Exercise 19.11 Consider a relation R that has three attributes ABC. It is decom- posed into relations R1 with attributes AB and R2 with attributes BC. 1. State the definition of a lossless-join...

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Extracted text: Exercise 19.11 Consider a relation R that has three attributes ABC. It is decom- posed 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 Rị and R2 lossless-join? Reconcile your answer with the observation that neither of the FDs R1n R2→ Rị nor R1 n 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?