EBK DATABASE SYSTEM CONCEPTS
EBK DATABASE SYSTEM CONCEPTS
7th Edition
ISBN: 9781260049268
Author: SILBERSCHATZ
Publisher: MCG COURSE
Expert Solution & Answer
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Chapter 7, Problem 35E

Explanation of Solution

Ensuring that the resulting decomposition is lossless:

  • Instead of using BCNF, it is possible to have a lossless decomposition into 3NF itself.
  • For this, it is required that one schema at least should have a candidate key for the schema that is being composed.
  • The set of functional dependencies F on a schema R is considered.
  • The decomposition of R which is dependency-preserving, sigma={R1,R2,...,Rn} is assumed.
  • Suppose X is a candidate key for R and a legal instance of R is r.
  • Suppose j=ProjectX(r) join ProjectR1(r) join ProjectR2(r) ... Join ProjectRn(r).
  • To prove that r=j, it is claimed that, t1 and t2 are two tuples, where t1=t2 such that t1[X]=t2[X].
  • The inductive argument is used for proving this claim.
  • Let F'=F1 union F2 union ... union Fn, where each Fi is the restriction of F to the schema Ri in sigma. Closure of X is computed under F'.
  • Induction is applied the number of times that the for loop in the algorithm executes.

Basis:

In the first step of the algorithm, result is assigned to X, and hence given that t1[X]=t2[X], it is known that t1[result]=t2[result] is true.

Induction Step:

  • Let t1[result]=t2[result] be true at the end of the kth execution of the for loop...

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