Functional Dependency Trivial

Question

If you have something on the LHS and RHS is it considered trivial if that's the only symbol on the RHS? For example:

ABC -> C

Could you break it down like this:

C -> C
A -> {}
B -> {}

where {} is the empty set. Or is this not valid?

This would make this rule useless and could it simply be dropped?

Answer 1

All FDs in which the RHS is a subset (not necessarily proper) of the LHS, are trivial.

Therefore, all the FDs mentioned in your question are trivial.

An FD such as {A} -> {} says that "if you know A, then you know at least nothing at all". An FD such as {ABC} -> {C} says that "if you know A and B and C, then you know at least C".

From the formal perspective of set theory, it's probably unwise to rule out the case of the empty set in FD theory, but at any rate, any trivial FD is typically uninteresting at best.

Hence, {ABC} -> {C} is exactly as "useless" as those with the empty RHS, and can be "dropped" equally well.