Why aren't normal forms defined more simply?

I believe that 2nd and 3rd normal form definitions could obviously be specified with less complexity, but they're not. The definition found in textbooks is something like this: **2NF: No partial FD between prime and non-prime attributes.** This means there is no such situation as:

-> B -> C

where

is a key,

its subset, and

a non-prime. As

-> B

always holds for every B, this could be simplified by removing

-> B

to get

-> C

, or: **2NF-new: No FD between a subset of a key and a non-prime.** and, **3NF: No indirect FD between a key and non-prime attributes.** This means there is no such thing as:

-> NP1 -> NP2

. Again, as

-> X

for every X, this could be simplified to get

-> NP2

leading to: **3NF-new: No FD between two non-primes.** However, I have not seen this simplified version anywhere, except > Equivalently, a transitive dependency exists when a nonprime > attribute determines another nonprime attribute. What makes me is more doubtful is the word *typically* in > A nonprime attribute determines another nonprime attribute. Here we > typically have a transitive dependency that violates 3NF. By Elmasri and Navathe. So why not *always*? Are my definitions wrong? If not, why are they not used?