Database Refinement - Minimum F Coverage (extraneous attributes)

Question

Pattern R = (A, B, C, D, E, F)

FD F = {ABC → D, CD → B, BCF → D, CDF → BE, BCDF → E}

Find Fc, minimum coverage (canonical coverage) F.

This method is used in my book:

Example: abc → xyz

a is redundant (extraneous) if (bc) + ⊇ a; x is redundant if (abc) + ⊇ x.

NOTE. Here, closures are calculated using F, with a or x being removed from abc → xyz, respectively.

I do not understand the last bold sentence.

one solution:

Consider CDF → BE

B is redundant: (CDF) + = (CDFBE) ⊇ ( B )

F becomes {ABC → D, CD → B, BCF → D, CDF → E }

but I do not understand.

according to this logic:

E may also be redundant,

sog:

Consider CDF → BE

E : (CDF) + = (CDFBE) ⊇ (E)

F {ABC → D, CD → B, BCF → D, CDF → B}

, . - , ?

Answer 1

r (R) , F, A R x- > Y

if A belongs to X and A is extraneous then
  (F - {X->Y}) U {(X-A) -> Y} is equivalent to F

if A belongs to Y and A is extraneous then
  F is equivalent to (F - {X->Y}) U {X -> (Y-A)}

, A X

1. Find (X-A)+ under F
2. If Y is a subset of (X-A)+ under F then A is extraneous

, , F. , A , FD.

, A Y

1. Find F' = (F - {X->Y}) U {X -> (Y-A)}
2. Find X+ under F'

3. X + F ' A, A Y

A , FD, A . , FD {X → (Y-A)} FS F X FD. , X + , , A Y A, F ', F' , F.

, F ' A, . , (X-A) X , , (X-A) + F (X-A) U Y . , F '= (F - {X- > Y}) U {(XA) → Y} (XA) F', (XA) , FD F ' (XA) U Y. , A X, F'. , .

, A Y, F ' A, Y, , X + F', XU (YA) , X F, XUY, , X , , - .

, FD . FD , FD . .

, FD , . FD, .

F = {ABC -> D, CD -> B, BCF -> D, CDF -> BE, BCDF -> E} - CDF->BE, B , E . , : F1 = {ABC -> D, CD -> B, BCF -> D, CDF -> B, BCDF -> E} F2 = {ABC -> D, CD -> B, BCF -> D, CDF -> E, BCDF -> E} ( CDF- > E , BCDF- > E CDF- > E)

/ . . , . .

AS - , / :

Fc = {AC->D, CD->B, CF->DE}

, .

EDIT1:

r(A, B, C)

FDs

 F = {A->BC, B->AC, C->AB}

, F, B A->BC. , 'C' A->BC F. , , B A->BC, B, A->C, : F1 = {A->C, B->AC, C->AB} C A->C F1. FD, , .

, 4 , .

                                         A->BC
                                         B->AC
                                         C->AB
                                           |
                         +-----------------+-----------------+
                         |                                   |
                       A->C                                A->B
                       B->AC                               B->AC
                       C->AB                               C->AB
                         |                                   |
                +--------+--------+                 +--------+--------+
                |                 |                 |                 |
              A->C            +---+---+         +---+---+           A->B
              B->A            | A->C  |         | A->B  |           B->AC
              C->AB           | B->C  |         | B->AC |           C->A
                |             | C->AB |         | C->B  |             |
                +             +-------+         +-------+         +---+---+
                |             |  Fc2  |         |  Fc3  |         | A->B  |
            +---+---+         +-------+         +-------+         | B->C  |
            | A->C  |                                             | C->A  |
            | B->A  |                                             +-------+
            | C->B  |                                             |  Fc4  |
            +-------+                                             +-------+
            |  Fc1  |
            +-------+

, , FD, . , FD A->BC, B C A->BC F, B FD A->C ( ), C A->BC FD, A->B ( ).