I have a set of relations {A, X, Y, Z}:

Given that the functional dependencies: AX→Y, AX→Z, Y→A, Z→X, I would think that this relation would be in the normal form 1NF, because there is a partial dependency for the composite key, YZ; Y determines A and Z determines X. However, I ran across this example on a university website claiming the relationship was in 3NF. Am I wrong about this relation being in 1NF?

On a similar note, given the same relation but with functional dependencies: A→XY, Y→AZ, would this be in 3NF or BCNF, I do not see any reason why this would not be in BCNF, but I have seen a similar example where there was a further decomposition, that was also BCNF. So is the decomposition wasteful because the original relation with given dependencies was already in BCNF?