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{SECT 0 {PARA 18 "" 0 "" {TEXT -1 24 "Solution to Assignment 2" }}
{PARA 19 "" 0 "" {TEXT -1 18 "Maple Programming " }}{PARA 256 "" 0 "" 
{TEXT -1 17 "Jeremy R. Johnson" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Question 1 (Set membership)" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Member := proc(x,S::set)" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "local y;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "for y in S do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " \+
 if (x = y) then RETURN(true); fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 
3 "od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "RETURN(false);" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%
'MemberGR6$%\"xG'%\"SG%$setG6#%\"yG6\"F-C$?&8$9%%%trueG@$/9$F0-%'RETUR
NG6#F2-F76#%&falseGF-F-F-" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "S := \{1,3,9,11,15,2\};" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"SG<
(\"\"\"\"\"#\"\"$\"\"*\"#6\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 12 "Member(2,S);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "Member(5,S);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#%&falseG" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 55 "Use a table to represent a set.  S[x] = t
rue if x in S." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 27 "Member2 := proc(x,S::table)" }}{PARA 0 "> " 0 "
" {MPLTEXT 1 0 59 "  if (S[x] = true) then RETURN(true) else RETURN(fa
lse) fi;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#>%(Member2GR6$%\"xG'%\"SG%&tableG6\"F+F+@%/&9%6#9$%%t
rueG-%'RETURNG6#F2-F46#%&falseGF+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 26 "Insert := proc(x,S::table)" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "  S[x] := true;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "
end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'InsertGR6$%\"xG'%\"SG%&tabl
eG6\"F+F+>&9%6#9$%%trueGF+F+F+" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "S2 := table();" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#S2G-%&TAB
LEG6#7\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "for x in S do" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "  Insert(x,S2);" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#%%trueG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%%tru
eG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "indices(S2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6(7#\"\"\"7#\"\"#7#\"\"$7#\"\"*7#\"#67#
\"#:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Member2(2,S2);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#%%trueG" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "Member2(5,S2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%&f
alseG" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "Question 2 (list inver
sion)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "ListInverse := proc
(L::list)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "local n;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 13 "n := nops(L);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "RETURN([seq(L[n-i+1],i=1..n)]);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 4 "end;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%,ListInverseGR6#'%\"LG%
%listG6#%\"nG6\"F,C$>8$-%%nopsG6#9$-%'RETURNG6#7#-%$seqG6$&F36#,(F/\"
\"\"%\"iG!\"\"F>F>/F?;F>F/F,F,F," }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "L := [1,2,3,4,5];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
>%\"LG7'\"\"\"\"\"#\"\"$\"\"%\"\"&" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "ListInverse(L);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'
\"\"&\"\"%\"\"$\"\"#\"\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
16 "ListInverse([]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "ListInverse2 := proc(L::list
)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "local Lp,M,x,n;" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 32 "Lp := L;  M := []; n := nops(L);" }}{PARA 0 ">
 " 0 "" {MPLTEXT 1 0 19 "while (Lp <> []) do" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "  M := [Lp[1],op(M)];" }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 17 "  Lp := Lp[2..n];" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "  n :=
