Find the functional dependencies that hold in a relation

Question

There is a question in my book that asks the following:

Suppose we have relation R ( A ,B ,C ,D ,E ) , with some set of FD’s, and we wish to project those FD’s onto relation S(A, B, C). Give the FD’s that hold in S if the FD’s for R are:

A->D
BD->E
AC->E
DE->B

In each case, it is sufficient to give a minimal basis for the full set of FD’s of S.

So I attempted to compute the closure of attributes for all of the subsets of (A,B,C). I could not find any minimal basis of FD's for S. I computer the closure of A, B, C, and AC, but I could not get all of the attributes. A only implies AD, B only implies B, C only implies C and AC only implies ACE. I could not find a functional dependency where there is a attribute on the left hand side that implies all of the attributes on the right hand side.

Answer 1

For convenience, define the (implied) numbering of your question:

1. A->D
2. BD->E
3. AC->E
4. DE->B

Then substituting 1 and 3 into 4 gives
(A)(AC) -> B
which reduces to just
5. AC -> B.