The Gateway to Computer Science Excellence
First time here? Checkout the FAQ!
x
+28 votes
4.2k views

Consider the following relational schemes for a library database:

Book (Title, Author, Catalog_no, Publisher, Year, Price)
Collection(Title, Author, Catalog_no)

with the following functional dependencies:

  1. $\text{Title Author }\rightarrow\text{ Catalog_no}$

  2. $\text{Catalog_no }\rightarrow\text{ Title Author Publisher Year}$

  3. $\text{Publisher Title Year}\rightarrow\text{ Price}$

Assume $\left\{\text{ Author, Title }\right\}$ is the key for both schemes. Which of the following statements is $\text{true}$?

  1. Both Book and Collection are in $\text{BCNF}$

  2. Both Book and Collection are in $3NF$ only

  3. Book is in $2NF$ and Collection in $3NF$

  4. Both Book and Collection are in $2NF$ only

asked in Databases by Veteran (59.5k points)
edited by | 4.2k views
0

@ worst_engineer  {Title , Author}  is the key for both schemes .what did u take candidate keys are : Catalog_no , {Title , Author}

0
@bikram sir see this pls
0
Ans is A, Both are in BCNF.
0

one point worth noting is that even though the key is mentioned but that simply refers to the primay key which is one of the key among candidate key ,so you have to find all  the candidate key first to determine the normal forms of the schema.

for collection schema:
fd's are:

catalog_no->title author

title author->catalog_no

also catalog_no and (title author) are candidate key for this schema.

so this is in bcnf.

for book schema:

fd's are:

  1. Title Author → Catalog_no

  2. Catalog_no → Title Author Publisher Year

  3. Publisher Title Year→ Price 

the candidate key's are catalog_no and (titlle,author) 

clearly 3rd is a transitive dependency hence 2nf and not 3nf or above.

4 Answers

+26 votes
Best answer

Answer: $C$

It is given that {Author , Title} is the key for both schemas.

The given dependencies are : 

  • {Title, Author} --> {Catalog_no}
  • Catalog_no --> {Title , Author , Publisher , Year }
  • {Publisher , Title , Year} --> {Price}

First , let's take schema Collection ( Title , Author , Catalog_no ) :

{Title , Author} --> Catalog_no

{Title, Author} is a candidate key and hence super key also and by definition of $\text{BCNF}$ this is in $\text{BCNF}$.

Now , let's see Book (Title , Author , Catalog_no , Publisher , Year , Price ) :

 {Title , Author}+ --> {Title , Author , Catalog_no , Publisher , Year , Price}

{Catalog_no}+--> {Title , Author , Publisher , Year , Price , Catalog_no}

So candidate keys are : Catalog_no , {Title , Author} 

But in the given dependencies , {Publisher , Title , Year} --> Price , which has Transitive Dependency. So , Book  is in $2NF$.

answered by Active (4.1k points)
edited by
0
Hello Sir, why it is in 2nf , please explain

Thank you
0
@Shekhar There is no partial dependency.
0
But in statement III ,FD III

a prime attribute Title is defining Price (non attribute) isn't this a partial dependency ?
+2

A functional dependency $\alpha \rightarrow \beta$ is called a partial dependency if there is a proper subset $\gamma$ of $\alpha $ such that $\gamma \rightarrow \beta $. We say that $\beta$ is partially dependent on $\alpha $.

- from book by Korth

0
Is this definition correct?
0
Yes. Of course. It's from a standard book.
0

Assume { Author, Title } is the key for both schemes

then candidate key is given. why are we adding Catalog_no as candidate key? 

is it necessary to find CKs even after it is specified in question?

0
Only collection is in BCNF too
+1

Book is in 2NF because:-

It is {Publisher Title Year } $\rightarrow$ Price

But not Title $\rightarrow$ Price

If the following Dependency

Title $\rightarrow$ Price

Holds Then Book will surely NOT be in 2NF, But it is present here.

