Normalization Using Join Dependencies Join dependencies constrain the set of legal relations over a schema R to those relations for which a given decomposition is a lossless-join decomposition. Let R be a relation schema and R1,R2,...,R be a decomposition of R.If R=RUR2....R,we say that a relation r(R)satisfies the join dependency*(R1,R2,...,R)if: r=ΠR1()凶ΠR2()凶.凶ΠRn() A join dependency is trivial if one of the R,is R itself. A join dependency *(R1,R2)is equivalent to the multivalued dependency R1∩R2>R2.Conversely,o-→>B is equivalent to*(aU(R-B),oUβ) However,there are join dependencies that are not equivalent to any multivalued dependency. Database System Concepts-7th Edition 28.8 ©Silberscha乜，Korth and SudarshanDatabase System Concepts - 7 28.8 ©Silberschatz, Korth and Sudarshan th Edition Normalization Using Join Dependencies ▪ Join dependencies constrain the set of legal relations over a schema R to those relations for which a given decomposition is a lossless-join decomposition. ▪ Let R be a relation schema and R1 , R2 ,..., Rn be a decomposition of R. If R = R1  R2  ….  Rn , we say that a relation r(R) satisfies the join dependency *(R1 , R2 ,..., Rn ) if: r =R1 (r) ⋈ R2 (r) ⋈ …… ⋈ Rn(r) A join dependency is trivial if one of the Ri is R itself. ▪ A join dependency *(R1 , R2 ) is equivalent to the multivalued dependency R1  R2 R2 . Conversely,   is equivalent to *( (R - ),   ) ▪ However, there are join dependencies that are not equivalent to any multivalued dependency