Relation R has eight attributes ABCDEFGH. Fields of R contain only atomic…

Concept

A candidate key is a minimal set of attributes whose attribute closure yields all attributes of the relation. Two facts drive the count: (1) an attribute that appears on no right-hand side of any FD is essential — it can be derived by nothing else, so it must lie in every candidate key; (2) when a group of attributes are mutually derivable (each reachable from the others), they are interchangeable, so swapping one for another in a key produces another key of the same size.

Setting up

• Attributes: {A, B, C, D, E, F, G, H}

• FDs: CH → G, A → BC, B → CFH, E → A, F → EG

• Attributes appearing on some RHS: A, B, C, E, F, G, H

D appears on no right-hand side, so D is essential and must lie in every candidate key. The attributes C, G and H, although some appear on a left-hand side, cannot determine any of A, B, E or F (for instance CH yields only G), so they can never start a minimal key — they only ever appear inside non-minimal superkeys.

Finding the interchangeable group

Trace the determines-relation among the remaining attributes:

• E → A

• A → BC, so A → B

• B → CFH, so B → F

• F → EG, so F → E

This forms the cycle E → A → B → F → E. Hence A, B, E, F are mutually derivable — any one of them reaches all four.

Application — verify a key

Pair the essential attribute D with one member of the cycle and compute the closure of {A, D}:

1. Start: {A, D}.

2. A → BC adds B, C → {A, B, C, D}.

3. B → CFH adds F, H → {A, B, C, D, F, H}.

4. F → EG adds E, G → {A, B, C, D, E, F, G, H} = all attributes.

So {A, D} is a key, and it is minimal (dropping either attribute fails to reach all). Because A, B, E, F are interchangeable, replacing A with B, E or F gives a key each time:

• {A, D}

• {B, D}

• {E, D}

• {F, D}

Cross-check

Could any other minimal key exist? C, G, H determine nothing outward, so no key can be built around them; D alone reaches nothing; and any pair already containing one of A/B/E/F together with D is one of the four above. No further minimal key is possible.

Number of candidate keys: 4.