Let R =(R; F) be a schema and S a set of attributes, such that S ? R. Let X ? Y be an FD such that X , Y ? S. Prove that
1. If X ? Y ? F+ then for every legal instance r of R its projection ?S(r) satis?es X ? Y .
2. Conversely, if X ? Y ? F+ then there is a legal instance r of R such that ?S(r) violates X ? Y .
1. Since X ? Y ? F+, this FD must be satis?ed by r. Since the attributes of this FD belong to S, it follows directly from the de?nition of FDs that X ? Y is satis?ed by ?X(r).
2. If X ? Y ? F+ then there is a legal instance of R, r, which satis?es F, but violates X ? Y . Again, by the de?nition of FDs, ?X(r) must violate this dependency.
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