# Identifying the normal form that a certain relation complies with

I have a little problem with a normalization exercise. I have the following relation:

``````R = {L,K,D,S,O,W}
``````

which presents the functional dependencies (FD) shown below:

``````F = { {D,S,K} -> {O},
{O} -> {K},
{O,L} -> {W}
{W,L,K} -> {O},
{D,S,W} -> {L}}
``````

I have to determine what normal form (NF) is satisfies. So I started from determining the candidate keys; they are:

``````{D,S,O,L}

{D,S,L,K}

{D,S,K,W}

{D,S,O,W}
``````

Then, I worked from-top-to-bottom (not sure if it is good strategy).

1. This relation isn't in BCNF because the left side of every FD does not contain key(s) e.g. `{O} -> {K}`

2. My definition for 3NF was that, for every nontrivial, simple FD, the left side contains a key or the right side is part of a key (or it is a key). Everything matches and I thought that the relation `R` is in 3NF.

Then I look on the conditions to be in 2NF and saw that:

1. This relation contains partial dependency e.g. `{O} -> {K}`

2. This relation contains transitive dependency.

Transitive dependency is for every FD `X -> A`, `X` is a subset or superset of the key (is that right, but if `X` is a subset of the key then this is a partial dependecy)?

This dependency doesn't work for me `{W,L,K} -> {O}` - this is transitive. So, is relation `R` in 1NF? This is something weird, either I have a bad definition of 3NF or this relation is in 3NF... but it isn't in 2NF, which is impossible. What's wrong?