Lossless-Join

No loss of information in comparison to the original schema before decomposition

Example

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-- Natural Join
select distinct Student, Assignment, Group, x.Mark as Mark
from SGM x, AM y
where x.Mark = y.Mark

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Spurious Tuples: Computes extra tuples

Decomposition { $R_{1}, R_{2}$ } of $R$ is a lossless-join if for any instance of $R$ the natural join of
$R_{1}$ and $R_{2}$ has no spurious tuples

Theorem: Decomposition ${R_{1}, R_{2}}$ of $R$ is a lossless-join decomposition if and only if the common Attributes of $R_{1}$ and $R_{2}$ form a superkey for either schema

Logical Consequence of the integrity constraints

$R_{1} \cap R_{2} \to R_{1}$
$R_{1} \cap R_{2} \to R_{2}$