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).

$$R(A,B,C,D)$$ 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?

Consider the schema $$R = \left( {S\,\,T\,\,U\,\,V} \right)$$ and the dependencies $$S \to T,\,\,T \to U,\,\,U \to V$$ and $$V \to S$$ let $$R =$$ $$(R1$$ and $$R2)$$ be a decomposition such that $$R1 \cap R2 \ne \phi .$$ The decomposition is: