GATE CSE 2006 | Question: 70

Question

The following functional dependencies are given:

$ AB\rightarrow CD,AF\rightarrow D,DE\rightarrow F,$$C\rightarrow G,F\rightarrow E,G\rightarrow A $

Which one of the following options is false?

• A. $ \left \{ CF \right \}^{*}=\left \{ ACDEFG \right \}$
• B. $ \left \{ BG \right \}^{*}=\left \{ ABCDG \right \}$
• C. $ \left \{ AF \right \}^{*}=\left \{ ACDEFG \right \}$
• D. $ \left \{ AB \right \}^{*}=\left \{ ABCDG \right \}$

Comments

from option D $\left \{ AB \right \}^{*}=\left \{ ABCDG \right \}$ the attribute $F$ has been removed it used to be $\left \{ ABCDFG \right \}$

this was done to let option C be the distinct one from all others.

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Detailed Video Solution: https://youtu.be/l8OLJIw8Dq0?t=2517&feature=shared

Normalization in DBMS Complete Playlist: https://youtube.com/playlist?list=PLIPZ2_p3RNHhJjQGnZB2jORa0JxLdMp4X&feature=shared

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Option C is wrong

Answer 1

$\left \{ AF \right \}$*$ =\left \{ AFDE \right \}.$

Hence, option C is wrong.

Comments

yes. But what about {AB}*?

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{AB}* are also looks correct. it is {ABCDG}

F cannot be derived from AB.

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Yes. It might be a typo in the question. I removed F to make it correct.

Answer 2

All options correct except $C$ option.

(a) ${\text{CF}}^*= {CFGADE} = {ACDEFG}$

(b) ${\text{BG}}^ *= {BGACD} = {ABCDG}$

(d) ${\text{AB}}^ *= {ABCDG}$

but in $(C)$ option , ${\text{AF}}^*={AFDE}$
so, option $C$ is false.

Correct answer : $C$

Answer 3

It is given in the question that:

$1.$  $AB → CD$ 

$2.$  $AF → D$ 

$3.$  $DE → F$ 

$4.$  $C → G$

$5.$  $F → E$ 

$6.$  $G → A$

We will go through each option one by one.

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Option A: $\left \{C F\right \}^{*}$ = $\left \{ACDEFG\right \}$

$=>$ $\left \{C F\right \}^{*}$ = $\underbrace{\left \{CGA\right \}}_\textrm{Closure of C using 4 and 6} + \underbrace{\left \{FDE\right \}}_\textrm{Closure of F using 2 and 5}$

$=>$ $\left \{C F\right \}^{*}$ = $\left \{CGAFDE\right \}$

If we just rearrange the terms:

$=>$ $\left \{C F\right \}^{*}$ = $\left \{ACDEFG\right \}$

Hence $\color{green} True$

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Option B: $\left \{BG\right \}^{*}$ = $\left \{ABCDG\right \}$

$=>$ $\left \{BG\right \}^{*}$ = $\underbrace{\left \{GA\right \}}_\textrm{Closure of G using 6} + \underbrace{\left \{BCD\right \}}_\textrm{Closure of B using 1: Note that A from G’s closure is being used}$

$=>$ $\left \{BG\right \}^{*}$ = $\left \{GABCD\right \}$

If we just rearrange the terms:

$=>$ $\left \{BG\right \}^{*}$ = $\left \{ABCDG\right \}$

Hence $\color{green} True$

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Option C: $\left \{AF\right \}^{*}$ = $\left \{ACDEFG\right \}$

$=>$ $\left \{AF\right \}^{*}$ = $\underbrace{\left \{A\right \}}_\textrm{Closure of A} + \underbrace{\left \{FDE\right \}}_\textrm{Closure of F using 2, 3 and 5: Note that A from A’s closure is being used}$

$=>$ $\left \{AF\right \}^{*}$ = $\left \{AFDE\right \}$

If we just rearrange the terms:

$=>$ $\left \{AF\right \}^{*}$ = $\left \{ADEF\right \}$

But this is not equal to $\left \{AF\right \}^{*}$ = $\left \{ACDEFG\right \}$

Hence $\color{red} False$

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Option D: $\left \{AB\right \}^{*}$ = $\left \{ABCDG\right \}$

$=>$ $\left \{AB\right \}^{*}$ = $\underbrace{\left \{ABCD\right \}}_\textrm{Closure of AB using 1} + \underbrace{\left \{G\right \}}_\textrm{Closure of C using 4: Note that C from AB’s closure is being used}$

$=>$ $\left \{AB\right \}^{*}$ = $\left \{ABCDG\right \}$

Hence $\color{green} True$

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Hence only False option C (Answer).

Answer 4

Here Option C Is False 

 

Answer 5

In option C CG is not preserved while taking closure of AF 

option c is false