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Inclusion Dependencies
Let X = (A1, ..., An) and Y = (B1, ... , Bn) be sequences of attributes of R. The inclusion dependency X Yholds in R if and only if in each legal instance r for R:
- t1 r, t2r, i in 1..n: t1[Ai] = t2[Bi]
A simple inclusion dependency A B met A, B R holds in R if and only if in each legal instance r for R:
- t1r, t2r: t1[A] = t2[B]
Give inference rules for ids and sids and prove that they are correct.
Notes:
The inclusion dependencies are something special because the order of the attributes plays an important role. In simple inclusion dependencies we do not yet see this.
For sid's the rules are reflexivity and transitivity.
For id's reflexivity and transitivity also hold, but then there is also the projection and permutation rule (which can be given as one rule or as two rules).
An attribute with the same index may be omitted on both sides, and the order of the attributes may be changed in the same way on the left and right sides.
We can only prove the correctness of the rules today as we are going to see the proofs of completeness later.