Prove the following generalized transitivity rule :If Z ? Y ,then X ? Y and Z ? W entail X ?W . Try to prove this rule in two ways:
1. Using the argument that directly appeals to the de?nition of FDs.
2. By deriving X ? W from X ? Y and Z ? W via a series of steps using Armstrong’s axioms
1. Let t , s be arbitrary tuples in a relation that satis?es the above conditions, and suppose t , s agree on X . Then, due to X ? Y , they agree on Y and thus also on Z . But then, due to Z ? W , t , s must agree on W .
Therefore, any pair of tuples that agrees on X in such a relation must also agree on W , i.e., X ?W must hold.
2. (a) Z ? W Given
(b) Y ? WY By Augmentation of (1) and because Z ? Y
(c) X ? Y Given
(d) X ? WY From (3),(2) by transitivity
(e) WY ? W By the Reflexivity rule
(f) X ? W From (4),(5) by Transitivity
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