Show the following rule for MVDs The attributes are arbitrar

Show the following rule for MVD\'s. The attributes are arbitrary sets X, Y, Z and the other unnamed attributes of the relation in which these dependencies hold. The Difference Rule. If X? Y, and Z are sets of attributes, X - > - > Y, and X - > - > Z, then X - > - > (Y-Z).

Solution

For the relation R whose schema is the set of attributes {A, B, C, D} with functional dependencies A B and B C, the result obtained by decomposing R according to A B and the result obtained by expanding the BCNF violation to A BC is same. It is same because both A B and A BC has A on the left side and part of the decomposition process involves finding {A}+ to form one decomposed relation and A plus the rest of attributes not in {A}+ as the second relation. The decomposed relations in both the cases are same.