What is the difference between F+ (closure of F) and F* (cover of F) for Functional Dependencies?

Question

F+ and F* are defined as follows:

• F+: closure of F

  • F+ = {fd | F |= fd}
  • Set of all FDs deduced from inference rule (normally: Armstrong axioms)

• F*: cover of F

  • F* = {fd | F |- fd}
  • Set of all FDs entailed by F (all FDs that are true)

So my question is: What is the difference between F+ and F*? Can you also give an example to demonstrate the difference.

Answer 1

An important property of the Armstrong’s axioms, (as well as of similar set of axioms), it that they are sound and complete (for a proof see for instance this).

This amount to say that F⁺ = F^*. In other words, all the FD derived from those axioms are logically entailed by F, as well as all the FD dependencies logically entailed by F can be derived by repeatedly applying the axioms.