 n-1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "RETURN(M);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end;
" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%-ListInverse2GR6#'%\"LG%%listG6&
%#LpG%\"MG%\"xG%\"nG6\"F/C'>8$9$>8%7\">8'-%%nopsG6#F3?(F/\"\"\"F=F/0F2
F6C%>F57$&F26#F=-%#opG6#F5>F2&F26#;\"\"#F8>F8,&F8F=F=!\"\"-%'RETURNGFF
F/F/F/" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "ListInverse2(L);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#7'\"\"&\"\"%\"\"$\"\"#\"\"\"" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 28 "Question 3 (Polynomial GCDs)" }}{PARA 0 "" 0 "" {TEXT 
-1 196 "Euclidean Algorithm for rational polynomials.  Simply replace \+
the mod operation in either of the integer versions (mygcd, or mygcdit
) presented in class with the polynomial remainder function rem." }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "# A recursive procedure for \+
computing the gcd of two polynomials with rational coefficients." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "# Inputs:  a, b : polynomials." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "#          x : variable name." }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "# Output:  polynomial = gcd(a,b).  \+
Note that the gcd is only defined upto a scalar multiple." }}{PARA 0 "
> " 0 "" {MPLTEXT 1 0 45 "mygcd := proc(a::polynom,b::polynom, x::name
)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "  if b = 0 then RETURN(a) fi;
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "  RETURN(mygcd(b, rem(a,b,x),x)
);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 " end;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%&mygcdGR6%'%\"aG%(polynomG'%\"bGF)'%\"xG%%nameG6\"F/F
/C$@$/9%\"\"!-%'RETURNG6#9$-F66#-F$6%F3-%$remG6%F8F39&F@F/F/F/" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Ab := randpoly(x);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"AG,.*$)%\"xG\"\"&\"\"\"!#&)*&\"#bF*)F(\"
\"%F*!\"\"*&\"#PF*)F(\"\"$F*F0*&\"#NF*)F(\"\"#F*F0*&\"#(*F*F(F*F*\"#]F
*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "Bb := randpoly(x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,.*$)%\"xG\"\"&\"\"\"\"#z*&\"#cF
*)F(\"\"%F*F**&\"#\\F*)F(\"\"$F*F**&\"#jF*)F(\"\"#F*F**&\"#dF*F(F*F*\"
#f!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "C := randpoly(x,
degree=3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG,**$)%\"xG\"\"$\"
\"\"\"#X*&\"\")F*)F(\"\"#F*!\"\"*&\"#$*F*F(F*F0\"##*F*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A := expand(Ab*C);" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6#>%\"AG,4*$)%\"xG\"\")\"\"\"!%DQ*&\"%&z\"F*)F(\"\"(
F*!\"\"*&\"%!o'F*)F(\"\"'F*F**&\"%%)RF*)F(\"\"&F*F0*&\"%EIF*)F(\"\"%F*
F**&\"%D8F*)F(\"\"$F*F**&\"&TE\"F*)F(\"\"#F*F0*&\"%uUF*F(F*F*\"%+YF*" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "B := expand(Bb*C);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"BG,4*$)%\"xG\"\")\"\"\"\"%bN*&\"%)
)=F*)F(\"\"(F*F**&\"%!f&F*)F(\"\"'F*!\"\"*&\"%.XF*)F(\"\"&F*F**&\"%cEF
*)F(\"\"%F*F**&\"%iWF*)F(\"\"$F*F4*&\"$n*F*)F(\"\"#F*F**&\"&J2\"F*F(F*
F*\"%GaF4" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "gcd(A,B);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#,**$)%\"xG\"\"$\"\"\"\"#X*&\"\")F()F&
\"\"#F(!\"\"*&\"#$*F(F&F(F.\"##*F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "G := mygcd(A,B,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6
#>%\"GG,*#\"Lv&eqDFW8*[$ph6ThIURS91%)\"J%)f'oBW#Q(GxqqO$QXK;m'4?\"\"\"
*&#\"MD62UsAG[I+%H#Ri%HRbN!oW;\"JORYZp(H&\\\"4$GoM`\")Hlk'Q!)F))%\"xG
\"\"$F)F)*&#\"KDg/ssiSMn')R4$ewJXBX[l$\"J#*HV=@7pV'QNNo\"pA;3L[+\"F)*$
)F/\"\"#F)F)!\"\"*&#\"MD.G+jLy*H1wSxAAa9\"og+*R$F-F)F/F)F8" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "simplify(G/C);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6##\"KDg/ssiSMn')R4$ewJXBX[l$\"JORYZp(H&\\\"4$GoM`\")Hlk'
Q!)" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 46 "Question 4 (Probability \+
of Relative Primality)" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 12 "Integer
 Case" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Samples := 10000;" 
}}{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "for n from 5 by 5 to 20 do" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "  N := 10^n:  count := 0:  print(\"
N = \", N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "  myrand := rand(1..
