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 (YUZ)

Solution

Suppose W be the set of attributes not in X, Y, or Z. Consider 2 tuples xyzw, xy\'z\'w\' are in the relation R . Since X®®Y, swap the y\'s, so xy\'zw and xyz\'w\' are in R. Because X®®Z, take the pair of tuples xyzw and xyz\'w\' and swap the z\'s to get xyz\'w and xyzw\'. Similarly, take the pair xy\'z\'w\' and xy\'zw and swap Z\'s to get xy\'zw\' and xy\'z\'w. In conclusion, started with tuples xyzw and xy\'z\'w\' and showed that xyzw\' and xy\'z\'w should also be in the relation. That is correctly the statement of the MVD X®®(Y Z).