]⊳⊲\text{t}\left[{\text{R}}_2\right]⊳⊲\mathrm{.....}⊳⊲\text{t}\left[{\text{R}}_n\right]\\ \in ({\sigma }_{\beta }\left(\text{t}\left[{\text{R}}_1\right]\times \text{t}\left[{\text{R}}_2\right]\right)\times \mathrm{}\end{array}$