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Chapter 10: Problem 29

Consider a relation \(R(A, B, C, D, E)\) with the following dependencies: \\[ A B \rightarrow C, C D \rightarrow E, D E \rightarrow B \\] Is \(A B\) a candidate key of this relation? If not, is \(A B D ?\) Explain your answer.

Short Answer

Expert verified
'AB' is not a candidate key because it cannot identify every attribute in the relation. However, 'ABD' is a candidate key because it can uniquely identify every possible tuple in the relation.

Step by step solution

01

Test 'AB' as Candidate Key

First let's check whether 'AB' is a candidate key. Doing this involves determining the closure of AB, denoted as \(AB^+\). We start by adding the attributes that are part of the candidate key itself, so \(AB^+\) initially is AB. We then apply the functional dependencies. By applying the dependency \(AB \rightarrow C\), we get \(AB^+ = ABC\). There are no more dependencies that we can apply with the attributes in \(AB^+\), so we stop here.
02

Analyze Results

We look at the resulting \(AB^+\) and compare it to the original relation \(R(ABDE)\). Since AB does not uniquely identify D and E in every possible tuple of the relation, 'AB' is not a candidate key.
03

Test 'ABD' as Candidate Key

Now let's test whether 'ABD' is a candidate key. Using the same process, we start with \(ABD^+\ = ABD\). Applying the functional dependencies, the dependency \(CD \rightarrow E\) can be added because we now have \(D\) in \(ABD^+\). So, \(ABD^+ = ABDE\). We also apply the dependency \(AB \rightarrow C\), which doesn't add a new attribute as \(C\) is already in \(ABD^+\). No other dependencies can be added.
04

Analyze Results

We again compare the resulting \(ABD^+\) to the original relation \(R(ABCDE)\). This time, \(ABD^+\) contains all the attributes of \(R\), so 'ABD' can uniquely identify every possible tuple of the relation and therefore, 'ABD' is a candidate key.

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Most popular questions from this chapter

What does the term unnormalized relation refer to? How did the normal forms develop historically from first normal form up to Boyce-Codd normal form?

What is a minimal set of functional dependencies? Does every set of dependencies have a minimal equivalent set? Is it always unique?

Suppose that we have the following requirements for a university database that is used to keep track of students' transcripts: a. The university keeps track of each student's name (SNAME), student number (SNUM), social security number (SSN), current address (SCADDR) and phone \((\mathrm{SCPHONE}),\) permanent address \((\mathrm{SPADDR})\) and phone \((\mathrm{SPPHONE}),\) birth date \((\mathrm{BDATE})\) \(\operatorname{sex}(\operatorname{sex}), \text { class (cLass) (freshman, sophomore, } \ldots, \text { graduate }),\) major depart ment (MAJORCODE), minor department (MINORCODE) (if any), and degree program \(\left(p_{R O C}\right)(B, A,, B, S, \ldots, P H, D,) .\) Both sss \(N\) and student number have unique val. ues for each student. b. Each department is described by a name (DNAME), department code (DCOOE), office number (DOFFICE), office phone (DPHONE), and college (DCOLLECE). Both name and code have unique values for each department. c. Each course has a course name (cNAME), description (cDESC), course number (CNUM), number of semester hours (cREDIT), level (LEVEL), and offering depart. ment (coept). The course number is unique for each course. d. Each section has an instructor (INAME), semester (SEMESTER), year (YEAR), course (seccourse), and section number (secwum). The section number distinguishes different sections of the same course that are taught during the same semester/ year; its values are \(1,2,3, \ldots,\) up to the total number of sections taught during each semester. e. \(A\) grade record refers to a student \((\operatorname{ss} N),\) a particular section, and a grade \((\mathrm{CRADE})\) Design a relational database schema for this database application. First show all the functional dependencies that should hold among the attributes. Then design relation schemas for the database that are each in \(3 \mathrm{NF}\) or BCNF. Specify the key attributes of each relation. Note any unspecified requirements, and make appropriate assumptions to render the specification complete.

What is meant by the closure of a set of functional dependencies? Illustrate with an example.

Discuss attribute semantics as an informal measure of goodness for a relation schema.
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