The following functional dependencies are given: AB → CD, AF → D, DE → F, C →…

Concept

The closure X+ of an attribute set X is the smallest set containing X that is closed under all given functional dependencies.

Begin with X itself. Whenever every attribute on the left side of a dependency is present, add its right-side attributes; repeat until a full pass adds nothing.

Application

1. {C,F}: Start with {C,F}; C → G adds G, F → E adds E, G → A adds A, and AF → D adds D. No dependency then adds another attribute, so CF+ = {A,C,D,E,F,G}.

2. {B,G}: Start with {B,G}; G → A adds A, and AB → CD then adds C and D. No dependency can introduce E or F, so BG+ = {A,B,C,D,G}.

3. {A,F}: Start with {A,F}; AF → D adds D and F → E adds E. DE → F adds nothing new, and no applicable dependency introduces C or G, so AF+ = {A,D,E,F}.

4. {A,B}: Start with {A,B}; AB → CD adds C and D, C → G adds G, and G → A adds nothing new. No dependency can introduce E or F, so AB+ = {A,B,C,D,G}.

Cross-check

Each claimed set can be checked in both directions: every listed attribute must be derivable, and every derivable attribute must be listed. The claims for CF+, BG+, and AB+ pass both checks, while the AF+ claim includes C and G even though neither can be derived from {A,F}.

Result

The false statement is AF+ = {ACDEFG}; its closure is {A,D,E,F}.