A relation R can decompose into many relations R1, R2, ... Rn to eliminate data redundancy and anomalies through Functional and Multivalued Dependencies Axioms Rules. Closure is a set of functional dependencies that can denote F and certain other functional dependencies that are logically implied by set F of functional dependencies that can denote F+.
The relation R and each decomposed relation R1, R2, ... Rn are hold a set of functional dependency F that are logically implied by set of F of functional dependencies and symbolize F(F+). Iff every instance r of the relation R that satisfies with set of functional dependency of R1, R2, ... Rn is called lossless join.
An instance, a relation R(A B C) is decomposed into relation R1(A B) and R2(B C) or R1(A B) and R2(A C). The functional dependency F is A ® B, attribute A is hold attribute B value that belongs to F+. The following theorem is helps to test lossless join decomposition status.
iff R1 ∩ R2 ® R1 - R2 or R1 ∩ R2 ® R2 - R1 that belongs to F+.
The following examples are determined about Lossy and Lossless Join Decomposition of relation R.