Logical Consequence

Functional Dependency Closure

Given a relation $R$ with Attributes ${A_{1}, . . ., A_{k}}$ and a set $F \cup {X \to Y}$ of functional dependencies over $R$
$F$ logically implies $X \to Y$ whenever $X \to Y$ holds in all instances of $R$ that satisfies each FD in $F$
The closure $F^{+}$ of $F$ is the set of all functional dependencies that are logically implied by $F$

Observation:

F \subseteq F^{+}

If $F = {A \to B, B \to C}$ then
${A \to B, B \to C, A \to C} \subseteq F^{+}$