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.

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

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

The redundancy that we found in "Multivalued Dependencies" to be caused by MVD's can be removed if we use these dependencies in a new decomposition algorithm for relations. In this section we shall introduce a new normal form, called "fourth normal form". In this 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

As we have pointed out, 4NF implies BCNF, which in turn implies 3NF. Therefore, the sets of relation schemas (including dependencies) satisfying the three normal forms are related as in the following figure. That is, if a relation with certain dependencies is in 4NF, it is also in