EBK DATABASE SYSTEM CONCEPTS
EBK DATABASE SYSTEM CONCEPTS
7th Edition
ISBN: 9781260049268
Author: SILBERSCHATZ
Publisher: MCG COURSE
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Chapter 7, Problem 8PE

Explanation of Solution

Proof:

Consider the set of attributes be α .

F is considered to be the set of functional dependencies (FDs).

The set of attributes is considered to be functionally determined by α under F and is defined to be close of α under F(α+).

Using the Figure 7.19 and Figure 7.8:

The algorithm provided in the figure 7.19 will correctly compute the (α+) as shown below:

  • Computing what will happen after adding A to result.
  • Proving αA
  • Since αα is trivially which means α is considered to be a part of the result.
  • Now, after adding Aα to the resultant there should be existence of functional dependency that is βγ where Aγ and where βresult.
  • Or else the value of the “fdcount” will not become zero, then the related of condition of the algorithm will not change to be the value “false”.
  • When Aα+, then it will added to the result for sure.
  • Proving this by induction by utilizing the Armstrong’s axiom on the length of proof on αA are shown below:
  • The add- in procedure gets called with the argument say “c”, then all the attribute present in the “C” gets added to the result.
  • Similarly in addition to it when the functional dependency “fdcount” value is zero, then all the attributes of its tail is surely added to the resultant too.
  • Utilizing the induction to the base case that is when Aα then Aα+ is considered to be true because the very first call that is made to the add-in procedure is made with the argument α

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