Is relation following given characteristic already in 2NF, 3NF and BCNF?

Question

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

1. $α$ is a superkey of $R$, or
2. $β$ 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?

Answer 1

Consider a relation schema R(A, B, C), with the functional dependencies:

A → B
B → C

In this schema there is only one candidate key (A), and it has a unique attribute. But the schema is not in BCNF, since B is not a superkey, neither in 3NF, since C is not a prime attribute.

So your intuition is correct.