#OK to post
#Rebecca Embar, Homework 12

read `C12.txt`:

#2
CheckFMBM:=proc(n,K) local f:
	f:=RG(n,K):
	EqualEntries(f,FM(BM(f,n),n)):
end:
CheckBMFM:=proc(n,K) local f:
	f:=RG(n,K):
	EqualEntries(f,BM(FM(f,n),n)):
end:

#3
#From observation, I believe the answer here is that Maple is 
#treating each table as a distinct object. In the real world, 
#if I put the same exact items in two different boxes, I would
#still consider them to be different boxes. Here Maple is saying 
#that although two tables may contain the same exact entries, 
#they are still different tables.
#
#Although this may seem pedantic, this is actually an important 
#distinction. For example, if I define,
#f := table([1=1]);
#g := f;
#evalb(eval(f)=eval(g));
#This will output true, becase f and g refer to the same "box"
#If I then add an item to this table, for example,
#f[2] := 2
#eval(g) will now return table([1=1,2=2]). This is because g
#refers to the same "box" as f.
#However, if I define,
#f := table([1=1]);
#g := table([1=1]);
#evalb(eval(f)=eval(g));
#This will return false. This time, if I add an item to f,
#f[2] := 2;
#eval(g) will still return table([1=1]). This is because g refers 
#to a different "box"
#
#The correct function to use here is EqualEntries(f,BM(FM(f,n),n))

#4
CheckSuperAdditivity:=proc(f,n) local PN,S,SC,PSC,T,i:
	PN := powerset(n):
	for S in PN do
		SC := {seq(i,i=1..n)} minus S:
		PSC := powerset(SC):
		for T in PSC do
			if f[S union T] < f[S]+f[T] then
				return false:
			fi:
		od:
	od:
	true:
end:
#Ran CheckSuperAdditivity(FM(RG(5,20),5),5) a decent amount of times
#and it output true every time

#5
Too difficult to type up as txt, attached as separate pdf