Can one apply the augmentation rule in functional dependency in reverse order?

Question

The augmentation rule for functional dependencies in relational database theory specifies that if A -> B then, for any X, AX -> BX.

If the functional dependency given is of the form AX -> BX then can we determine A -> B?

Answer 1

No. Given AX→BX we cannot infer that A→B.

Example: given A=DateOfBirth, B=Sex, X=PatientId,

{DateOfBirth,PatientId} → {Sex,PatientId}

does not imply that

{DateOfBirth} → {Sex}