List all possible F+

The closure F⁺ is the set of all functional dependencies logically implied by F. It is built by applying Armstrong's axioms again and again.

• Reflexivity: if Y ⊆ X then X → Y
• Augmentation: if X → Y then XZ → YZ
• Transitivity: if X → Y and Y → Z then X → Z
• Derived: union (X→Y, X→Z ⇒ X→YZ), decomposition (X→YZ ⇒ X→Y, X→Z), pseudotransitivity (X→Y, YW→Z ⇒ XW→Z)

Given R = (A, B, C, D, E, F, G) and F = {A → B, ABCD → E, EF → G}.

A complete listing is not practical: R has 7 attributes, and F⁺ contains 3357 dependencies X → Y with Y non-empty (counting the trivial ones). So F⁺ is written by its structure, with all the important members listed.

1. Members of F itself

A → B, ABCD → E, EF → G

2. Trivial dependencies (reflexivity)

X → Y for every Y ⊆ X, for example A → A, AB → A, ABCDEFG → any subset of R.

3. Dependencies from the rules

• Augmentation of A → B: AC → BC, AD → BD, AF → BF, ACD → BCD, ACDF → BCDF, and so on with any Z.
• Pseudotransitivity: A → B and ABCD → E give ACD → E. Check: A → B augmented with CD gives ACD → BCD, so ACD → ABCD (add the trivial ACD → ACD and take the union); then ABCD → E gives ACD → E by transitivity.
• From EF → G and ACD → E: ACDF → EF (union/augment), then ACDF → G by transitivity.
• Union of these: ACDF → ABCDEFG.
• Augmentation of derived ones: ACDE → E (trivial), ACDF → any subset of {A,B,C,D,E,F,G}, for example ACDF → BE, ACDF → EG.

4. Attribute closures that generate F⁺

The closure X⁺ gives every right side for a left side X:

So F⁺ = { X → Y : Y ⊆ X⁺ }. For example, ACD → AB, ACD → BE, ACD → ABCDE, A → B, EF → G, EF → E, BEF → G.

Candidate key

A, C, D and F never appear on the right side of any FD, so they must be in every key. ACDF⁺ = ABCDEFG = R, so ACDF is the only candidate key.