Let R1 and R2 be sets of attributes and R = R1 R2. Let r be a relation on R. Prove that r? ? R1 (r) ? R2 (r). Generalize this result to decompositions of R into n > 2 schemas.
What will be an ideal response?
If t ? r,then t1= ?R1(t ) and t2= ?R2(t ) match over the attributes in R1? R2, so this tuple will be in the natural join of ?R1(r) and ?R2(r).
For the general case, it is easy to see that
```
?R1(r) ... ?Rn (r) =
{t | t is a tuple over R and ?R1(t ) ? ?R1(r), . . . , ?Rn(t ) ? ?Rn(r)}
```
Clearly, every tuple in r satis?es the above conditions and thus is in the above set.
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