## Find whether the given decomposition is Lossless or lossy decomposition

__Question:__
Suppose you are given a relation

*R(A,B,C,D)*. let us assume that set of functional dependencies F = { B → C, D → A} and R is decomposed into R1(B,C), and R2(A,D).
(a) Find the candidate key(s) for

*R*. (b) State whether or not the proposed decomposition of*R*into smaller relations is a good decomposition and briefly explain why or why not.

__Solution:__
a) To find the candidate keys, we have to
find the closure of attributes of LHS (left hand side) of all functional dependencies.
If the closure includes all the attributes of the given relation, then that
attribute (or set of attributes) is the candidate key.

B

^{+}= BC. It does not include all attributes of R.
D

^{+}= DA. It does not include all attributes of R.
(BD)

^{+}= BCDA. It includes all attributes of R, hence BD is the candidate key.
b) The decomposition of R into R1 and R2 is
lossy because there is no common attribute between R1 and R2 (R1∩R2 = ∅).
Hence, the join of R1 and R2 will result in Cartesian product of these two
relations which is not the base relation R.

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