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Consider the following relation instance:

A B C
1 2 3
2 3 5
1 4 3
2 3 6

 $$\begin{array}{|c|c|c|} \hline  A & B &C \\ \hline 1 & 2 & 3 \\ \hline  2 & 3 & 5  \\ \hline  1 & 4 & 3 \\ \hline 2 & 3 & 6  \\ \hline  \end{array}$$

Which of the following Functional Dependencies are satisfied by the above relation instance ?

  1.  $A\rightarrow B$
  2. $A\rightarrow C$
  3. $C\rightarrow A$
  4. $AB\rightarrow C$
  5. $BC\rightarrow A$
  6. $AC\rightarrow B$
  1.     II, III & V only
  2.     I, IV & VI only
  3.     III & V only
  4.     V only
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1 Answer

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 $$\begin{array}{|c|c|c|} \hline  A & B &C \\ \hline 1 & 2 & 3 \\ \hline  2 & 3 & 5  \\ \hline  1 & 4 & 3 \\ \hline 2 & 3 & 6  \\ \hline  \end{array}$$

For the given relation instance 

  1.  $A\rightarrow B\implies 1\rightarrow2,1\rightarrow 4.$ For given relation instance,it is not possible.
  2. $A\rightarrow C\implies 2\rightarrow 5,2\rightarrow 6.$ For given relation instance,it is not possible.
  3. $C\rightarrow A\implies 3\rightarrow 1,5\rightarrow 2,6\rightarrow 2.$ For given relation instance it is possible.
  4. $AB\rightarrow C\implies 23\rightarrow 5,23\rightarrow 6.$ For given relation instance,it is not possible.
  5. $BC\rightarrow A\implies 23\rightarrow 1,35\rightarrow 2,43\rightarrow 1,36\rightarrow 2.$ For given relation instance it is possible.
  6. $AC\rightarrow B\implies 13\rightarrow 2,13\rightarrow 4.$ For given relation instance,it is not possible.

$(III)$ and $(V)$ are true.

So, the correct answer is $(C)$. 

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