Chapter 7, Problem 14PE

Explanation of Solution

• Let F be a set of functional dependencies that hold on a schema R.
• Let σ= {R1, R2, . . . , Rn} be a dependency-preserving 3NF decomposition of R. Let X be a candidate key for R.
• Consider a legal instance r of R. Let j = Π_X(r) ⋈ΠR1 (r) ⋈ΠR2 (r)    ⋈ΠRn (r). We want to prove that r = j.
• The user can claim that if t1 and t2 are two tuples in j such that t1[X] = t2[X], then t1 = t2. To prove this claim, we use the following inductive argument-
• Let F ′ = F1 ∪ F2 ∪ . . . ∪ Fn, where each Fi is the restriction of F to the schema Ri in s.
• Consider the use of the algorithm given in Figure 8.8 to compute the closure of X under F ′ . We use induction on the number of times that the f or loop in this algorithm is executed.
• Basis : In the first step of the algorithm, result is assigned to X, and hence given that t1[X] = t2[X], we know that t1[result] = t2[result] is true.
• Induction Step : Let t1[result] = t2[result] be true at the end of the k th execution of the f or loop...