N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "  for i from 1 to Samples do
" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "    a := myrand():  b := myrand
():" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "    c := igcd(a,b):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "    if (c = 1) then count := count \+
+ 1:  fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od:" }}{PARA 0 "> " 
0 "" {MPLTEXT 1 0 69 "  prob := count/Samples:  print(\"prob that gcd \+
= 1 is \",evalf(prob)):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "od:" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(SamplesG\"&++\"" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6$Q%N~=~6\"\"'++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q6
prob~that~gcd~=~1~is~6\"$\"++++Ph!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "
6$Q%N~=~6\"\",+++++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q6prob~that~g
cd~=~1~is~6\"$\"++++%4'!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q%N~=~6
\"\"1+++++++5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q6prob~that~gcd~=~1~i
s~6\"$\"++++-h!#5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q%N~=~6\"\"6+++++
+++++\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q6prob~that~gcd~=~1~is~6\"$
\"++++!>'!#5" }}}{PARA 0 "" 0 "" {TEXT -1 84 "It can be shown that if \+
a and b are integers chosen at random, the probability that " }
{XPPEDIT 18 0 "gcd(a,b) = 1" "6#/-%$gcdG6$%\"aG%\"bG\"\"\"" }{TEXT -1 
13 " is equal to " }{XPPEDIT 18 0 "6/Pi^2" "6#*&\"\"'\"\"\"*$%#PiG\"\"
#!\"\"" }{TEXT -1 33 " which is approximately equal to " }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "evalf(6/Pi^2);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+;5Fzg!#5" }}}{PARA 0 "" 0 "" {TEXT -1 85 "Assume th
at there is a well defined probability called p.  Then the probability
 that " }{XPPEDIT 18 0 "gcd(a,b) = d" "6#/-%$gcdG6$%\"aG%\"bG%\"dG" }
{TEXT -1 13 " is equal to " }{XPPEDIT 18 0 "p*(1/d)^2" "6#*&%\"pG\"\"
\"*$*&F%F%%\"dG!\"\"\"\"#F%" }{TEXT -1 76 ".  Summing these probabilit
ies over all d leads to the following equation:  " }{XPPEDIT 18 0 "1 =
 Sum(p/(d^2),d = 1 .. infinity);" "6#/\"\"\"-%$SumG6$*&%\"pGF$*$%\"dG
\"\"#!\"\"/F+;F$%)infinityG" }{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 52 "Sum(p/d^2,d=1..infinity) = sum(p/d^2,d=1..infinity)
;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*&%\"pG\"\"\"*$)%\"dG\"
\"#F)!\"\"/F,;F)%)infinityG,$*&F(F))%#PiGF-F)#F)\"\"'" }}}{PARA 0 "" 
0 "" {TEXT -1 19 "which implies that " }{XPPEDIT 18 0 "p = 6/Pi^2" "6#
/%\"pG*&\"\"'\"\"\"*$%#PiG\"\"#!\"\"" }{TEXT -1 1 "." }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 15 "Polynomial Case" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 16 "Samples := 1000;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 26 "for n from 5 by 5 to 20 do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 
"  count := 0:  print(\"deg = \", n):  " }}{PARA 0 "> " 0 "" {MPLTEXT 
1 0 28 "  for i from 1 to Samples do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 
0 70 "    a := randpoly(x,dense,degree=n):  b := randpoly(x,dense,degr
ee=n):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "    c := gcd(a,b):" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "    if (degree(c,x) = 0) then count
 := count + 1:  fi:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od:" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "  prob := count/Samples:  print(\"p
rob that deg(gcd) = 0 is \",evalf(prob)):" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 3 "od:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%(SamplesG\"%+
5" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$Q'deg~=~6\"\"\"&" }}{PARA 11 "" 
1 "" {XPPMATH 20 "6$Q;prob~that~deg(gcd)~=~0~is~6\"$\"\"\"\"\"!" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6$Q'deg~=~6\"\"#5" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$Q;prob~that~deg(gcd)~=~0~is~6\"$\"\"\"\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$Q'deg~=~6\"\"#:" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$Q;prob~that~deg(gcd)~=~0~is~6\"$\"\"\"\"\"!" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6$Q'deg~=~6\"\"#?" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6$Q;prob~that~deg(gcd)~=~0~is~6\"$\"++++!***!#5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 23 "Question 5 (Partitions)" }}{PARA 0 "" 0 "" {TEXT -1 24 
"See Partition worksheet." }}}}{MARK "8" 0 }{VIEWOPTS 1 1 0 1 1 1803 
1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }