The definitions of 2NF, 3NF and BCNF goes like this:
2 Normal Form (NF) definition
A relation is in second normal form if and only if it is in first normal form and all the non-key attributes are fully functionally dependent on the candidate key.
3 NF - definition
A relation schema $R$ is in third normal form (3NF) if, whenever a nontrivial functional dependency $α→β$ holds in $R$, either
- $α$ is a superkey of $R$, or
- $β$ is a prime attribute of $R$.
BCNF (Boyce–Codd normal form) - definition
A relation schema $R$ is in BCNF, if whenever a nontrivial functional dependency $α→β$ holds in $R$, $α$ is a superkey in $R$.
Based upon these decision, I want to know if the relation having below characteristic is already in some normal form.
All candidate keys are of single attributes: this one is definitely in 2NF as there cannot be any partial dependency, but full functional dependency on CK. I am more confused about 3NF and BCNF. I feel its not in 3NF and BCNF since there can exist $\alpha \rightarrow \beta$, such that both $\alpha$ and $\beta$ are non key, violating 3NF and BCNF definitions.
Am I right with this?