According to the textbook by Silberschatz,
A relation schema R is in third normal form with respect to a set $F$ of functional dependencies if, for all functional dependencies in $F^+$ of the form $\alpha$ → $\beta$, where $\alpha$ ⊆ R and $\beta$ ⊆ R, at least one of the following holds:
$\bullet$ $\alpha$ → $\beta$ is a trivial functional dependency.
$\bullet$ $\alpha$ is a superkey for R.
$\bullet$ Each attribute A in $\beta - \alpha$ is contained in a candidate key for R.
Another definition from an online video lecture series by a reputed college that I am following states,
A relation is in third normal form if:
$\bullet$ It is in second normal form,
(or in other words, no non-prime attribute of R is dependent on any proper subset of any candidate key of R).
$\bullet$ No non-prime attribute of R is functionally dependent on any other non-prime attribute of R.
I'm having a hard time understanding how they are equivalent. I can't understand how they seem so distinct from each other, and yet, at their core, are saying the same thing.
Can someone explain why saying one thing is the same as saying the other?