Attributes

The Transitive Rule

The transitive rule lets us cascade two FD's. ● If A1A2 . . . An → B1B2 . . . Bm and B1B2 . . . Bm → C1C2 . . . Ck hold in relation R, then A1A2 . . . An → C1C2 . . . Ck also holds in R.

Closing Sets of Functional Dependencies

As we have seen, given a set of FD's, we can sometimes infer some other FD's, including both trivial and nontrivial FD's. We shall, in later sections, want to differentiate between given FD's that are stated initially for a relation and derived FD's that are inferred using one of the

Projecting Functional Dependencies

When we learn design of relation schemas, we shall also have need to answer the following question about FD's. Assume we have a relation R with some FD's F, and we "project" R by removing certain attributes from the schema. Consider S is the relation that results from R if we

Boyce-Codd Normal Form

The objective of decomposition is to replace a relation by several that do not exhibit anomalies. There is, it turns out, a simple condition under which the anomalies discussed above can be guaranteed not to exist. This condition is called Boyce-Codd

Third Normal Form

Sometimes, one encounters a relation schema and its FD's that are not in BCNF but that one doesn't want to decompose further. The following example is typical.

Multivalued Dependencies

A "multivalued dependency" is an assertion that two attributes or sets of attributes are independent of one another. This condition is, as we shall see, an overview of the notion of a functional dependency, in the sense that every FD implies a corresponding multivalued dependency.

Definition of Multivalued Dependencies

A multivalued dependency (often abbreviated MVD) is a statement about some relation R that when you fix the values for one set of attributes, then the values in certain other attributes are independent of the values of all the other attributes in the relation. More specifically, we say the MVD

Reasoning About Multivalued Dependencies

There are a many rules about MVD's that are similar to the rules we learned for FD's in "Rules About Functional Dependencies". For instance, MVD's obey

Decomposition into Fourth Normal Form

The 4NF decomposition algorithm is quite similar to the BCNF decomposition algorithm. We find a 4NF violation, say A1A2 . . . An → B1B2 . . . Bm , where {A1, A2, . . . , An} is not a superkey. Note this MVD could be a true MVD, or it could be derived from the equivalent FD A1A2 . . . An

The Type System

An object-oriented programming language offers the user a rich collection of types. Starting with atomic types, such as integers, real numbers, booleans, and character strings, one may construct new types by using type constructors. Usually, the type constructors let us

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