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

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.

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.