# Determining minimal cover of a relation

I am having trouble understanding why a certain set of functional dependencies is not minimal. We have a relation `R(A,B,C,D,E,F,G)` with the following dependencies F:

``````1. A->CDE
2. B->FG
3. AB->CDEFG
``````

Minimal cover of F is just dependency 1 and 2. It is intuitive that attributes CDEFG are already determined by A and B separately. Hence, no new attribute is determined by union of AB. Is there an exact rule that determines that such dependency (union of AB->CDEFG) is redundant?

The Armstrong rules listed so far in the book are:

``````    IR1 (reflexive rule): If X ⊇ Y, then X →Y.
IR2 (augmentation rule): {X → Y} |=XZ → YZ.
IR3 (transitive rule): {X → Y, Y → Z} |=X → Z.
``````

and

``````    IR4 (decomposition, or projective, rule): {X → YZ} |=X → Y.
IR5 (union, or additive, rule): {X → Y, X → Z} |=X → YZ.
IR6 (pseudotransitive rule): {X → Y, WY → Z} |=WX → Z.
``````

I am not sure which rule is responsible for redundancy of `AB->CDEFG`. Union rule seem close, but the left-hand side attributes are listed as being the same (both X), which I can't say for `FD1 U FD2` in my case.

• F cannot be a minimal cover. You seem to be trying to say that it is not a minimal cover but you don't & also you do say that it is a minimal cover. Do you mean, it is a cover? Where are you stuck using what definition or algorithm? Where are you stuck applying what definition of "redundant"? May 9, 2021 at 18:52

``````1. A → CDE (given)