Closure of a Set of FDs

Given a relation, R, and its associated minimal FD cover set, F,
we can use the FD Inference Rules to construct the closureof F (for R): F+

Example #1

R (A, B, C)

F = {A → B,
     C → B}

F+ for R

Trivial
A       → A
A, B    → A
A, C    → A
A, B, C → A
B       → B
B, A    → B
B, C    → B
B, A, C → B
C       → C
C, A    → C
C, B    → C
C, A, B → C
      
Wasteful
A, C → B
      
Derived
none
      
Minimal   
A → B
C → B

Wasteful does not include FDs already in Trivial
Derived does not include FDs already in Trivial or Wasteful

As with F, F all the FDs in our F+ have only one attibute in the RHS.
Therefore, we do not use the Union FD Inference Rule to generate Redundant FDs such as

     A, B, C → A, B
     A, B, C → A, C
     A, B, C → A, B, C

Example #2

   F = {A → B,
        B → Y}

F+, the closure of F (for R)

Wasteful does not include FDs already in Trivial
Derived does not include FDs already in Trivial or Wasteful

As with F, F all the FDs in our F+ have only one attibute in the RHS.
Therefore, we do not use the Union FD Inference Rule to generate Redundant FDs such as

A, B, Y → A, B
A, B, Y → A, Y
A, B, Y → B, Y
A, B, Y → A, B, Y

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