CS 4250: Database Management Systems

Fall 2024 - Homework 3

Due Friday, 8 November 2024, at midnight

This is an individual assignment. All work must be entirely your own. You should not look at any other student's work (in whole or in part, on paper or on screen), nor allow anyone else to look at yours, nor discuss the assignment with other students, during the course of this assignment. Nor may you submit the work of software programs as your own.

(Strongly prefer plain text. Will accept PDF or MSWord documents, if typed.) Upload to Canvas on or before midnight on 8 November 2024, or turn in a typed hardcopy during class time on the due date.

1. Consider a relation R with attributes ABCDEF. You are given the following dependencies B -> D, B -> A, BC -> E.

   1. Attribute closure of B?

   2. Attribute closure of D?

   3. Attribute closure of BC?

   4. Attribute closure of DC?

   5. Attribute closure of ABCDEF?

   6. Identify which, if any, of the above are candidate keys.

2. Consider a relation R with attributes ABCDE. You are given the following dependencies B -> E, B -> D, B -> C, B -> A, A -> B.

   1. List all (candidate) keys for R.

   2. Is R in 3NF? Why or why not?

   3. Is R in BCNF? Why or why not?

3. Consider a relation R with attributes ABCDE. You are given the following dependencies AD -> B, AD -> C, AD -> E, C -> A.

   1. List all (candidate) keys for R.

   2. Is R in 3NF? Why or why not?

   3. Is R in BCNF? Why or why not?

4. Consider a relation R with attributes ABCDE. You are given the following dependencies C -> A, D -> E, C -> B, C -> D, C -> E.

   1. List all (candidate) keys for R.

   2. Is R in 3NF? Why or why not?

   3. Is R in BCNF? Why or why not?

5. Suppose that we have the following three tuples in a legal instance of a relation schema S with attributes primary_key, A, B and C. (The primary_key field contains the primary key of the table. The 'attribute names' column is not part of the relation; it is only there to name rows and columns so your answers can clearly discuss the rows and columns.)
   

   attribute names	primary_key	A	B	C
   tuple 1	1	x	h	k
   tuple 2	2	x	p	m
   tuple 3	3	y	z	n

   
   
   1. For each of the following dependencies, can you infer whether or not it holds over schema S? Why or why not?
      1. A -> B

      2. B -> C

   2. Can you identify any functional dependencies that hold over fields A, B and / or C? If so, please name one. (N.B. primary_key is not on the list of A, B and / or C. FDs involving the primary_key field should not be in your answer.)

6. Suppose you are given a relation R(A, B, C, D). For each of the following sets of FDs, assume they are the only dependencies that hold for R. Do the following: a) identify the candidate key(s) for R. b) State whether or not the proposed decomposition of R into smaller relations is a good decomposition and briefly explain why or why not.
   1. A -> D, C -> B; decompose into AD and BC.
   1. Candidate keys:
   2. Analysis:
   2. D -> C, C -> B, B -> A; decompose into AB and ACD.
   1. Candidate keys:
   2. Analysis:
   3. C -> A, D -> ABC; decompose into AC and BCD.
   1. Candidate keys:
   2. Analysis:
   4. AB -> C, D -> B; decompose into ABC and BD.
   1. Candidate keys:
   2. Analysis:

7. Suppose you are given a relation R with attributes ABCDEF. You are given functional dependencies EF -> D, C -> AB, B -> A, and D -> B. Using the decomposition techniques described in class, decompose this relation into relations in "good" normal forms. For each of the new relations you create, identify the normal form it is in.

   If you choose not to decompose the relations into the best normal form we discussed, explain why you made that choice. If there are any interesting issues or potential problems with your decomposition, explain what they are.