Canonical cover

In the following example, F_c is the canonical cover of F.

Given the following, we can find the canonical cover: R = (A, B, C, G, H, I), F = {A→BC, B→C, A→B, AB→C}

1. {A→BC, B→C, A→B, AB→C}
2. {A → BC, B →C, AB → C}
3. {A → BC, B → C}
4. {A → B, B →C}

F_c =  {A → B, B →C}

An attribute is extraneous in a functional dependency if its removal from that functional dependency does not alter the closure of any attributes.^[2]

Given a set of functional dependencies ${\displaystyle F}$ and a functional dependency ${\displaystyle A\rightarrow B}$ in ${\displaystyle F}$, the attribute ${\displaystyle a}$ is extraneous in ${\displaystyle A}$ if ${\displaystyle a\subset A}$ and any of the functional dependencies in ${\displaystyle (F-(A\rightarrow B)\cup {(A-a)\rightarrow B})}$ can be implied by ${\displaystyle F}$ using Armstrong's Axioms.^[2]

Using an alternate method, given the set of functional dependencies ${\displaystyle F}$, and a functional dependency X → A in ${\displaystyle F}$, attribute Y is extraneous in X if ${\displaystyle Y\subseteq X}$, and ${\displaystyle (X-Y)^{+}\supseteq A}$.^[3]

For example:

• If F = {A → C, AB → C}, B is extraneous in AB → C because A → C can be inferred even after deleting B. This is true because if A functionally determines C, then AB also functionally determines C.
• If F = {A → D, D → C, AB → C}, B is extraneous in AB → C because {A → D, D → C, AB → C} logically implies A → C.

Given a set of functional dependencies ${\displaystyle F}$ and a functional dependency ${\displaystyle A\rightarrow B}$ in ${\displaystyle F}$, the attribute ${\displaystyle a}$ is extraneous in ${\displaystyle A}$ if ${\displaystyle a\subset A}$ and any of the functional dependencies in ${\displaystyle (F-(A\rightarrow B)\cup \{A\rightarrow (B-a)\})}$ can be implied by ${\displaystyle F}$ using Armstrong's axioms.^[3]

A dependent attribute of a functional dependency is extraneous if we can remove it without changing the closure of the set of determinant attributes in that functional dependency.^[2]

For example:

• If F = {A → C, AB → CD}, C is extraneous in AB → CD because AB → C can be inferred even after deleting C.
• If F = {A → BC, B → C}, C is extraneous in A → BC because A → C can be inferred even after deleting C.