#OK to post homework
#Kent Mei, 9/20/20, Assignment 4

#---------------------------------
#Part 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

#Note: In the Maple Code for Lecture 4, I had to rename CheckDecomp to CheckAdd to get the desired output.
#CheckAdd({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

#---------------------------------
#Part 2

#With how the OEIS works, A257 (aka the natural numbers) only appears up to n = 77 where there are 11 hits.
#The smallest positive integer n that returns one hit is n = 106 where the only hit is the sequence A262530: the numbers such that digits occur at most twice in decimal representation.

#---------------------------------
#Part 3

#U = {1,...,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}
#Comp(U,A1) intersect Comp(U,A2) = {1,2,4,7,8,11,13,14}
#|Comp(U,A1) intersect Comp(U,A2)| = 8

#A1 intersect A2 = {15}
#|U|- |A1|-|A2|+ |A1 intersect A2| = 15 - 5 - 3 + 1 = 8

#|Comp(U,A1) intersect Comp(U,A2)| = |U|- |A1|-|A2|+ |A1 intersect A2| as desired.


PIE2:=proc(U,A1,A2) local x, y:

if not type(U, set) then
 print(U, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A1, set) then
 print(A1, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A2, set) then
 print(A2, `is not a set.`):
 RETURN(FAIL):
fi:

if not ((A1 subset U) and (A2 subset U)) then
 print(A1, `and/or`, A2, `is not a subset of`, U):
 RETURN(FAIL):
fi:

x := nops((U minus A1) intersect (U minus A2)):
y := nops(U) - nops(A1) - nops(A2) + nops(A1 intersect A2):

RETURN([x,y]):

end:
	


#---------------------------------
#Part 4

PIE3:=proc(U,A1,A2,A3) local x, y:

if not type(U, set) then
 print(U, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A1, set) then
 print(A1, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A2, set) then
 print(A2, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A3, set) then
 print(A3, `is not a set.`):
 RETURN(FAIL):
fi:

if not ((A1 subset U) and (A2 subset U) and (A3 subset U)) then
 print(A1, `and/or`, A2, `and/or`, A3, `is not a subset of`, U):
 RETURN(FAIL):
fi:

x := nops((U minus A1) intersect (U minus A2) intersect (U minus A3)):
y := nops(U) - nops(A1) - nops(A2) - nops(A3) + nops(A1 intersect A2) + nops(A1 intersect A3) + nops(A2 intersect A3) - nops(A1 intersect A2 intersect A3):

RETURN([x,y]):

end:

#---------------------------------
#Part 5

PIE4:=proc(U,A1,A2,A3,A4) local x, y:

if not type(U, set) then
 print(U, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A1, set) then
 print(A1, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A2, set) then
 print(A2, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A3, set) then
 print(A3, `is not a set.`):
 RETURN(FAIL):
fi:

if not type(A4, set) then
 print(A4, `is not a set.`):
 RETURN(FAIL):
fi:

if not ((A1 subset U) and (A2 subset U) and (A3 subset U) and (A4 subset U)) then
 print(A1, `and/or`, A2, `and/or`, A3, `and/or`, A4, `is not a subset of`, U):
 RETURN(FAIL):
fi:

x := nops((U minus A1) intersect (U minus A2) intersect (U minus A3) intersect (U minus A4)):
y := nops(U) - nops(A1) - nops(A2) - nops(A3) - nops(A4) + nops(A1 intersect A2) + nops(A1 intersect A3) + nops(A1 intersect A4) + nops(A2 intersect A3) + nops(A2 intersect A4) + nops(A3 intersect A4) - nops(A1 intersect A2 intersect A3) - nops(A1 intersect A2 intersect A4) - nops(A1 intersect A3 intersect A4) - nops(A2 intersect A3 intersect A4) + nops (A1 intersect A2 intersect A3 intersect A4):

RETURN([x,y]):

end: