A set of attributes X is closed under functional dependencies if X+ = X. Consider now a relation R(A,B,C,D) where the
set of functional dependencies is unknown to us. 1. What are the fd's if we know that every attribute set is closed? This means we have only trivial functional dependencies! The empty set {} is thus a consistent set of fd's. 2. What if we know that the only closed sets are {} and {A,B,C,D} ? One solution could be: {A -> B, B -> C, C -> D, D -> A}. What are other possible consistent sets of fd's ? Can you think of something different than permutations of the above solution?
3. What if the closed sets are {}, {A,B} and {A,B,C,D} ?
First, notice that {A}+ is not {A}, and since {A,B}+ = {A,B}, it must be that A -> B is an fd. With a similar argument, we have
that B -> A is another fd we must include. A possible solution is as follows: {C -> ABD, D -> ABC, A -> B, B -> A}. Is there any other solution ?
