> #ok to post
;
> #Yifan Zhang, 09/16/2020, hw4
;
> 
;
> #Q1.
;
> CP:=proc(A,B) local a,b:
> {seq(seq([a,b],a in A),b in B)}:
> end:
> 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]}

;
> Mynops:=proc(A) local a:
> 
> if not type(A,set) then
>  print(A, `is not a set `):
>  RETURN(FAIL):
> fi:
> 
> add(1, a in A):
> end:
> CheckMult:=proc(A,B)
> evalb(Mynops(CP(A,B))=Mynops(A)*Mynops(B)):
> end:
> CheckMult({1,2},{3,4});
true

;
> CheckAdd:=proc(A,B):
> 
> 
> if not type(A,set) then
>  print(A, `is not a set `):
>  RETURN(FAIL):
> fi:
> 
> if not type(B,set) then
>  print(B, `is not a set `):
>  RETURN(FAIL):
> fi:
> 
> if A intersect B<>{} then
>  print(A,B, `have the following common elements `, A intersect B):
>  RETURN(FAIL):
> fi:
> 
> #This checks that if A and B are disjoint finite sets, then
> #|A U B|= |A| + |B|
> evalb(Mynops(A union B)=Mynops(A) + Mynops(B)):
> end:
> CheckAdd({1,2,3},{3,4,5});
{1, 2, 3}, {3, 4, 5}, `have the following common elements `, {3}
FAIL

;
> Words:=proc(A,n) local W1,w,a:
> option remember:
> if n=0  then
>  RETURN({[]}):
> else
> W1:=Words(A,n-1):
> RETURN({seq(seq([op(w),a], a in A),w in W1)}):
> fi:
> end:
> member([d,o,r,o,n],Words({d,o,n,r},5));
true

;
> 
;
> 
;
> 
;
> #Q2.
;
> #1 is the smallest positive integer.
;
> 
;
> 
;
> 
;
> #Q3. 
;
> #U = {1, ..., 15}
;
> #A1 = {3,6,9,12,15} => Comp(U,A1) = {1,2,4,5,7,8,10,11,13,14}
;
> #A2 = {5,10,15}=> Comp(U,A2) = {1,2,3,4,6,7,8,9,11,12,13,14}
;
> #LHS = Comp(U,A1) intersect Comp(U,A2) = {1,2,4,7,8,11,13,14}=> nops(Comp(U,A1)) = 8
;
> #nops(U) = 15; nops(A1) = 5; nops(A2) = 3; nops(A1 interset A2) = 1
;
> #RHS = 15-5-3+1 = 8
;
> #LHS = RHS. 
;
> 
;
> 
;
> #Maple Procedure.
;
> PIE2:= proc(U,A1,A2) local A,B:
> option remember:
> 
> if not (type(A1,set) and type(A2,set) and type(U,set) )then
>  RETURN(1):
> fi:
> 
> if not (A1 subset U and A2 subset U)then
> RETURN (2):
> fi:
> 
> A:=nops((U minus A1) intersect (U minus A2));
> B:=nops(U) - nops(A1) - nops(A2) + nops(A1 intersect A2);
> [A,B]:
> end:
> 
;
> PIE2({1,2,3,4,5}, {1,2,3},{2,3,5});
[1, 1]

;
> 
;
> 
;
> 
;
> #Q4.    
;
> PIE3:= proc(U,A1,A2,A3) local A,B:
> option remember:
> 
> if not (type (A1,set) or type(A2,set) or type(A3, set) or type(U, set)) then
> RETURN (FAIL):
> fi:
> 
> if not (A1 subset U or A2 subset U or A3 subset U) then
> RETURN (FAIL):
> fi:
> 
> A:=nops((U minus A1) intersect (U minus A2) intersect (U minus A3)):
> B:=nops(U)- nops(A1)-nops(A2)-nops(A3) +nops(A1 intersect A2 ) + nops(A2 intersect A3) + nops(A1 intersect A3) - nops(A1 intersect A2 intersect A3):
> [A,B]:
> end:
> 
;
> #For example: 
> U:={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15};A1:={2,4,6,8,10,12,14};A2:={3,6,9,12,15};A3:={4,8,12};
Typesetting:-mprintslash([(U := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15})],[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}])
Typesetting:-mprintslash([(A1 := {2, 4, 6, 8, 10, 12, 14})],[{2, 4, 6, 8, 10, 
12, 14}])
Typesetting:-mprintslash([(A2 := {3, 6, 9, 12, 15})],[{3, 6, 9, 12, 15}])
Typesetting:-mprintslash([(A3 := {4, 8, 12})],[{4, 8, 12}])

;
> PIE3(U,A1,A2,A3);
[5, 5]

;
> #My maple procedure's answer is equal to the hand write answer. 
;
> 
;
> 
;
> 
;
> #Q5.
;
> PIE4:= proc(U,A1,A2,A3,A4) local A,B:
> option remember:
> 
> if not (type(A1,set) or type(A2,set) or type(A3,set) or type(A4,set) or type(U,set)) then
> RETURN(FAIL):
> fi:
> 
> if not (A1 subset U or A2 subset U or A3 subset U or A4 subset U) then 
> RETURN(FAIL):
> fi:
> 
> A:=nops((U minus A1) intersect (U minus A2) intersect (U minus A3) intersect (U minus A4)):
> B:=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(A2 intersect A3 intersect A4) + nops(A1 intersect A2 intersect A3 intersect A4):
> [A,B]:
> end:
> 
;
> 
;
> U:= {1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}; A1:={2,4,6,8,10,12,14}; A2:={3,6,9,12,15}; A3:={4,8,12}; A4:={5,10,15};
Typesetting:-mprintslash([(U := {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,
15})],[{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15}])
Typesetting:-mprintslash([(A1 := {2, 4, 6, 8, 10, 12, 14})],[{2, 4, 6, 8, 10, 
12, 14}])
Typesetting:-mprintslash([(A2 := {3, 6, 9, 12, 15})],[{3, 6, 9, 12, 15}])
Typesetting:-mprintslash([(A3 := {4, 8, 12})],[{4, 8, 12}])
Typesetting:-mprintslash([(A4 := {5, 10, 15})],[{5, 10, 15}])

;
> PIE4(U,A1,A2,A3,A4);
[4, 4]

;
> 
;