Part 1: Validating Functional Dependencies (FDs)

We will check if the following FDs hold in $F = \{ AB \to C, B \to D, CD \to E, CE \to GH, G \to A \}$:

1. $AB \to E$:
   To verify, we check if $AB$ can derive $E$.

   • From $AB \to C$, $AB$ determines $C$.
   • From $CD \to E$, we need $D$ and $C$ to determine $E$.
   • $D$ can be obtained using $B \to D$. Hence, $AB \to C$, $B \to D$, and $CD \to E$ together imply $AB \to E$.
     Conclusion: $AB \to E$ is correct.

2. $BG \to C$:
   To verify, we check if $BG$ can determine $C$.

   • $AB \to C$ tells us $AB$ determines $C$, but $BG$ does not give access to $AB$.
   • $G \to A$ means $G$ determines $A$, so $BG$ could be expanded to $ABG$. However, since $B$ and $G$ together don't directly lead to $AB$, we cannot derive $C$.
     Conclusion: $BG \to C$ is incorrect.

3. $AB \to G$:
   To verify, we check if $AB$ determines $G$.

   • From the given FDs, $AB \to C$, and $C$ is needed for $CE \to GH$.
   • However, $AB \to G$ is not directly derivable as $AB$ does not guarantee $E$, which is required to derive $G$.
     Conclusion: $AB \to G$ is incorrect.

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Part 2: Candidate Keys of $F$

To determine the candidate keys, we need to identify the minimal set(s) of attributes that can determine all attributes in $R(A, B, C, D, E, G, H)$.

Step 1: Determine the closure of attributes.

Start by finding what each attribute (or set of attributes) can determine using the given FDs:

• $AB$:
  $AB \to C$. From $C$, $CD \to E$. From $E$, $CE \to GH$. From $G$, $G \to A$. Thus, $AB \to \{A, B, C, D, E, G, H\}$.

• $B$:
  $B \to D$. No further attributes can be derived from $B$ alone.

• $G$:
  $G \to A$. With $A$, no further attributes can be derived from $G$.

Step 2: Identify candidate keys.

A candidate key must:

1. Determine all attributes in $R$.
2. Be minimal (no proper subset should be a candidate key).

• $AB$: From above, $AB^+ = \{A, B, C, D, E, G, H\}$. Thus, $AB$ is a candidate key.
• No proper subset of $AB$ (e.g., $A$, $B$) determines all attributes.

Step 3: Verify alternative combinations.

Testing other combinations (e.g., $BG$, $AG$, etc.), none of them cover all attributes without requiring $AB$.

Conclusion: The only candidate key is $AB$.

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Final Answers:

1. Are the FDs correct?

   • $AB \to E$: Correct.
   • $BG \to C$: Incorrect.
   • $AB \to G$: Incorrect.

2. Candidate Key(s):
   The only candidate key is $AB$.

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Do you want details or have questions? Here are some related questions:

1. How is the closure of attributes determined?
2. Can there be more than one candidate key in a relation?
3. What is the role of minimality in candidate key selection?
4. How do transitive dependencies affect FDs?
5. Why are candidate keys important in normalization?

Tip: Always compute attribute closures systematically to avoid missing dependencies when determining keys or validating FDs.