Relation $$R$$ is decomposed using a set of functional dependencies, $$F,$$ and relation $$S$$ is decomposed using another set of functional dependencies, $$G.$$ One decomposition is definitely $$BCNF,$$ the other is definitely. $$3NF,$$ but it is not known which is which. To make a guaranteed identification, which one of the following tests should be used on the decompositions? (Assume that the closures of $$F$$ and $$G$$ are available).

From the following instance of a relation schema $$R(A, B, C),$$ we can conclude that:

$$R(A,B,C,D)$$ is a relation. Which of the following does not have a lossless-join, dependency preserving $$BCNF$$ decomposition?

Given the following relation instance
$$\eqalign{ & X\,\,\,\,\,Y\,\,\,\,\,Z \cr & \,\,1\,\,\,\,\,\,4\,\,\,\,\,\,2 \cr & \,\,1\,\,\,\,\,\,5\,\,\,\,\,\,3 \cr & \,\,1\,\,\,\,\,\,6\,\,\,\,\,\,3 \cr & \,\,3\,\,\,\,\,\,2\,\,\,\,\,\,2 \cr} $$

Which of the following functional dependencies are satisfied by the instance?