#OK to post homework
#Karnaa Mistry, 09/20/20, Assignment HW #4

with(combinat):


# 1. 

# CP({a,b,c},{b,c,d})  =  {[a, b], [a, c], [a, d], [b, b], [b, c], [b, d], [c, b], [c, c], [c, d]}
# CheckMult({1,2},{3,4})  =  true
# CheckAdd appears to reference proc CheckDecomp, so:
#		CheckDecomp({1,2,3},{3,4,5})  =  {1, 2, 3}, {3, 4, 5}, `have the following common elements `, {3}   FAIL
# member([d,o,r,o,n],Words({d,o,n,r},5))  =  true




# 2.  I found that n = 106




# 3.

# Checking PIE by hand:  U = {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15};  A1 = {3,6,9,12,15};  A2 = {5,10,15};
# Comp(U,A1) = {1,2,4,5,7,8,10,11,13,14};  Comp(U,A2) = {1,2,3,4,6,7,8,9,11,12,13,14};
# A1 intersect A2 = {15};
# Comp(U,A1) intersect Comp(U,A2) = {1,2,4,7,8,11,13,14};
# |Comp(U,A1) intersect Comp(U,A2)| = 8
# |U| = 15;  |A1| = 5;  |A2| = 3;  |A1 intersect A2|  =  1
# Does 8 = 15 - 5 - 3 + 1?

# Yes, 8 = 8, and so PIE for these sets are verified

#PIE2(U,A1,A2): that inputs three sets U, A1, A2. After checking that U, A1,A2, are indeed sets (returns FAIL if not)
#AND that A1 and A2 are subsets of U (returns FAIL otherwise), computes PIE in two ways. The first way by directly 
#finding |Comp(U,A1) intersect Comp(U,A2)| and the second way by using the formula
PIE2:=proc(U,A1,A2) local i,A1comp,A2comp:
option remember:

if (not type(U,set)) or (not type(A1,set)) or (not type(A2,set)) then
 RETURN(FAIL):
fi:

if (not A1 subset U) or (not A2 subset U) then
 RETURN(FAIL):
fi:

A1comp:={}:
A2comp:={}:

for i from 1 to numelems(U) do
 if not (U[i] in A1) then
  A1comp:=A1comp union {U[i]}:
 fi:
od:

for i from 1 to numelems(U) do
 if not (U[i] in A2) then
  A2comp:=A2comp union {U[i]}:
 fi:
od:

RETURN([numelems(A1comp intersect A2comp), numelems(U)-numelems(A1)-numelems(A2)+numelems(A1 intersect A2)]):

end:




# 4. optional PIE3(U,A1,A2,A3)

#PIE3(U,A1,A2,A3): similar to PIE2 but for 3 sets, using an appropriate PIE formula to prevent overcounting
PIE3:=proc(U,A1,A2,A3) local i,A1comp,A2comp,A3comp:
option remember:

if (not type(U,set)) or (not type(A1,set)) or (not type(A2,set)) or (not type(A3,set)) then
 RETURN(FAIL):
fi:

if (not A1 subset U) or (not A2 subset U) or (not A3 subset U) then
 RETURN(FAIL):
fi:

A1comp:={}:
A2comp:={}:
A3comp:={}:

for i from 1 to numelems(U) do
 if not (U[i] in A1) then
  A1comp:=A1comp union {U[i]}:
 fi:
od:

for i from 1 to numelems(U) do
 if not (U[i] in A2) then
  A2comp:=A2comp union {U[i]}:
 fi:
od:

for i from 1 to numelems(U) do
 if not (U[i] in A3) then
  A3comp:=A3comp union {U[i]}:
 fi:
od:

RETURN([numelems(A1comp intersect A2comp intersect A3comp), numelems(U)-numelems(A1)-numelems(A2)
-numelems(A3)+numelems(A1 intersect A2)+numelems(A2 intersect A3)+numelems(A3 intersect A1)
-numelems(A1 intersect A2 intersect A3)]):

end:




# 5. optional PIE4(U,A1,A2,A3,A4)

#PIE4(U,A1,A2,A3,A4): similar to PIE2 but for 4 sets, using an appropriate PIE formula to prevent overcounting
PIE4:=proc(U,A1,A2,A3,A4) local i,A1comp,A2comp,A3comp,A4comp:
option remember:

if (not type(U,set)) or (not type(A1,set)) or (not type(A2,set)) or (not type(A3,set)) or
(not type(A4,set)) then
 RETURN(FAIL):
fi:

if (not A1 subset U) or (not A2 subset U) or (not A3 subset U) or (not A4 subset U) then
 RETURN(FAIL):
fi:

A1comp:={}:
A2comp:={}:
A3comp:={}:
A4comp:={}:

for i from 1 to numelems(U) do
 if not (U[i] in A1) then
  A1comp:=A1comp union {U[i]}:
 fi:
od:

for i from 1 to numelems(U) do
 if not (U[i] in A2) then
  A2comp:=A2comp union {U[i]}:
 fi:
od:

for i from 1 to numelems(U) do
 if not (U[i] in A3) then
  A3comp:=A3comp union {U[i]}:
 fi:
od:

for i from 1 to numelems(U) do
 if not (U[i] in A4) then
  A4comp:=A4comp union {U[i]}:
 fi:
od:

RETURN([numelems(A1comp intersect A2comp intersect A3comp intersect A4comp), numelems(U)
-numelems(A1)-numelems(A2)-numelems(A3)-numelems(A4)+numelems(A1 intersect A2)
+numelems(A2 intersect A3)+numelems(A3 intersect A4)+numelems(A4 intersect A1)
+numelems(A1 intersect A3)+numelems(A2 intersect A4)
-numelems(A1 intersect A2 intersect A3)-numelems(A1 intersect A3 intersect A4)
-numelems(A2 intersect A3 intersect A4)-numelems(A1 intersect A2 intersect A4)
+numelems(A1 intersect A2 intersect A3 intersect A4)]):

end: