#OK to post homework
#TaerimKim,9/20/2020,Assignment 4

#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

#2.
#For A27, the value for n was 77, but there were 22 total results from seq(i,i=1..77)
#To get the smallest n with only 1 hit, the answer for n = 106.
#seq(i,i=1..106) in oeis gave only 1 result of A262530(Numbers such that digits occur at most twice in decimal representation.)

#3. 
#3-1. Doing so by hand,
#Given U={1,..,15},
#A1={3,6,9,12,15}
#A2={5,10,15}
#A1 intersect A2 ={15}
#|Comp(U,A1) intersect Comp(U,A2)|= |U|- |A1|-|A2|+ |A1 intersect A2|
#={1,2,4,7,8,11,13,14}, total of 8

#3-2. Formulating the maple code,
#1st Method
Comp := proc(U, A):
U minus A; 
end;

PIE2 := proc(U, A1, A2): 
if not type(U, set) then 
RETURN(FAIL);
end if;
if not type(A1, set) then 
RETURN(FAIL); 
end if; 
if not type(A2, set) then 
RETURN(FAIL); 
end if; 
if A1 intersect U = {} or A2 intersect U = {} then 
RETURN(FAIL); 
end if; 

nops(Comp(U, A1) intersect Comp(U, A2)); 
end proc;
#Check by comparing with answers by hand
PIE2({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {3, 6, 9, 12, 15}, {5, 10, 15});
                  8
#3-2. THe 2nd Method
PIE2 := proc(U, A1, A2): 
if not type(U, set) then 
RETURN(FAIL);
end if;
if not type(A1, set) then 
RETURN(FAIL); 
end if; 
if not type(A2, set) then 
RETURN(FAIL); 
end if; 
if A1 intersect U = {} or A2 intersect U = {} then 
RETURN(FAIL); 
end if; 

nops(U) - nops(A1) - nops(A2) + nops(A1 intersect A2):
end:
#check
PIE22({1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}, {3, 6, 9, 12, 15}, {5, 10, 15});
                  8

#4. optional Challenge
PIE3 := proc(U, A1, A2,A3): 
if not type(U, set) then 
RETURN(FAIL);
end if;
if not type(A1, set) then 
RETURN(FAIL); 
end if; 
if not type(A2, set) then 
RETURN(FAIL); 
end if; 
if not type(A3, set) then 
RETURN(FAIL); 
end if; 
if A1 intersect U = {} or A2 intersect U = {} or A3 intersect U = {} then 
RETURN(FAIL); 
end if; 

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):
end:
#check
PIE3({1, 2, 3, 4, 5, 6}, {1, 2, 3}, {2, 3, 5}, {5, 6});
                               1