# Multivalued Dependency

**DOI:**https://doi.org/10.1007/978-1-4899-7993-3_1248-2

## Synonyms

## Definition

A *multivalued dependency (MVD)* over a relation schema *R*[*U*], is an expression of the form *X* ↠ *Y*, where *X*,*Y* ⊆ *U*. An instance *I* of *R*[*U*] satisfies *X* ↠ *Y*, denoted by *I*⊧*X* ↠ *Y*, if for every two tuples *t*_{1},*t*_{2} in *I* such that *t*_{1}[*X*] = *t*_{2}[*X*], there is another tuple *t*_{3} in *I* such that *t*_{3}[*X*] = *t*_{1}[*X*] = *t*_{2}[*X*], *t*_{3}[*Y* ] = *t*_{1}[*Y*], and *t*_{3}[*Z*] = *t*_{2}[*Z*], where *Z* = *U* − *XY* (*XY* represents *X* ∪ *Y*). In other words, for every value of *X*, the value of attributes in *Y* is independent of the value of attributes in *Z*. A multivalued dependency*X* ↠ *Y* is a special case of a *join dependency* expressed as ⋈[*XY*,*X*(*U* − *XY*)], which specifies that the decomposition of any instance *I* satisfying ⋈ [*XY*,*X*(*U* − *XY*)] into π _{XY} (*I*) and π _{X(U−XY)}(*I*) is lossless, i.e., *I* = π _{XY} (*I*) ⋈ π _{X(U−XY)}(*I*).

## Key Points

Multivalued dependencies, like functional dependencies, can cause redundancy in relational databases. For instance, in the following table, each director of the movie *The Matrix*is recorded once per actor of the...

## Recommended Reading

- 1.Abiteboul S, Hull R, Vianu V. Foundations of databases. Reading: Addison-Wesley; 1995.Google Scholar