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mes" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 1 
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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 9 "Lecture 4" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 38 "Maple 6 and 7 versus Previous Versions" }
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 50 "Let's discuss once more some diff
erences between  " }{TEXT 256 9 "Maple  6 " }{TEXT 398 3 "and" }{TEXT 
399 3 " 7 " }{TEXT 259 23 "and previous versions. " }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "In previous versions of M
aple :\n\ninstead of  " }{TEXT 345 6 "end if" }{TEXT -1 15 " one must \+
use  " }{TEXT 346 2 "fi" }{TEXT -1 14 ",\ninstead of  " }{TEXT 347 3 "
end" }{TEXT -1 1 " " }{TEXT 348 4 "proc" }{TEXT -1 15 " one must use  \+
" }{TEXT 349 3 "end" }{TEXT -1 14 ",\ninstead of  " }{TEXT 350 6 "end \+
do" }{TEXT -1 15 " one must use  " }{TEXT 351 2 "od" }{TEXT -1 1 "." }
}{PARA 0 "" 0 "" {TEXT -1 12 "instead of  " }{TEXT 352 16 "return stat
ement" }{TEXT -1 14 " one must use " }{TEXT 353 17 "RETURN(statement)
" }{TEXT -1 3 ".\n\n" }{TEXT 359 7 "Maple 6" }{TEXT -1 6 " and  " }
{TEXT 400 1 "7" }{TEXT -1 52 " do, however, allow one to  use the old \+
short forms " }{TEXT 354 2 "od" }{TEXT -1 2 ", " }{TEXT 355 2 "fi" }
{TEXT -1 5 " and " }{TEXT 356 3 "end" }{TEXT -1 10 ". In fact " }
{TEXT 357 3 "end" }{TEXT -1 65 " alone will work for any of these so-c
alled delimiters.  Thus in " }{TEXT 257 7 "Maple 6" }{TEXT -1 4 " or \+
" }{TEXT 401 1 "7" }{TEXT -1 24 " we would usually write:" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 96 "f:=p
roc(x) \n  if x > 0 then \n   return sin(x); \n  else \n   return cos(
x); \n  end if; \nend proc:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 7 "B
ut in " }{TEXT 258 7 "Maple V" }{TEXT -1 47 " we MUST write the same  \+
procedure as follows:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "
f:=proc(x) \n  if x > 0 then \n   RETURN(sin(x)); \n  else \n   RETURN
(cos(x)); \n  fi; \nend:\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The
 latter will also work in " }{TEXT 358 7 "Maple 6" }{TEXT -1 5 " and \+
" }{TEXT 402 7 "Maple 7" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }}{PARA 0 "" 0 "" {TEXT -1 35 "Another major inovation began with
 " }{TEXT 405 7 "Maple 6" }{TEXT -1 18 " and continued in " }{TEXT 
406 7 "Maple 7" }{TEXT -1 20 " is the new package " }{TEXT 403 15 "Lin
earAlgebra. " }{TEXT -1 21 "In previous versions " }{TEXT 407 5 "Maple
" }{TEXT -1 22 " has a package called " }{TEXT 404 6 "linalg" }{TEXT 
-1 515 " which was the only way to deal with linear algebra problems s
uch as computing rank, determinants, jordan canonical forms, etc.  The
 difference between these two ways of doing linear algebra in Maple 6 \+
and 7 will be discussed later.  [If you are impatient to get started o
n linear algebra you may type with(linalg); or with(LinearAlgebra); an
d this will produce a list of procedures that may be used after each o
f these commands is executed. Use ?command to see how command works fo
r any command in either package.]" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 27 "The Concate
nation Operator " }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Another differ
ence is the " }{TEXT 260 22 "concatenation operator" }{TEXT 268 2 ". \+
" }{TEXT -1 19 " In Maple 6 and 7  " }{TEXT 360 6 "x || y" }{TEXT -1 
42 " gives the concatenation of strings x and " }{TEXT 361 1 "y" }
{TEXT -1 48 ". In previous versions of Maple it was given by " }{TEXT 
266 3 "x.y" }{TEXT -1 92 ".  Concatenation is useful for forming a lar
ge number of variables. Here's a simple example:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 37 "for i from 1 to 5 do\n  x||i:=i^2;\nod;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "We have created the four variables
 x1,x2,x3,x4 and assigned to xi the value " }{XPPEDIT 18 0 "i^2;" "6#*
$%\"iG\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "x1,x2,x3,x4;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 269 78 "The Mapl
e help page for || states that the use of this operator is discourage.
