Consider a relation R that has three attributes ABC. It is decomposed into relations R1 with attributes AB and R2 with attributes BC.
1. State the definition of a lossless-join decomposition with respect to this example. Answer this question concisely by writing a relational algebra equation involving R, R1, and R2.
2. Suppose that B → C. Is the decomposition of R into R1 and R2 lossless-join?
Reconcile your answer with the observation that neither of the FDs R1 ∩ R2 → R1 nor R1 ∩ R2 → R2 hold, in light of the simple test offering a necessary and sufficient condition for lossless-join decomposition into two relations in Section 15.6.1.
3. If you are given the following instances of R1 and R2, what can you say about the instance of R from which these were obtained? Answer this question by listing tuples that are definitely in R and tuples that are possibly in R.
Instance of R1 = {(5,1), (6,1)}
Instance of R2 = {(1,8), (1,9)}
Can you say that attribute B definitely is or is not a key for R?