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 Union Rule. If X, Y, and Z are sets of attributes, X rightarrow rightarrow Y, and X rightarrow rightarrow Z, then X rightarrow rightarrow (Y Z)

Solution

Q9)

We will do this by using a proof by contradiction.

Assume that (X⁺)⁺ X⁺. Then for (X⁺)⁺, it must be that some Functional Dependency allowed additional attributes to be added to the original set X⁺.

For instance, X^{+ ->}A where A is some attribute not in X⁺.

If this was the case, then X⁺ wouldn’t be the closure of X. The closure of X should have to include A as well.

This contradicts the fact that we were given the closure of X, X⁺.

Hence, it must be that (X⁺)⁺ = X⁺ or else X⁺ is not the closure of X.