" }{TEXT -1 124 "  In Lecture 2 we discussed how to use indexed variab
les. In the next section we say a little more about indexed variables.
 " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 8 "Usually " }{TEXT 408 9 "cat(x,y) " }{TEXT -1 14 "is the sa
me as" }{TEXT 409 7 " x || y" }{TEXT -1 16 ". The  function " }{TEXT 
267 3 "cat" }{TEXT -1 124 " may also be useful to get rid of extra com
mas when printing combinations of text and integers, as in the followi
ng example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 123 "for i from 1 to 4 do\n#prin
t(`If i = `, i, ` then i^2 = `, i^2);\nprint(cat(`If i = `, i, ` then \+
i^2 = `, i^2, `.`));\nend do;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 13 "But actually " }{TEXT 411 6 "printf" }
{TEXT -1 56 " which we will discuss later is better for such matters.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
14 "Note that the " }{TEXT 410 3 "cat" }{TEXT -1 72 " operator will no
t work nicely with floating point numbers. For example:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x||1.0:=3;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 14 "cat(x,1.0):=3;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 20 "But this works fine:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
9 "x||1:=3; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "But note that if \+
we use " }{TEXT 412 3 "cat" }{TEXT -1 25 " we get an error message:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "restart:\ncat(x,1):=3; " }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Check out " }{TEXT 362 4 "?cat" 
}{TEXT -1 19 " for more examples." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 17 "Indexed Var
iables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 56 "An alternative way to create several variable is to use \+
" }{TEXT 261 17 "indexed variables" }{TEXT -1 34 ". These are variable
s of the form " }{TEXT 262 4 "x[i]" }{TEXT -1 25 " which Maple exhibit
s as " }{XPPEDIT 263 0 "x[i];" "6#&%\"xG6#%\"iG" }{TEXT 264 2 " ." }
{TEXT -1 15 "  That is, the " }{TEXT 265 1 "i" }{TEXT -1 150 " becomes
 a subscript.  This is the same in Maple 6 and 7 as in previous versio
ns of Maple.  [Althought the notation is similar to that for arrays, a
n " }{TEXT 365 16 "indexed variable" }{TEXT -1 11 " is not an " }
{TEXT 364 5 "array" }{TEXT -1 80 ". We will discuss the distinction la
ter. Indexed variables are actually of type " }{TEXT 363 5 "table" }
{TEXT -1 2 ".]" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "for i from 1 to 5 do\n  x
[i]:=i^2;\nod;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "One may also us
e doubly, triply, etc., subscribted variables as in the following exam
ples:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "A[2,5]:=2;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "z[1,2,2,3]:=x^2 + sin(x);" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 11 "Sequences, " }{TEXT 283 25 "(max,
 min, seq, ithprime)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 27 "Perhaps the most important " }{TEXT 366 5 "type
s" }{TEXT -1 4 " or " }{TEXT 367 7 "classes" }{TEXT -1 25 " of objects
 in Maple are " }{TEXT 274 16 "sequences, sets " }{TEXT -1 4 "and " }
{TEXT 275 5 "lists" }{TEXT -1 46 ". It is vital to understand these th
oroughly. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 31 "We start with some examples of " }{TEXT 270 9 "sequences
" }{TEXT -1 4 ": A " }{TEXT 276 8 "sequence" }{TEXT -1 3 "  (" }{TEXT 
341 3 "aka" }{TEXT -1 4 " an " }{TEXT 339 10 "expression" }{TEXT -1 1 
" " }{TEXT 340 8 "sequence" }{TEXT -1 151 ") is just a bunch of expres
sions separated by commas: The expressions may be numbers, variables, \+
functions, etc...--and they need not all be different." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "sequence1:=1,2,1,A,3,x,sin(t);" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 3 "If " }{TEXT 277 9 "sequence1" }{TEXT -1 20 " is a s
equence then " }{TEXT 278 12 "sequence1[i]" }{TEXT -1 47 " is the i-th
 term of the sequence, for example:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "sequence1[5];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "sequence1[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "se
quence2:=55,7,33,-1,4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 90 "The max
imum and minimum of a sequence or real numbers is obtained by use of t
he functions " }{TEXT 279 3 "max" }{TEXT -1 5 " and " }{TEXT 280 3 "mi
