#Ok to post homework
#Tifany Tong, September 20th, 2020, HW #4

# Question 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, n, o, r}, 5)) = true

# Question 2:

# There is no sequence that will result in A27 as the only hit. When I search up the entire sequence (seq(i,i=1..77)), I still get 11 search hits.

# Question 3:

# Checking 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}
# The length of (Comp(U,A1) intersect Comp(U,A2)) is the length of {1,2,4,7,8,11,13,14}, which is 8.
# |U| - |A1| - |A2| + |A1 intersect A2| = 15 - 5 - 3 + (length of {15}) = 15-8+1 = 8. 
# Both values come out to 8, showing that these two values are equal for the given case of U,A1, and A2.

Comp := proc(n, z) local S, S1, i, l, j:
option remember:
S := {}:
l := nops(n):
for i to l do 
  if not member(n[i], z) 
    then S := S union {n[i]}:
  fi:
od: 
S:
end:

PIE2 := proc(U, A1, A2) local subprob1, subprob2, S, Z:
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 nops(A1 intersect U) = nops(U) then print(A1, 'is a subset of', U):
 RETURN(FAIL):
fi:
if nops(A2 intersect U) = nops(U) then print(A2, 'is a subset of', U):
 RETURN(FAIL):
fi:
subprob1 := nops(U) - nops(A1) - nops(A2) + nops(A1 intersect A2):
S := Comp(U, A1):
Z := Comp(U, A2):
subprob2 := nops(S intersect Z):
RETURN([subprob1, subprob2]):
end: