For each FD set below, do the following:

1. Compute the minimal basis

2. Identify all candidate keys for the relation

3. Use the minimal basis to decompose the relation into subrelations
   and project the minimal basis onto each resulting sub-relation.

(see the example below for ideas on how to proceed)
   
Problem 1:
	B -> C
	B C D G -> A
	B E -> C
	C E F -> B H


Problem 2:
	A F H -> E
	D F G H -> A
	E -> C
	F H -> A D
	H -> A F G
        
==============================        

Example:
	B -> C D G
	C F -> G
	C F G -> D H
	D -> A B H
	D H -> B

NOTE: I'm being painfully exhaustive here to be sure nobody misses
anything. In practice you can probably do fine with far less writing
(and you only need one minimal basis per answer).

When in doubt, though, the slow way always works.

Minimal basis:

	Closure of LHS:

                B+	= ABCD..GH
                CF+	= ABCD.FGH (key)
                CFG+	= (superkey, see CF+)
                D+	= ABCD..GH
                DH+	= (same as D+)
        
        Strategy 1: remove cols from RHS

	        Remove B -> CDG? Can't, B not in any other LHS

                Remove CF -> G? Can't, C, F, CF not in any other LHS

                Remove CFG -> D?
                	CF -> G has matching LHS, but RHS adds nothing
                        ==> Can't remove CFG -> D

                Remove CFG -> H?
                	CFG -> D and CF -> G have matching LHS
                        CFG+	= CDFG
                        ==> D has matching LHS
                        	= ABCDFGH (same as before!)
                        => CAN remove CFG -> H

                Remove D -> A? No, A not in any other RHS

                Remove D -> B?
                	D -> ADH has matching LHS
                        D+	= ADH
                        ==> DH -> B has matching LHS
                        	= ABDH
                        ==> B -> CDG has matching LHS
                        	= ABCDGH (same as before!)
                        ==> CAN remove D -> B

                Remove D -> H?
                	D -> ABD has matching LHS
                        D+	= ABD
                        ==> B -> CDG has matching LHS
                        	= ABCDG (missing H)
                        ==> Can't remove D -> H

                Remove DH -> B?
                	D -> AH has matching LHS (recall D->B is deleted)
                        DH+	= ADH (missing B)
                        ==> Can't remove DH -> B

        FD set remaining:
        	B -> CDG
                CF -> G
                CFG -> D
                D -> ABH
                DH -> B

	Strategy 2: remove cols from LHS

        	Remove C from CF -> G?

                F+ before = F 
		==> no G in F+
                ==> unsafe

                Remove F from CF -> G?
                C+ before = C
                ==> no F in C+
                ==> unsafe

                Remove C from CFG -> DH?
                FG+ before = FG
                ==> no C in FC+
                ==> Unsafe

                Remove F from CFG -> DH?
                CG+ before = CG
                ==> No F in CG+
                ==> unsafe

                Remove G from CFG -> DH?
                CF+ before = key
                ==> G in CF+
                ==> Replace CFG -> DH with CF -> DH

                Remove D from DH -> B?
                H+ before = H
                ==> no D in H+
                ==> unsafe

                Remove H from DH -> B?
                D+ before = ABDH
                ==> H in D+
                ==> Replace DH -> B with D -> B

        FD set remaining:
        	B -> CDG
                CF -> DG
                D -> ABH (notice D->B came back!)

        Strategy 1 (again): remove cols from RHS

        	Remove B -> C? No, C not in any other RHS

                Remove B -> D? No, B+ becomes BCG (missing D)

                Remove B -> G? No, B+ becomes ABCDH (missing G)

                Remove CF -> D? No, CF+ becomes CFG (missing D)

                Remove CF -> G? CF+ becomes ABCDFGH (contains G!)
                ==> Remove!

                Remove D -> A? No, A not in other RHS

                (Same applies to D->BH)

	FD set remaining:
        	B -> CDG
                CF -> D
                D -> ABH

        Strategy 2 (again): remove cols from LHS

        	Remove C from CF -> D? No, F+ = F

                Remove F from CF -> D? No, C+ = C

        No change in either strategy. FD set above is minimal basis!

Candidate keys?

	B+ = .B...... (now have B)
           = .BCD..G. (now have D)
           = ABCD..GH (missing F, which is not on any RHS)
        ==> BF is one key

        CF+ = ..C..F.. (now have CF)
            = ..CD.F.. (now have D)
            = ABCD.F.H (now have B)
            = ABCD.FGH (key!)
        ==> CF is another key

        D+ = ...D.... (now have D)
           = AB.D...H (now have B)
           = ABCD..GH (missing F, which is not on any RHS)
        ==> DF is a third key

Projection

	R1(B, C, D, G)
        R2(C, F, D)
        R3(D, A, B, H)

        Projected FD set for R1(B, C, D, G):

                B -> CDG comes for free
		Ignore CF -> D (F not in R1)
                D -> ABH comes also

                B+ = .BCD..G.
                   = ABCD..GH

                Add B -> BCDG (AH not in R1)

                D+ = AB.D...H
                   = ABCD..GH

                Add D -> BCDG (AH not in R1)

                No need to consider DB+ (B already in D+)

        FD set:
        	B -> CDG
                D -> BCG

	Minbase of R1:

		(no need for Strategy 2, all LHS are single col)
                
                Remove C from B -> CDG? B+ becomes BCDG (C present)
                ==> Remove

                Remove D from B -> DG? B+ becomes BG (D not present)
                ==> keep

                Remove G from B -> DG? B+ becomes BCDG (G present)
                ==> Remove

                Remove B from D -> BCG? B not in other RHS.
                ==> keep

                (same applies for C and G)

                FDs remaining:
                	B -> D
                        D -> BCG
                        
                (NOTE: B -> CDG and D -> B also a valid minbase)

	Projected FD set for R2(C, F, D):

        	Ignore B -> CDG (B not in R)
                CF -> D comes for free
                D -> ABH comes (D in R2)

                D+ = AB.D...H

                Do not add D -> D (ABH not in R2, D trivial)

                CF+ = ..CD.F..

                Add CF -> D

                FDs remaining:
                	CF -> D 

                (already in minbase form)

	Projected FD set for R3(D, A, B, H):

        	B -> CDG comes (B in R3)
                Ignore CF -> D (CF not in R3)
                D -> ABH comes for free

                B+ = .BCD..G. (now have D)
                   = ABCD..GH

                Add B -> ADH (G not in R3)

                D+ = AB.D...H (now have B)
                   = ABCD..GH

                Add D -> ABH

	FD set:

        	B -> ADH
                D -> ABH

	Minbase for R3:

        	(ignore strategy 2, all LHS are single col)

                Remove A from B -> ADH? B+ becomes BDHA (A present)
                ==> Remove

                Remove D from B -> DH? B+ becomes BH (D not present)
                ==> keep

                Remove H from B -> DH? B+ becomes BDHA (H present)
                ==> Remove

                Remove A from D -> ABH? A no longer in other RHS
                ==> keep

                (same reasoning applies for B, H)

                FDs remaining:

                B -> D
                D -> ABH

                NOTE: other possibilities include at least the following:
                D -> B and B -> ADH
                B -> DH and D -> AB