n" }{TEXT -1 31 ", as in the following examples:" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 31 "max(sequence2);\nmin(sequence2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "N
ote that the solution to a " }{TEXT 368 5 "solve" }{TEXT -1 51 " comma
nd as in the following example is given as a " }{TEXT 369 8 "sequence
" }{TEXT -1 58 ". We name it so we can manipulate the different soluti
ons:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "p:=x^3-60*x^2 + 110
0*x - 6000;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "sequence3:=s
olve(p=0,x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sequence3[1
]; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sequence3[2];" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sequence3[3];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT 413 13 "The function " }{TEXT 273 3 "seq" }
{TEXT 414 187 " is a very important way to form sequences. It can be u
sed to do many things that can be done with do loops and when that's p
ossible should be done. Generally this is a faster way to go. " }
{TEXT -1 96 "For example, if we wish to apply some function f to the n
umbers -1, 2, 3, . . . , 10 we can use " }{TEXT 281 3 "seq" }{TEXT -1 
12 " as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "sequenc
e4:=seq(f(i),i=-1..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 16 "For ex
ample, if " }{XPPEDIT 18 0 "f(x) = x^2;" "6#/-%\"fG6#%\"xG*$F'\"\"#" }
{TEXT -1 9 " we have:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "se
quence4:=seq(i^2,i=-1..10);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "W
e could also form this sequence with the following do loop.  We use s \+
to indicate the sequence and we start by setting s to be the NULL sequ
ence:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "s:=NULL:\nfor i fr
om -1 to 10 do\n  s:=s,i^2:\nend do:\nsequenc4:=s;" }}}{EXCHG {PARA 0 
"" 0 "" {TEXT -1 47 "In Maple ithprime(i) is the i-th prime number. " 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "ithprime(1);\nithprime(2)
;\nithprime(3); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "We can get a \+
sequence consisting of the first  20 primes by the following method:" 
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 36 "sequence5:=seq(ithprime(k),k=1..20);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 8 "One can " }{TEXT 272 11 "concatenate" }{TEXT -1 23 " seq
uences as follows:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "seq
uence4;\nsequence5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "sequ
ence6:=sequence4,sequence5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "A \+
simpler example would be:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "seqA:=a,b,c;\nseqB:=x,y,z,w;\nseqC:=seqA,seqB,seqA;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "\nNow consider the following sequence:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "sequence7:=a,b,c,d,e,f,g,h
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "One can select more than one
 element of a sequence at a time. For example:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "sequence7[2..4];" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 42 "To get the first element on the right use:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "sequence7[-1];" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 45 "To get the second element from the right use:" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 14 "sequence7[-2];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "We can us
e this to reverse a sequence:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 18 "S1:=seq(i,i=1..7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 
"S2:=seq(S1[-i],i=1..7);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The e
mpty sequence is given by " }{TEXT 271 8 "NULL -- " }{TEXT 282 23 "as \+
we aready say above." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "S3:
=NULL;\n\nS4:=x,x,S3,y;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 5 "Sets " }{TEXT 284 47 "(union
, intersect, subset, nops, minus, member)" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 17 "One can obtain a " }{TEXT 289 3 "set" }{TEXT -1 8 " from \+
a " }{TEXT 290 8 "sequence" }{TEXT -1 109 " by placing braces around i
t. We illustrate this below. We also illustrate some basic set operati
ons such as " }{TEXT 287 48 "union, intersect, minus, subset, nops and
