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Suppose that we have a set of functional dependencies F on a relation schema Whenever a user performs an update on the relation, the database system must ensure that the update does not violate any functional dependencies, that is, all the functional dependencies in F are satis ed in the new database state The system must roll back the update if it violates any functional dependencies in the set F We can reduce the effort spent in checking for violations by testing a simpli ed set of functional dependencies that has the same closure as the given set Any database that satis es the simpli ed set of functional dependencies will also satisfy the original set, and vice versa, since the two sets have the same closure However, the simpli ed set is easier to test We shall see how the simpli ed set can be constructed in a moment First, we need some de nitions An attribute of a functional dependency is said to be extraneous if we can remove it without changing the closure of the set of functional dependencies The formal de nition of extraneous attributes is as follows Consider a set F of functional dependencies and the functional dependency in F Attribute A is extraneous in if A , and F logically implies (F { }) {( A) } Attribute A is extraneous in if A , and the set of functional dependencies (F { }) { ( A)} logically implies F For example, suppose we have the functional dependencies AB C and A C in F Then, B is extraneous in AB C As another example, suppose we have the functional dependencies AB CD and A C in F Then C would be extraneous in the right-hand side of AB CD Beware of the direction of the implications when using the de nition of extraneous attributes: If you exchange the left-hand side with right-hand side, the implication will always hold That is, (F { }) {( A) } always logically implies F, and also F always logically implies (F { }) { ( A)}.

As I mentioned in 7, DVD titles and DVD menus are closely related You can, in fact, use the title editor to create menus manually if you are so inclined As a practical matter, though, I think that you will find that it really is a lot easier to use the specialized menu creation and editing tools to create your menus Before we proceed, I d like to point out that even though some digital video editing programs place arbitrary limits on the number of buttons you can have on each menu, aesthetic considerations will probably determine the actual number used That is, you ll want to keep your menus fairly simple in order to make them easier to use Packing too many buttons onto a single menu can make it difficult for someone using a set-top DVD player to navigate the menu A reasonable alternative is to create multiple linked menus (as you ll learn how to do a bit later in this section)

The McGraw Hill Companies, 2001

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