GATE DS&AI 2024 | Question: 36

Question

​​​​​​Given the relational schema $R=(U, V, W, X, Y, Z)$ and the set of functional dependencies:

\[
\{U \rightarrow V, U \rightarrow W, W X \rightarrow Y, W X \rightarrow Z, V \rightarrow X\}
\]

Which of the following functional dependencies can be derived from the above set?

• A. $V W \rightarrow Y Z$
• B. $W X \rightarrow Y Z$
• C. $V W \rightarrow U$
• D. $V W \rightarrow Y$

Comments

Detailed Video Solution: https://youtu.be/zBYzl4wvJ9g

Answer 1

Given a set of functional dependencies derived from an attribute iff with the help of its closure, we can determine it.

$(VW)^+=VWXYZ$ we can determine $V,W,X,Y,Z$

$(WX)^+=WXYZ$ we can determine $Y,Z$

but
$(VW)^+$ can not determine $U$

Option $(A,B,D) $ true.

Answer 2

Given FDs

U → V U → W V → X WX → Y WX → Z 

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Option-wise verification

Option 1: VW → YZ ✅ TRUE
 

From V → X, get VW → X (augmentation).
 

With W and X, use WX → Y and WX → Z.
 

So (VW)⁺ = {V, W, X, Y, Z} ⇒ VW → YZ.

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Option 2: WX → YZ ✅ TRUE
 

Given WX → Y and WX → Z.
 

By union, WX → YZ.

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Option 3: VW → U ❌ FALSE
 

(VW)⁺ = {V, W, X, Y, Z} does not contain U.
 

So VW → U does not follow.

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Option 4: VW → Y ✅ TRUE
 

From Option 1, since VW → YZ, by decomposition ⇒ VW → Y.

Answer 3

Options A, B and D.