 member" }{TEXT -1 2 ". " }{TEXT 288 79 "Note, however, that the user \+
has no control over the order of elements of a set" }{TEXT -1 89 ". To
 control the order of elements we must use sequences or lists --as we \+
will see below:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "set1:=\{4,3,3,2,1\};" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "set2:=\{4,4,5,5,6\};" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "set3:=set1 union set2;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "set4:=set1 intersect set2;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "set5:=set1 minus set2;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "set5 subset set3;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "set5 subset set4;" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "Let's form a sequence and then tur
n the sequence into a set." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
21 "Z[10]:=seq(i,i=0..9);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "set6:=\{Z[10]\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Here's an
other example." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq(x^j-1
,j=1..5);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "set7:=\{seq(x^
j-1,j=1..5)\};" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 
292 9 "empty set" }{TEXT -1 19 " is denoted by \{\};\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "\{1,2,3\} intersect \{4,5,6\};" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 285 4 "nops" }{TEXT -1 49 " is used to fi
nd the number of elements in a set:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "nops(set7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "nops(\{\});" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "Note that nop
s will not work for " }{TEXT 291 9 "sequences" }{TEXT -1 1 "!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "nops(1,2,3,4);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 4 "But " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "nops(\{1,2,3,4\});" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "t:=1,1,3,4,4;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 
"Note that the following does not give the number of elements in the s
equence t" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(t);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "But we can convert to to a list an
d get the number of elements in the list as follows:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "nops([t]);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 286 "If S is a set then S[i] is the i-th element of the set, \+
but you have no way of knowing what order Maple assigns to the element
s of the set S. And the order may change in a single worksheet session
. So do NOT make any assumptions about the order even if you see the o
rder on the monitor." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "set7
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "for i from 1 to nops(s
et7) do i,set7[i]; end do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 286 10 "Me
mbership" }{TEXT -1 31 " in a set is tested as follows:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "set7;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "member(x^2-1,set7);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "member(x,set7);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 61 "if member(x,set7) = false then set7:=set7 union \{x\}
; end if;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 85 "Recall the example in the previous section on the use o
f solve. Let's repeat it here:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "p:=x^3-60*x^2 + 1100*x - 6000;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "sequence3:=solve(p=0,x);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "sequence3[1]; " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
112 "Now we make a small change in the solve command. Namely we replac
e x  by \{x\}: See the difference in the output. " }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 26 "sequence4:=solve(p=0,\{x\});" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "sequence4[1];" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 64 "This will be convenient for some purposes as we sh
all see later." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 6 "Lists " }{TEXT 293 43 "(sort,  nop
s,  op,  concatenation of lists)" }}{EXCHG {PARA 0 "" 0 "" {TEXT 297 
