For which of the following set of functional dependencies does the relation R(A,B,C,D) has AB,CD as closed sets

Question

For which of the following set of functions dependencies does the relation $R(A, B, C, D)$ has $AB, CD$ as closed sets?

1. $A \rightarrow B, B \rightarrow A, C \rightarrow D$
2. $A \rightarrow B, B \rightarrow C, C \rightarrow D, D \rightarrow A$
3. $A \rightarrow B, B \rightarrow A, C \rightarrow A, D \rightarrow A$
4. $A \rightarrow B, B \rightarrow A, C \rightarrow D, D \rightarrow C$

Comments

closed here means that if we do the closure of (AB)+ than it will be only AB similalry for (CD)+=CD !! by the way wt the ans ??

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yes  but  I think (B) also closed

means (A),(B),(D) closed and (C) not closed

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answer is (B)

got it..

AB, CD are closed sets imply they are keys..

Answer 1

Answer is option (B)

AB, CD are closed sets imply, does AB, CD cover all the attributes of R (attribute closure)
 

In option (A), 

AB closure is : { A B }
CD closure is : { C D }

In option (B), 

AB closure is : { A B C D }
CD closure is : { A B C D }

 

In option (C), 

AB closure is : { A B }
CD closure is : { A B C D }

 

In option (D), 

AB closure is : { A B }
CD closure is : { C D }

 

Thus, answer is option (B)

Comments

So, they asking basically whether AB , CD are superkeys in which FD set.

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yes. only in option (B) they cover all attributes of R.

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From where u verified that (B) is correct ?

Answer 2

answer should be D)

set of attributes are closed under functional dependencies iff closure of attribute set is set itself.

(A,B)+ = {A,B}

(C,D)+ = {C,D}

and inside closed set , if  C ->D then D ->C should also need to satisfy.

Only D satisfies

example :to say  { a,b,c,d } is closed set.

{a,b,c,d}+ = {a,b,c,d}

let X , Y be any subsets of {a,b,c,d}

if X -> Y then Y -> X should present  (directly or indirectly which we can infer from given functional dependencies.

Comments

what about option (A) ??

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Why not A ?

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please check now , hope it will be clear.