#OK to post

#Soham Palande, Assignment 4, 09/19/2020


#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


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

# 106 is the smallest positive integer such that 1, 2, 3, ....., n (in Maple: seq(i,i=1..n) ) only returns ONE hit


#PART 3

# U={1,..., 15}, A1= subset of U that are multiples of 3, A2=subsets of U that are multiples of 5
# A1={3,6,9,12,15}
# A2={5,10,15}

# A1^ = Complement of A1={1,2,4,5,7,8,10,11,13,14}
# A2^ = COmplement of A2={1,2,3,4,6,7,8,9,11,12,13,14}

# A1^ intersect A2^ = {1,2,4,7,8,11,13,14} , |A1^ intersect A2^|= 8
# A1 intersect A2 = {15}, |A1 intersect A2|= 1

#Verifying Formula: |Comp(U,A1) intersect Comp(U,A2)|= |U|- |A1|-|A2|+ |A1 intersect A2|

#		      		8 = 15 - 5 - 3 + 1
# It works!


#PIE2(U,A1,A2): |Comp(U,A1) intersect Comp(U,A2)|= |U|- |A1|-|A2|+ |A1 intersect A2|

PIE2:=proc(U,A1,A2) local n1,n2,A1comp, A2comp:


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 A2 subset U then
 print(A2, `is not a subset of universal set`):
 RETURN(FAIL):
fi:

if not A1 subset U then
 print(A1, `is not a subset of universal set`):
 RETURN(FAIL):
fi:


#FIRST WAY
A1comp:=U minus A1:
A2comp:=U minus A2:

n1:=numelems(A1comp intersect A2comp):

#SECOND WAY
n2:=numelems(U)-numelems(A1)-numelems(A2)+numelems(A1 intersect A2):

print(n1):
print(n2):

end:

#EXAMPLE

U:={seq(i,i=1..15)}
U := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
A1:={3,6,9,12,15}
A1 := {3, 6, 9, 12, 15}
A2:={5,10,15}
A2 := {5, 10, 15}
PIE2(U,A1,A2)
8
8




#PART 4

#PIE3(U,A1,A2,A3): 
PIE3:=proc(U,A1,A2,A3) local n1,n2,A1comp, A2comp,A3comp:


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 A2 subset U then
 print(A2, `is not a subset of universal set`):
 RETURN(FAIL):
fi:

if not A1 subset U then
 print(A1, `is not a subset of universal set`):
 RETURN(FAIL):
fi:

if not A3 subset U then
 print(A3, `is not a subset of universal set`):
 RETURN(FAIL):
fi:


#FIRST WAY
A1comp:=U minus A1:
A2comp:=U minus A2:
A3comp:=U minus A3:

n1:=numelems(A1comp intersect A2comp intersect A3comp):

#SECOND WAY
n2:=numelems(U)-numelems(A1)-numelems(A2)-numelems(A3)+numelems(A1 intersect A2)+numelems(A1 intersect A3) + numelems(A2 intersect A3) - numelems(A1 intersect A2 intersect A3):

print(n1):
print(n2):

end:

U:={seq(i,i=1..15)}
U := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
A1:={3,6,9,12,15}
A1 := {3, 6, 9, 12, 15}
A2:={5,10,15}
A2 := {5, 10, 15}
A3:={1,2,3,4,5,6,7}
A3 := {1, 2, 3, 4, 5, 6, 7}
PIE3(U,A1,A2,A3)
4
4


#PART 5

#PIE4(U,A1,A2,A3,A4): 
PIE4:=proc(U,A1,A2,A3,A4) local n1,n2,A1comp, A2comp,A3comp,A4comp:


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 A2 subset U then
 print(A2, `is not a subset of universal set`):
 RETURN(FAIL):
fi:

if not A1 subset U then
 print(A1, `is not a subset of universal set`):
 RETURN(FAIL):
fi:

if not A3 subset U then
 print(A3, `is not a subset of universal set`):
 RETURN(FAIL):
fi:
if not A4 subset U then
 print(A4, `is not a subset of universal set`):
 RETURN(FAIL):
fi:

#FIRST WAY
A1comp:=U minus A1:
A2comp:=U minus A2:
A3comp:=U minus A3:
A4comp:=U minus A4:

n1:=numelems(A1comp intersect A2comp intersect A3comp intersect A4comp):

#SECOND WAY
n2:=numelems(U)-numelems(A1)-numelems(A2)-numelems(A3)-numelems(A4)+numelems(A1 intersect A2)+numelems(A1 intersect A3)+numelems(A1 intersect A4)+numelems(A2 intersect A4)+numelems(A3 intersect A4) + numelems(A2 intersect A3) - numelems(A1 intersect A2 intersect A3)- numelems(A1 intersect A2 intersect A4)- numelems(A1 intersect A3 intersect A4)- numelems(A2 intersect A3 intersect A4)+numelems(A1 intersect A2 intersect A3 intersect A4):

print(n1):
print(n2):

end:

#EXAMPLE
U:={seq(i,i=1..15)}
U := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}
A1:={3,6,9,12,15}
A1 := {3, 6, 9, 12, 15}
A2:={5,10,15}
A2 := {5, 10, 15}
PIE4(U,A1,A2,A3,A4)
3
3
A3:={1,2,3,4,5,6,7}
A3 := {1, 2, 3, 4, 5, 6, 7}
A4:={2,4,6,8,10}