5 "Lists" }{TEXT 300 1 " " }{TEXT -1 68 "may be formed by placing a se
quence in square brackets, as follows: " }{TEXT 299 36 "In this case o
rder is all important." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "L
ist1:=[4,0,2,2,3,1,4];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 301 5 "sort " }{TEXT -1 58 " is useful for rearranging a list in
 non-decreasing order." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "s
ort(List1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 296 4 "nops" }{TEXT -1 
43 "(List1) gives the length of the list List1:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "N:=nops(List1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 22 "Compare the following:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "nops([1,3,3,3,3]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "nops(\{1,3,3,3,3\});" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 20 "The i-th element in " }{TEXT 302 5 "List1" }{TEXT -1 13 "
 is given by " }{TEXT 303 9 "List1[i]:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 40 "for i from 1 to N do i,List1[i]; end do;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "List1[2..4];" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 28 "List1;\nList1[-1];\nList1[-2];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 3 "To " }{TEXT 298 11 "concatenate" }{TEXT 
-1 81 " lists we need to first recover the underlying sequence by appl
ying the function " }{TEXT 295 2 "op" }{TEXT -1 49 ". This also works \+
for sets, as we now illustrate:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "List1; \nop(List1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "set1;\nop(set1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "Lis
t1;\nList2:=[x,y,z];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "Lis
t3:=[op(List1),op(List2)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
35 "List4:=[op(List3),`a new element`];" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 4 "The " }{TEXT 294 10 "empty list" }{TEXT -1 20 " is denoted
 by [ ] ;" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 97 "Let's make a list of the primes numbers up to 100: We wil
l put them in a list we will call List5:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 89 "List5:=[]:\nfor i from 1 to 100 do\nif isprime(i) the
n List5:=[op(List5),i]; fi;\nod;\nList5;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 46 "Another way to create this list is as follows:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 76 "seq5:=NULL:\nfor i from 1 to 100 do\nif ispr
ime(i) then seq5:=seq5,i; fi;\nod;\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "seq5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Li
st5:=[seq5];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 194 "As mentioned abo
ve one can also select elements of a list in reverse order. For exampl
e, the last element of the list List is found by List[-1]. The second \+
from last is given by List[-2], etc..." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 63 "List:=[seq(i, i=1..10)];\na:=List[-1];\nb:=List[-2];
\nc:=List[-3];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "We can obtain t
he reverse of the list " }{TEXT 304 4 "List" }{TEXT -1 18 " above as f
ollows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "revList:=[seq(Li
st[-i],i=1..10)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 48 "Converting from list to set and f
rom set to list" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "One can convert
 lists to sets and vice versa by using the function " }{TEXT 305 7 "co
nvert" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "res
tart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "L:=[1,1,1,2,3];\nA
:=convert(L, set);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "B:=\{
1,2,3,4,5\};\nLL:=convert(B,list);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 20 "We can convert from " }{TEXT 306 3 "set" }{TEXT -1 5 " and " }
{TEXT 307 4 "list" }{TEXT -1 4 " to " }{TEXT 308 8 "sequence" }{TEXT 
-1 11 " by use of " }{TEXT 309 2 "op" }{TEXT -1 16 " as shown above:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "op(L);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 6 "op(B);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 42 "Right Click
ing (Option Clicking with  Mac)" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 
"With Windows " }{TEXT 415 11 "right click" }{TEXT -1 182 " on equatio
