Suppose that R =(R; F) is a schema and that R 1 =(R 1 ; F 1 ), R 2 =(R 2 ; F 2) is a binary decomposition such that neither (R 1 ? R 2 ) ? R 1 nor (R 1 ? R 2) ? R 2 is implied by F. Construct a relation, r, such that r ? ? R1 (r) ? R 2 (r), where ? denotes strict subset. (This relation, therefore, shows that at least one of these FDs is necessary for the decomposition of R to be
lossless.)
What will be an ideal response?
Let R = ABC , R1= AB ,and R2= BC .Let F be empty and r be the following relation:
Clearly, this relation satis?es the conditions in the statement of the problem. It is easy to see that the join ?R1(r) ?R2(r) contains (among others) the tuple 1, 0, 4,which is not in r.
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