and , then The above rules are sound and complete. A decomposition of R into (X, Y) and (X, R − Y) is a lossless-join decomposition if and only if X Y holds in R. Every FD is an MVD because if X Y, then swapping Y's between tuples that agree on X doesn't create new tuples. Splitting Doesn't Hold. Like FD's, we cannot generally split the left side of an MVD.But unlike FD's, we cannot split the right side either, sometimes you have to leave several attributes on the right side. Closure of a set of MVDs is the set of all MVDs that can be inferred using the following rules (Armstrong's axioms): Complementation: If X Y, then X R - Y Augmentation: If X Y and Z W, then XW YZ Transitivity: If X Y and Y Z, then X Z - Y Replication: If X Y, then X Y Coalescence: If X Y and W s.t. W Y = , W Z, and Z Y, then X Z Definitions full constraint A constraint which expresses something about all attributes in a database. (In contrast to an embedded constraint.) That a multivalued dependency is a full constraint follows from its definition, as where it says something about the attributes . tuple-generating dependency A dependency which explicitly requires certain tuples to be present in the relation. trivial multivalued dependency 1 A multivalued dependency which involves all the attributes of a relation i.e.. A trivial multivalued dependency implies, for tuples and , tuples and which are equal to and . trivial multivalued dependency 2 A multivalued dependency for which . References External links Multivalued dependencies and a new Normal form for Relational Databases (PDF) - Ronald Fagin, IBM Research Lab On the Structure of Armstrong Relations for Functional Dependencies (PDF) - CATRIEL BEERI (The Hebrew University), MARTIN DOWD (Rutgers University), RONALD FAGIN (IBM Research Laboratory) AND RICHARD STATMAN (Rutgers University) On a problem of Fagin concerning multivalued dependencies in relational databases (PDF) - Sven Hartmann, Massey University Data modeling Database constraints