n, matrix, etc. to obtain a list of commands that one may apply to tha
t object. Note the distinction between clicking on the input and click
ing on the output. On a Mac do " }{TEXT 416 14 "option + click" }
{TEXT -1 25 " instead of right click.\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "sin(x) + x^2 + 1;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}
{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 " round,  floor, ceil" }}{PARA 0 "
" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 3 "" 0 "" {TEXT 370 5 "round" }
{TEXT 30 2 " (" }{TEXT 376 1 "x" }{TEXT 30 3 ") -" }{TEXT 375 23 " the
 nearest integer to" }{TEXT 30 1 " " }{TEXT 378 1 "x" }}{PARA 3 "" 0 "
" {TEXT 371 9 "floor(x) " }{TEXT 30 3 " - " }{TEXT 373 20 "the largest
 integer " }{XPPEDIT 18 0 "n <= x;" "6#1%\"nG%\"xG" }{TEXT -1 1 "." }}
{PARA 3 "" 0 "" {TEXT 372 4 "ceil" }{TEXT 30 1 "(" }{TEXT 377 1 "x" }
{TEXT 30 4 ") - " }{TEXT 374 33 "the smallest integer n such that " }
{XPPEDIT 18 0 "x <= n;" "6#1%\"xG%\"nG" }{TEXT -1 1 "." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 23 "Here are some exampl
es:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 41 "L:=[-3.4,-3.6, 3.6,3.5, 3.4, 3.5, 3, -3]:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "for x in L do\nprint(`if x = `, x \+
, `   round(x) = `, round(x),  `   floor(x) = `, floor(x), `   ceil(x)
 = `, ceil(x));\nend do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 76 "Somet
imes one must apply evalf before invoking these functions. For example
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "floor(sin(10^8));" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Note that floor cannot evaluate th
e following." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "floor(sin(1
0^10));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 42 "We must first apply ev
alf to make it work." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "flo
or(evalf(sin(10^10)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "f
loor(sqrt(10^9 + 1));" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "However,
 compare the following:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "
floor(sqrt(10^10 + 1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "
floor(evalf(sqrt(10^10+1)));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "floor(evalf(sqrt(10^100 + 1)));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 14 "The Digits \+
of " }{XPPEDIT 18 0 "pi;" "6#%#piG" }{TEXT -1 1 " " }}{EXCHG {PARA 
256 "" 0 "" {TEXT 317 140 "Here we write a procedure to explore the qu
estion of whether or not the digits 0 to 9 are uniformly distributed a
mong the first N digits of " }{XPPEDIT 319 0 "pi;" "6#%#piG" }{TEXT 
-1 0 "" }{TEXT 318 1 "." }{TEXT -1 1 "\n" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 16 "Use the command " }{TEXT 310 13 "?convert/base" }{TEXT 
-1 37 " to find out more about the function " }{TEXT 311 7 "convert" }
{TEXT -1 25 " used below.  Actually,  " }{TEXT 312 7 "convert" }{TEXT 
-1 60 " has many uses in maple, as we will see later in the course." }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 313 6 "
Digits" }{TEXT -1 129 " is an \"environmental\" variable that tells ho
w many digits are carried in floating point operations. The default is
 10 as we see:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "Digits;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 79 "The following sets the number of digit carried by floatin
g point numbers to 30:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "D
igits:=30;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(Pi);" }
}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "Now we set the number of digits \+
carried by floating point numbers to 3. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "n:=3:\nDigits:=3; " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "X:=evalf(Pi);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "10^(n-1)*X;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 22 "X:=floor(10^(n-1)*X); " }{TEXT -1 51 "This con
verts X to an integer with the same digits." }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 24 "X:=convert(X,base, 10); " }{TEXT -1 178 "Note the \+
digits appear in reversed order, but they are now in a list which make
s them easier to handle. There are alternative ways to get at the digi
ts that we may discuss later." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "
