Multivalued dependency

In database theory, a multivalued dependency is a full constraint between two sets of attributes in a relation.

In contrast to the functional dependency, the multivalued dependency requires that certain tuples be present in a relation. Therefore, a multivalued dependency is a special case of tuple-generating dependency. The multivalued dependency plays a role in the 4NF database normalization.

A multivalued dependency is a special case of a join dependency, with only two sets of values involved, i.e. it is a binary join dependency.

A multivalued dependency exists when there are at least 3 attributes (like X,Y and Z) in a relation and for value of X there is a well defined set of values of Y and a well defined set of values of Z. However, the set of values of Y is independent of set Z and vice versa.

The formal definition is given as follows.

Let ${\displaystyle R}$ be a relational schema and let ${\displaystyle \alpha \subseteq R}$ and ${\displaystyle \beta \subseteq R}$ (subsets). The multivalued dependency
${\displaystyle \alpha \twoheadrightarrow \beta }$
(which can be read as ${\displaystyle \alpha }$ multidetermines ${\displaystyle \beta }$) holds on ${\displaystyle R}$ if, in any legal relation ${\displaystyle r(R)}$, for all pairs of tuples ${\displaystyle t_{1}}$ and ${\displaystyle t_{2}}$ in ${\displaystyle r}$ such that ${\displaystyle t_{1}[\alpha ]=t_{2}[\alpha ]}$, there exist tuples ${\displaystyle t_{3}}$ and ${\displaystyle t_{4}}$ in ${\displaystyle r}$ such that
${\displaystyle t_{1}[\alpha ]=t_{2}[\alpha ]=t_{3}[\alpha ]=t_{4}[\alpha ]}$
${\displaystyle t_{3}[\beta ]=t_{1}[\beta ]}$
${\displaystyle t_{3}[R-\beta ]=t_{2}[R-\beta ]}$
${\displaystyle t_{4}[\beta ]=t_{2}[\beta ]}$
${\displaystyle t_{4}[R-\beta ]=t_{1}[R-\beta ]}$

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