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. Removing attributes shared by left and right side. If X - > - > Y holds, then X - > - > (Y-X) holds.

Solution

Assume W be the set of attributes not in X or Y, V be the set of attributes that X and Y have in common, Y1 be the set of attributes of Y not in V and X1 be the set of attributes of X not in V. Consider the two tuples x1vy1w and x1vy1\'w\'. Because X®®Y, swap the y\'s so tuples x1vy1\'w and x1vy1w\' are in R. In finally, started with tuples x1vy1w and x1vy1\'w\' and showed that the x1vy1\'w and x1vy1w\' should also bein the relation. That is accurately the statement of the MVD X®®(Y – X).