Let's do this again with n = 100 instead of just 3:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "n:=100:\n
Digits:=n;\nX:=evalf(Pi,n);\nX:=floor(10^(n-1)*X);\nX:=convert(X,base,
 10);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "nops(X);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 85 "Now we shall count the number of t
imes each digit appears in the first 100 digits of " }{XPPEDIT 18 0 "P
i;" "6#%#PiG" }{TEXT -1 142 ". We will let c[0] be the number of times
 0 appears, c[1] be the number of times 1 appears, etc... First we ini
tialize 10 variables c[i] to 0." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 33 "for i from 0 to 9 do c[i]:=0; od:" }}{PARA 0 "" 0 "" {TEXT -1 
1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "for k from 1 to n d
o\n  j:=X[k]; \n  c[j]:=c[j] + 1;\nod:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "Now we exhibi
t the values of the c[i]'s in a sequence as follows:" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "[seq(c[j],j=0..9)];" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 78 "Another way to do the countinng is as follows: Fir
st set the c[i]'s back to 0:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 33 "for i from 0 to 9 do c[i]:=0; od:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "for k from 1 to n do \n  c[X[k]]:=c[X[k]] + 1;\nod:" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "[seq(c[j],j=0..9)];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 104 "Now we construct a procedure to c
ompute the number of times each digit appears in the first n digits of
 " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 113 ". Before constructing
 the procedure we do it step by step for a small value of n so we can \+
see what is happening." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "re
start:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "n:=40;\nDigits:
=n;\nX:=evalf(Pi,n);\nX:=floor(10^(n-1)*X);\nX:=convert(X,base, 10);\n
  \n \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for i from 0 to \+
9 do c[i]:=0; end do:\n  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
60 "for k from 1 to n do\n   j:=X[k]; \n   c[j]:=c[j] + 1;\nend do:" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "[seq(c[j],j=0..9)];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Now we make this into a procedure:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 252 "numdigits:=proc(n)\nlo
cal X,c,i,k,j;\n  Digits:=n:\n  X:=evalf(Pi,n):\n  X:=floor(10^(n-1)*X
):\n  X:=convert(X,base, 10):\n  for i from 0 to 9 do c[i]:=0; end do:
\n  for k from 1 to n do\n   j:=X[k]; \n   c[j]:=c[j] + 1;\n  end do:
\n [seq(c[j],j=0..9)];\nend proc:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "numdigits(100);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "numdigits(1000);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "numdigits(10000);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}{PARA 0 "" 0 "" {TEXT -1 15 "At the webpage " }}{PARA 0 "" 0 "" 
{TEXT -1 3 "   " }}{PARA 259 "" 0 "" {TEXT 315 59 "http://www.cs.ruu.n
l/wais/html/na-dir/sci-math-faq/pi.html " }{TEXT -1 0 "" }{TEXT 314 1 
" " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 93 "you
 can find out how mathematicians have been able to compute the first 5
1 billion digits of " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 125 ". \+
Maple will not be able to do this using the method we have just descri
bed. More sophisticated techniques are necessary. At " }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {TEXT 316 42 "http://mathworld.wo
lfram.com/PiDigits.html" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}{PARA 0 "" 0 "" {TEXT -1 89 "you will also see some data on the d
istribution of the digits 0-9 in the known digits of " }{XPPEDIT 18 0 
"pi;" "6#%#piG" }{TEXT -1 113 ". It appears to be true that the digits
 appear with approximately equal frequency--but this has not been prov
ed. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "    " }}}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 41 "Assignment 4 - Due Wednesday, February 13
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 328 
10 "Problem 1." }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 56 "(a)  Prove that every prime p greater tha
n 2 satisfiesn " }{TEXT 327 11 "p mod 8 = r" }{TEXT -1 127 " where r i
s 1, 3, 5, or 7. Thus primes > 2 may be divided into four classes depe
