STEP 1: After splitting the FD's we get : AC->G, D->E, D->G, BC->D, CG->B, CG->D, ACD->B, CE->A, CE->G
Step 2: now removing extraneous attributes:
a) D is extraneous in ACD as AC->D by AC->G and CG->D therefore ACD will be AC
b) A is extraneous in ACD as CD->A by D->E and CE->A therefore ACD will be CD
Step 3: removing redundant FD's:
A) removing ACD->B , CE->G, CG->D or CG->B FD's
1) Considering a) of Step 2 FD's are : AC->G, D->E, D->G, BC->D, CG->B, CG->D, AC->B, CE->A, CE->G
--> AC->B is redundant as AC->G and CG->B by transitivity AC->B so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CG->D, CE->A, CE->G
-->1)CG->D is redundant as CG->B and BC->D by transitivity CG->D, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CE->A, CE->G
or
2) CG->B is redundant as CG->D, D->E, CE->A and AC->B by transitivity CG->B, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, CE->A, CE->G
-->CE->G is redundant as CE->A and AC->G by transitivity CE->G, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CE->A--------------------------------------(1) minimal cover
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, CE->A--------------------------------------(2) minimal cover
2)Considering b) of Step 2 FD's are : AC->G, D->E, D->G, BC->D, CG->B, CG->D, CD->B, CE->A, CE->G
-->CD->B is redundant as D->G and CG->B by transitivity CD->B so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CG->D, CE->A, CE->G
-->1)CG->D is redundant as CG->B and BC->D by transitivity CG->D, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CE->A, CE->G
or
2) CG->B is redundant as CG->D, D->E, CE->A and AC->B by transitivity CG->B, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, CE->A, CE->G
-->CE->G is redundant as CE->A and AC->G by transitivity CE->G, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CE->A--------------------------------------(1) minimal cover
or
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, CE->A--------------------------------------(2) minimal cover
B) removing CG->B or CG->D , CE->G, AC->G
1)Considering a) of Step 2 FD's are : AC->G, D->E, D->G, BC->D, CG->B, CG->D, AC->B, CE->A, CE->G
-->1) CG->B is redundant as CG->D, D->E, CE->A and AC->B by transitivity CG->B, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, AC->B, CE->A, CE->G
or
2) CG->D is redundant as CG->B and BC->D by transitivity CG->D, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, AC->B, CE->A, CE->G
-->CE->G is redundant as CE->A and AC->G by transitivity CE->G, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, AC->B, CE->A
or
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, AC->B, CE->A
-->AC->G is redundant as AC->B, BC->D and D->G by transitivity AC->G, so it should be removed.
now remaining FD's are D->E, D->G, BC->D, CG->B, AC->B, CE->A-------------(3) minimal cover
or
now remaining FD's are D->E, D->G, BC->D, CG->D, AC->B, CE->A--------------(4) minimal cover
2) Considering b) of Step 2 FD's are : AC->G, D->E, D->G, BC->D, CG->B, CG->D, CD->B, CE->A, CE->G
-->1) CG->B is redundant as CG->D and CD->B by transitivity CG->B, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, CD->B, CE->A, CE->G
or
2) CG->D is redundant as CG->B and BC->D by transitivity CG->D, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CD->B, CE->A, CE->G
-->CE->G is redundant as CE->A and AC->G by transitivity CE->G, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CD->B, CE->A
or
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, CD->B, CE->A
-->CD->B is redundant as D->G and CG->B by transitivity CD->B, so it should be removed.
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->B, CE->A----------------(1) minimal cover
or
now remaining FD's are AC->G, D->E, D->G, BC->D, CG->D, CD->B, CE->A-------(6) minimal cover
Finally the minimal clovers are
AC->G, D->E, D->G, BC->D, CG->B, CE->A
AC->G, D->E, D->G, BC->D, CG->D, CE->A
D->E, D->G, BC->D, CG->B, AC->B, CE->A
D->E, D->G, BC->D, CG->D, AC->B, CE->A
AC->G, D->E, D->G, BC->D, CG->D, CD->B, CE->A
Please correct me if i am wrong anywhere