Problem Breakdown:

1. Relation: $R(A, B, C, D, E, G, H)$
2. Functional Dependencies (FDs): $F = \{ AB \to C, B \to D, CD \to E, CE \to GH, G \to A \}$

We need to:

1. Determine if $AB \to E$, $BG \to C$, and $AB \to G$ hold, providing justification.
2. Find the candidate keys of the relation.

---

Step 1: Are the FDs $AB \to E$, $BG \to C$, and $AB \to G$ correct?

Testing $AB \to E$:

• $AB \to C$ is given.
• Using $CD \to E$, we can see that if we know $C$ and $D$, we can derive $E$.
• To derive $D$, we note that $B \to D$ is given.
• Thus, knowing $AB$, we can derive:
  • $AB \xrightarrow{AB \to C} C$,
  • $B \xrightarrow{B \to D} D$,
  • $CD \xrightarrow{CD \to E} E$.
• Therefore, $AB \to E$ holds.

Testing $BG \to C$:

• $G \to A$ is given.
• $AB \to C$ is given.
• Knowing $BG$, we can derive:
  • $G \xrightarrow{G \to A} A$,
  • $AB \xrightarrow{AB \to C} C$.
• Thus, $BG \to C$ holds.

Testing $AB \to G$:

• To check $AB \to G$, we need to determine if $AB$ can uniquely determine $G$.
• From the given FDs, there is no direct or indirect derivation of $G$ from $AB$. Therefore, $AB \to G$ does not hold.

---

Step 2: Finding the Candidate Keys

Definition:

A candidate key is a minimal set of attributes that can uniquely determine all attributes in the relation.

Steps:

1. Identify the closure of attribute sets to find which can determine all attributes ($\{A, B, C, D, E, G, H\}$).
2. Use the given FDs.

Analysis:

• $AB \to C$: Knowing $AB$ determines $C$.
• $B \to D$: Knowing $B$ determines $D$.
• $CD \to E$: Knowing $C$ and $D$ determines $E$.
• $CE \to GH$: Knowing $C$ and $E$ determines $G$ and $H$.
• $G \to A$: Knowing $G$ determines $A$.

Let us calculate the closures of possible combinations:

1. $AB^+$:

   • $AB \to C$,
   • $B \to D$,
   • $CD \to E$,
   • $CE \to GH$,
   • $G \to A$.
   • $AB^+ = \{A, B, C, D, E, G, H\}$.

   Thus, $AB$ is a candidate key.

2. Check smaller subsets of $AB$: $A$ or $B$:

   • $A^+ = \{A\}$: Cannot determine all attributes.
   • $B^+$: Only determines $\{B, D\}$. Not all attributes.

3. Check other combinations:

   • $BG^+$:
     • $G \to A$,
     • $AB \to C$ (from $BG$),
     • Follows similar derivations as $AB^+$.
     • $BG^+ = \{A, B, C, D, E, G, H\}$.

   Thus, $BG$ is another candidate key.

Candidate Keys:

The candidate keys are $\{AB, BG\}$.

---

Final Answers:

1. Are the FDs correct?

   • $AB \to E$: Yes (proved through derivation).
   • $BG \to C$: Yes (proved through derivation).
   • $AB \to G$: No (cannot derive $G$ from $AB$).

2. Candidate Keys:

   • $\{AB, BG\}$.

---

Would you like further details or have questions?

Related Questions:

1. Can you explain how to find closures systematically?
2. What is the difference between a candidate key and a superkey?
3. How do FDs impact database normalization?
4. Can you illustrate $AB^+$ and $BG^+$ in more depth?
5. How can you verify that no smaller subsets of a candidate key are keys?

Tip:

To ensure your candidate keys are correct, verify their minimality and uniqueness exhaustively!