0

@arjun Sir can you please clear my doubt ?

as mentioned in your comment 

Catalog_no --> {Title , Author , Publisher , Year } which could be split to

  • Catalog_no --> Title
  • Catalog_no --> Author
  • Catalog_no --> Publisher
  • Catalog_no --> Year

and according to wiki def for BCNF 

If a relational schema is in BCNF then all redundancy based on functional dependency has been removed, although other types of redundancy may still exist. A relational schema R is in Boyce–Codd normal form if and only if for every one of its dependencies X → Y, at least one of the following conditions hold:[2]

  • X → Y is a trivial functional dependency (Y ⊆ X)
  • X is a superkey for schema R

clearly Catalog_no is not SK for 'collection' relation hence it is not in BCNF but in 3NF 

and that's what answer C says but answer by  worst_engineer amd taught by my coaching instructor says it is in BCNF

+9 votes
(c)

in collection all the non prime attributes depend directly on candidate key - so BCNF and hence 3NF (actually collection has only prime attributes so it should, by default be at least in 3NF)

in book the non prime attr (price), depends indirectly on the candidate key (catalog_no) thus forming transitive dependency and hence not in 3NF. There is no partial dependency so - 2nf
answered by Active (3.3k points)
+3
why collection not in BCNF? Catalog_no is a key for Book rt?
+1
Yeah, Collection should be BCNF, And I think there answers are twisted, just to trick us
+2
Collection is in BCNF because Title, Author is a super key.
0
book is not in 2NF also...because in last FD... price is not fully functionally dependent on (title,author)....pls explain..
+4
Price is fully dependent on (title, author). Neither title nor author alone can determine price. So, no partial FD and hence 2NF condition satisfied.
0
How come price is fully functionally dependent on (title, author) as it is dependent on (publisher, title, year)?
+1
Note: Prime attribute Non prime attribute -> Non prime is in 2NF
0
ok therefore we are considering

{Publisher , Title , Year} --> {Price} in 2NF

since Title is prime Attribute and {publisher, year} is non prime attribute and {price} is also a non prime attribute right?

and also plz tell what is the NF of following dependency:-

{Prime, Non Prime }-> {Prime} ?
0
In you question Prime Non Prime -> prime

RHS becomes Prime attributes which is suffiecint for this FD to be in 3NF.
0
How come this
(prime,non-prime)-->(non-prime)
be in 2NF?
Since defn. says "every non-prime attribute A in relation R must be fully functionally dependent on the primary key of R"

but above says that only having some prime attributes and not the primary key could result in 2NF
0
I think

(prime, non prime) -> non prime

will be consider as

(non prime) -> non prime.

hence it is considered as 2nf.

ryt?
0

To be in 2NF "every non-prime attribute should fully functionally dependent on the primary key."

So how

non-prime --> non-prime    is in 2NF ??

0
Primary key -> any thing(prime, non prime, key) will be in BCNF.
0
sir collection has some other functional dependencies which u can derive if u take closure of title or author they give only title->title and author ->author

hence the rhs is not a super key but lhs is is a prime attribute hence the collection is only in 3NF but not in BCNF
0

@Arjun sir Cant we think like this that since key has only one attribute. so no part of key, no partial dependency . So book is in 2NF. We can easily see transitive dependency , so it is not 3nf.

+4 votes
Ans. C

In Relation Book there is a transitive dependency and hence not in 3NF. There is no partial dependency so in 2NF

In Relation Collection all the non prime attributes depend directly on candidate key. In fact, collection has only prime attributes. So BCNF and hence 3NF.
answered by Active (1.1k points)
0

Raushank2

catalog_no->Title Author Publisher year 

According to 3nf if X->Y

then 1) X is a super key or 2) y is a prime attribute 

in case of title ,it is prime attribute so by this it must be in 3 nf ?

https://gateoverflow.in/1260/gate2007-62-ugcnet-june2014-ii-47 see this

0 votes
one point worth noting is that even though the key is mentioned but that simply refers to the primay key which is one of the key among candidate key ,so you have to find all  the candidate key first to determine the normal forms of the schema.

for collection schema:
fd's are:

catalog_no->title author

title author->catalog_no

also catalog_no and (title author) are candidate key for this schema.

so this is in bcnf.

for book schema:

fd's are:

Title Author → Catalog_no

Catalog_no → Title Author Publisher Year

Publisher Title Year→ Price 

the candidate key's are catalog_no and (titlle,author) 

clearly 3rd is a transitive dependency hence 2nf and not 3nf or above.
answered by Junior (637 points)


Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true

36,992 questions
44,564 answers
126,750 comments
43,627 users