GATE IT 2007 | Question: 67

Question

Consider the following implications relating to functional and multivalued dependencies given below, which may or may not be correct.

1. if $A \rightarrow \rightarrow B$ and $A \rightarrow \rightarrow C$ then $A \rightarrow  BC$
2. if $A \rightarrow B$ and $A \rightarrow  C$ then $A \rightarrow \rightarrow BC$
3. if $A \rightarrow \rightarrow BC$ and $A \rightarrow  B$ then $A \rightarrow C$
4. if $A \rightarrow BC$ and $A \rightarrow  B$ then $A \rightarrow \rightarrow C$

Exactly how many of the above implications are valid?

• A. $0$
• B. $1$
• C. $2$
• D. $3$

Comments

only B is true.

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Please refer this video for better understanding of Multivalued dependencies.

Link

Now according to multivalued dependency definition.

SSN -> -> CollegeName means

If we have the below tuples

SSN	CollegeName	Hobby
100	Global Academy Of Technology	Music
100	SJBIT	Swimming

then we also need to have the below two tuples.

SSN	CollegeName	Hobby
100	Global Academy Of Technology	Swimming
100	SJBIT	Music

 

Counter example of Option (i)If A→→ B and A → → C then A → BC

A	B	C
100	X1	P1
100	X2	P2
100	X1	P2
100	X2	P1
200	Y	Q
300	Z	R

From this table we can see that A $\rightarrow \rightarrow$ B

and A $\rightarrow \rightarrow$ C.

But here A -> B and also A -> C do not hold true.

Counter example of Option (iii)If A → → BC and A → B then A → C

A	B	C	D(Rest)
100	X1	P1	D1
100	X1	P1	D2
200	Y1	Q1	D3
200	Y1	Q2	D4

 For the same value of A,B,C the rest of the attributes(D here) can be interchanged.

Here A -> -> BC and A -> B hold true but A -> C is not true.

Please correct me if I'm wrong here.

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8 statements are given in question while there should be only 4.. Which 4 statements were there in original question

Answer 1

a. If A → → B and A  → →C then A → BC . So FALSE
b. If A → B and A → C then A→ BC.   So   A → →BC    TRUE..
c. If A → → BC and A → B  here B is Subset of AB and (A intersection BC) is phi so
 A → B but not A → C so FALSE  (Coalescence rule )
d. If A → BC  then A → C   so  A → → C    TRUE
 if A → B then A → → B  holds but reverse not true.

Correct Answer: $C$

Comments

I think GATE asked very rare questions from multi-valued dependencies (4nf). So its not very important.

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@Lakshman Patel RJIT so if we ignore it completely it will be ok ?? i have seem questions from this topic in some test series that’s why i am worried.

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As we don't know what GATE can ask. It is rarely asked & most probably won't be asked. So it’s good to know the things.

Answer 2

Check answer to following question ->

http://dba.stackexchange.com/questions/123510/how-can-i-prove-disprove-if-a-%E2%86%A0-bc-and-a-%E2%86%92-b-then-a-%E2%86%92-c

Answer 3

Every FD is a MVD.

i.e suppose $x\rightarrow y$  $\Rightarrow$ $x\rightarrow \rightarrow y$

If y can be determined by x on y's single value then we can easily say x multi-determines y. as single value $\subseteq$ multiple value.

Now,

1. can't even possible.

2.  A -> BC which implies A -> -> BC. (true)

3. using given data we can't prove the then part.

4. given FDs are A->B & A->C, so using this we can say A->->C.(true)

here 2 implications are valid. i.e option C