Download Relational Database Theory: A Comprehensive Introduction by Paolo Atzeni;Carlo Batini;Valeria De Antonellis PDF

By Paolo Atzeni;Carlo Batini;Valeria De Antonellis

Written by way of across the world famous experts within the database box, this e-book provides an intensive dialogue of the rules of the relational version of database layout besides a scientific remedy of the formal conception for the version.

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1. 1 Only if. Assume there is an expression E mentioning C that produces r; that is, r = E(r). 5), we may assume that E involves only unions, renamings, differences, Cartesian products, projections, and selections with only atomic conditions (of the form A 1 = A 2 or A = c, with c E C). This part of the proof is completed by showing (by induction on the number of operators in E) that the result of E is invariant by every C-fixed automorphism of r. The basis (zero operators) is trivial: the expression is just a relation ri E r, which is invariant by every automorphism of r, since it is part of r.

3 The Expressive Power of Relational Algebra the attribute (or one of them, if there are more) such that t[Ck,] then, we claim that ra = PB, ... C. Ckd = Ci; (rw)) is the 0-fixed automorphism relation r0 . We prove this equality by showing containment in both directions. 1. ro g r,. Let to be a tuple in ro; by definition, it describes an automorphism h, such that h(ci) = to[Bc,] for every 1 < i < d. Since automorphisms transform tuples of r, into tuples of rl, and t is the combination of tuples of rl, it follows that the tuple h(t) obtained by applying h to t elementwise is also a combination of tuples of rl, and therefore belongs to rp, which contains all such combinations.

Let ta be a generic tuple in ra; we show that it belongs to ro. Let ti be the tuple in ra obtained from tuple t in rw; by construction, tl[Bci] = ci for every 1 < i < d. Also, let h be the function on Dr such that h(t 1 ) = ta; h is a bijection, since all the tuples in ra have the same pairwise equalities. Then, let tw be the tuple in rw from which ta originates: thus we have that tw = h(t), and so, since all tuples of rw are obtained as juxtapositions of tuples of rl, we have that h is an automorphism.

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