11,678 views
30 votes
30 votes

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?

  1. $ \left \{ CF \right \}^{*}=\left \{ ACDEFG \right \}$
  2. $ \left \{ BG \right \}^{*}=\left \{ ABCDG \right \}$
  3. $ \left \{ AF \right \}^{*}=\left \{ ACDEFG \right \}$
  4. $ \left \{ AB \right \}^{*}=\left \{ ABCDG \right \}$

3 Answers

Best answer
36 votes
36 votes

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

Hence, option C is wrong.

12 votes
12 votes
All options correct except c option.
(a) (CF)*= {CFGADE} = {ACDEFG}

(b) (BG)*= {BGACD} = {ABCDG}

(d) (AB)*= {ABCDG}

but in (c) option, (AF)*={AFDE}
so option c is false.
0 votes
0 votes

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.


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 $\color{green} True$


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 $\color{green} True$


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 $\color{red} False$


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 $\color{green} True$


Hence only False option C (Answer).

Answer:

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