nding on their remainders when divided by  8.  " }{TEXT 344 99 "No Map
le calculations are necessary here, but write out your proof using Map
le as a word processor." }{TEXT -1 2 "  " }{TEXT 321 6 "HINT: " }
{TEXT -1 24 "Can p > 2 be a prime if " }{TEXT 320 12 "p mod 8  = r" }
{TEXT -1 65 "  when r = 0, 2, 4 or 6?  (Take each case separately.) No
te that " }{TEXT 330 12 "p mod 8 =  r" }{TEXT -1 12 " means that " }
{TEXT 331 10 "p = 8q + r" }{TEXT -1 7 " where " }{TEXT 332 30 "r = 0, \+
1, 2, 3, 4, 5, 6, or 7." }{TEXT -1 3 ". \n" }}{PARA 0 "" 0 "" {TEXT 
-1 9 "(b)  Set " }{TEXT 333 6 "N:=100" }{TEXT -1 2 ". " }}{PARA 0 "" 
0 "" {TEXT -1 100 "     (i) Form four lists L[r] for r in  [1,3,5,7] w
here L[r] contains all primes 2< p < N such that " }{TEXT 329 11 "p mo
d 8 = r" }{TEXT -1 3 " . " }}{PARA 0 "" 0 "" {TEXT -1 62 "    (ii) Usi
ng Maple count the number of primes in each list.\n" }{TEXT 322 10 "Pr
oblem 2." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
88 "In the notation of Problem 1, make a procedure with input = arbitr
ary  positive integer " }{TEXT 334 1 "N" }{TEXT -1 23 " and output = t
he list " }{TEXT 326 48 "[nops(L[1]), nops(L[3]), nops(L[5]), nops(L[7
])]" }{TEXT -1 2 ". " }{TEXT 325 67 "Do some experiments to see for wh
ich (if any) r, L[r] is largest. ." }{TEXT -1 15 " [According to " }
{TEXT 323 56 "Dirichlet's Theorem on Primes in Arithmetic Progressions
" }{TEXT -1 13 " the numbers " }{TEXT 324 10 "nops(L[r])" }{TEXT -1 
38 " will be asymptotically the same.]  \n\n" }{TEXT 335 3 "(a)" }
{TEXT -1 29 " Test your procedure on some " }{TEXT 336 5 "small" }
{TEXT -1 13 " values of N " }{TEXT 337 52 "where you can check by hand
 if the answer is correct" }{TEXT -1 3 ".  " }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT 338 3 "(b)" }{TEXT -1 64 " Then try it
 on some larger values of N. Try, for example,  N = " }{XPPEDIT 18 0 "
10^i;" "6#)\"#5%\"iG" }{TEXT -1 16 ",  i = 2,3,4, 5." }}}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 342 10 "Problem 3." }
{TEXT -1 3 " \n " }}{PARA 0 "" 0 "" {TEXT -1 82 "(a) Write a procedure
 whose input is a positive integer n and whose output is the " }{TEXT 
384 4 "set " }{TEXT -1 74 "of all squares of positive integers less th
an or equal to n.  Write it in " }{TEXT 385 18 "two different ways" }
{TEXT -1 12 ". (i) using " }{TEXT 382 3 "seq" }{TEXT -1 18 " and (ii) \+
using a " }{TEXT 383 7 "do loop" }{TEXT -1 54 ". Show output for n = 4
, 10, 20, 100 and 1000. [Hint: " }{XPPEDIT 18 0 "i^2 <= n;" "6#1*$%\"i
G\"\"#%\"nG" }{TEXT -1 18 " is equivalent to " }{XPPEDIT 18 0 "i <= sq
rt(n);" "6#1%\"iG-%%sqrtG6#%\"nG" }{TEXT -1 3 ".]\n" }}{PARA 0 "" 0 "
" {TEXT -1 22 "(b) Write a procedure " }{TEXT 388 6 "genseq" }{TEXT 
-1 55 " whose input is the integer n. and whose output is the " }
{TEXT 386 4 "list" }{TEXT -1 22 " of all integers  7*i " }{TEXT 379 4 
" mod" }{TEXT -1 112 " 11 for i from 1 to n.  Note that each of these \+
integers will be in the set \{0, 1, 2, . . .  , 10\}. Write it in " }
{TEXT 387 18 "two different ways" }{TEXT -1 11 " (i) using " }{TEXT 
380 3 "seq" }{TEXT -1 18 " and (ii) using a " }{TEXT 381 7 "do loop" }
{TEXT -1 42 ". Test each procedures for n = 11 and 110." }}{PARA 0 "" 
0 "" {TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT 343 10 "Problem 4." }{TEXT -1 3 "  \n" }}{PARA 0 "" 0 "" {TEXT 
-1 65 "(a) Write a procedure that will take as input a positive intege
r " }{TEXT 391 1 "n" }{TEXT -1 30 " and whose output will be the " }
{TEXT 397 4 "list" }{TEXT -1 1 " " }{TEXT 392 27 "[c[0], c[1], . . . ,
 c[10]]" }{TEXT -1 7 " where " }{TEXT 393 4 "c[i]" }{TEXT -1 51 " is t
he number of times that i appears in the list " }{TEXT 389 9 "genseq(n
)" }{TEXT -1 7 " where " }{TEXT 390 6 "genseq" }{TEXT -1 183 " is as d
escribed in Problem 3.  Test the procedure for n =  100, 1000 and 1000
0. Do the numbers 0, 1, . . ., 10 appear to be equally distributed?\n
\n(b) In the procedure genseq replace " }{TEXT 394 10 "7*i mod 11" }
{TEXT -1 4 " by " }{TEXT 395 10 "8*i mod 22" }{TEXT -1 48 ". Then repe
at the instructions in part (a) with " }{TEXT 396 69 "[c[0], c[1], . .
 . , c[10]] replaced by [c[0], c[1], . . . , c[21]]. " }{TEXT -1 65 "D
o the numbers 0, 1, . . ., 21 appear to be equally distributed? " }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 19 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "
" {TEXT -1 1 "\n" }}}}{MARK "13 0 0" 0 }{VIEWOPTS 0 0 0 1 1 1803 1 1 
1 1 }{PAGENUMBERS 1 1 2 33 1 1 }