GATE Overflow Test Series | Databases | Test 1 | Question: 24

Question

Which of the following is a minimal cover of the set of functional dependencies $\{A \rightarrow BC,
B \rightarrow CE,
A \rightarrow E,
AC \rightarrow H,
D \rightarrow B \}?$

• A. $\{A \rightarrow B,B \rightarrow C, B \rightarrow E, AC \rightarrow H, D \rightarrow B \}$
• B. $\{A \rightarrow B, B \rightarrow E, AC \rightarrow H, D \rightarrow B \}$
• C. $\{A \rightarrow BH,B \rightarrow CE, D \rightarrow B \}$
• D. $\{A \rightarrow B,B \rightarrow C, B \rightarrow E, A \rightarrow H, D \rightarrow B \}$

Answer 1

A set of functional dependencies $X$ is minimal if it satisfies the following conditions: 

1. Every dependency in $X$ has a single attribute on its right-hand side.
2. We cannot replace any dependency $A \to B$ in $X$ with dependency $C \to B,$ where $C$ is a proper subset of $A,$ and still have a set of dependencies that is equivalent to $X.$
3. We cannot remove any dependency from $X$ and still have a set of dependencies that is equivalent to $X.$

Option C is violating condition $1$ and hence cannot be a minimal cover. (But condition 1 is only in Navathe and not in books like Korth. So, marks are given for both)

In option A, we have $AC \to H$ but since we have $A \to C,$ this implies $A \to H$ and thus violating condition $2.$ So, option A also is not a minimal cover.

In option B, condition $3$ is violated as $A\to C$ cannot be inferred.

Option D is a minimal cover.

Option C,D also gets mark. 

Comments

no sir, still showing the same answer, I refreshed multiple times

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please update the result it’s showing c,d,d
please update @gatecse

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C and D are correct answers but I have not been awarded any marks...plz check and update it

Answer 2

Caption

Option D is the minimal cover.