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imes" 1 18 0 0 0 1 2 1 1 2 2 2 1 1 1 1 }3 1 0 0 12 12 1 0 1 0 2 2 19 
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imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }
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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 9 "Lecture 4" }}}{SECT 1 
{PARA 3 "" 0 "" {TEXT -1 33 " Deletion and Undoing a Deletion." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 151 "Usually \+
selecting something with the cursor and hitting the delete key will de
lete things.  But to delete some things I have found that  I need to u
se " }{TEXT 333 6 "Delete" }{TEXT -1 1 " " }{TEXT 334 9 "Paragraph" }
{TEXT -1 20 " from the pull down " }{TEXT 335 4 "Edit" }{TEXT -1 73 " \+
menu. Then the paragraph where your cursor is located will be deleted.
  " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 120 "Af
ter deleting or pasting something,  if you change your mind before doi
ng anything else,  you can go to the top of the " }{TEXT 336 4 "Edit" 
}{TEXT -1 10 " menu and " }{TEXT 337 4 "undo" }{TEXT -1 23 " the delet
ion or paste." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 239 "Also note that if you click the paragraph symbol at the \+
right end of the menu bar it will sometimes help to see what you shoul
d delete. This is especially true when dealing with text. To get back \+
to normal click the paragraph symbol again." }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 " Pseudo Random \+
Number Generators" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "
" 0 "" {TEXT 315 10 "rand(a..b)" }{TEXT -1 49 " is a procedure with no
 arguments that creates a " }{TEXT 317 23 "(pseudo) random integer" }
{TEXT -1 13 " in the range" }{TEXT 316 8 " a to b." }{TEXT -1 233 " To
 call the procedure one needs to add extra parenthensizes as in the fo
llowing examples. If you don't add the extra () after rand(a..b) you j
ust get the description of the procedure. We will use the custom of ca
lling the output of " }{TEXT 397 12 "rand(a..b)()" }{TEXT -1 3 " a " }
{TEXT 322 13 "random number" }{TEXT -1 214 ". But strictly speaking th
e numbers so give are not really random.  After all,  they are generat
e by a compute program. If you wish to learn more about such matters I
 recommend that you take a look at the book The " }{TEXT 318 27 "Art o
f Computer Programming" }{TEXT -1 1 " " }{TEXT 319 30 "Vol 2 Seminumer
ical Algorithms" }{TEXT -1 38 ", by Donald Knuth.  See Chapter Three \+
" }{TEXT 320 14 "Random Numbers" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 9 "restart;\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "rand(0..20)();" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
127 "If you want to use this procedure repeatedly you can give it a sh
orter name. I will use f here. Sometime people use names like " }
{TEXT 338 4 "dice" }{TEXT -1 4 " or " }{TEXT 339 3 "die" }{TEXT -1 14 
" (singular of " }{TEXT 340 4 "dice" }{TEXT -1 3 "), " }{TEXT 343 4 "r
oll" }{TEXT -1 6 ",  or " }{TEXT 341 4 "flip" }{TEXT -1 37 ". But you \+
can use any name you wish. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
15 "f:=rand(0..20):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "Note that \+
each time " }{TEXT 321 4 "f ()" }{TEXT -1 56 " is executed it gives a \+
usually different random number." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 38 "for i from 1 to 4 do print(i,f()); od;" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 61 "Or, we can create a sequence of 50 random
 numbers as follows:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "seq
(f(), i=1..50);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "Let's create a
 sequence of 50 random " }{TEXT 342 4 "bits" }{TEXT -1 14 " ( 0s and 1
s):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "b:=rand(0..1):\nseq(
b(), i=0..50);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 82 "To get a (pseudo) random number in the interval [0,1
] you might use the following:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "h:=rand(0..10^6):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "for i from 1 to 10 do\nprint
(evalf(h()/10^6));\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 30 " Generating Pseudo Random S
ets" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Here's one way to pick a " 
}{TEXT 311 22 "(pseudo) random subset" }{TEXT -1 26 " from the set \{1
,2,3,4,5\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "with(combina
t); #first load the package combinat" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "n:=5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S:
=powerset(\{seq(i,i=1..n)\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 8 "nops(S);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "f:=rand(1.
.nops(S)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 259 "\nThere are 2^5 = \+
32 subsets of \{1,2,3,4,5\}. The set S above contains these 32 sets. I
f we use f as just defined to get a random number i from 1 to 32. Then
 the corresponding set S[i] may be considered as a random subset of \{
1,2,3,4,5\}. Here are some examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "S[f()];" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "S[f()];" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 7 "S[f()];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 91 "A bett
er way to generate a random set is given in the homework assignment fo
r this lecture." }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 24 " The procedu
re numboccur" }}{PARA 0 "" 0 "" {TEXT -1 14 "The procedure " }{TEXT 
312 9 "numboccur" }{TEXT -1 150 " can be used to simplify the type of \+
computation made in the last lecture when counting the number of times
 a digit appears in  the first n digits of " }{XPPEDIT 18 0 "pi;" "6#%
#piG" }{TEXT -1 7 ".  The " }{TEXT 35 15 "numboccur(f, x)" }{TEXT -1 
43 " function returns the number of times that " }{TEXT 35 1 "x" }
{TEXT -1 13 " is found in " }{TEXT 35 3 "f, " }{TEXT 313 29 "where f m
ay be any expression" }{TEXT 314 2 ". " }{TEXT -1 23 "Here are some ex
amples:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "roll:=rand(1..6):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "L:=[seq(roll(),i=1..100
)];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "numboccur(L,2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "for i from 1 to 6 do\n   i,n
umboccur(L,i);\nend do;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
45 "p:=x^3+3*x*y+2*x^2 + 5*x + 1;\nnumboccur(p,x);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 29 "A:=matrix([[1,2,3],[1,3,6]]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 80 "Note that numboccur doesn't work with mat
rices unless we use first the function " }{TEXT 344 5 "evalm" }{TEXT 
-1 31 ", as in the following examples:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "numboccur(A,3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "numboccur(evalm(A),3);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 " Remarks on
 Packages" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg);" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 258 6 "linalg" }{TEXT -1 34 " is an example of what are called \+
" }{TEXT 259 8 "packages" }{TEXT -1 126 " in Maple. A package is a col
lection of  commands that must be loaded before they can be used. Othe
r examples of packages are " }{TEXT 260 56 "numtheory, combinat, group
s,  logic, geometry, finance, " }{TEXT -1 4 "and " }{TEXT 261 6 "plots
." }{TEXT -1 47 " To see a list of all packages use the command " }
{TEXT 262 14 "?index,package" }{TEXT -1 1 "." }{TEXT 263 1 " " }{TEXT 
-1 34 "To load a package use the command " }{TEXT 257 4 "with" }{TEXT 
-1 69 " as in the above example: Note that if you use a semicolon afte
r the " }{TEXT 264 12 "with(linalg)" }{TEXT -1 133 " command you will \+
get a list of all the commands in that package. If you don't want to s
ee all the commands, just use a colon. \n\n[In " }{TEXT 275 7 "Maple 6
" }{TEXT -1 52 " there are a number of new packages that are not in " 
}{TEXT 276 7 "Maple V" }{TEXT -1 36 ". For example, there is the packa
ge " }{TEXT 265 13 "LinearAlgebra" }{TEXT -1 29 " -- but Maple 6 also \+
retains " }{TEXT 266 6 "linalg" }{TEXT -1 10 ". The new " }{TEXT 267 
13 "LinearAlgebra" }{TEXT -1 54 " package in Maple 6 has a lot of the \+
same features as " }{TEXT 268 6 "linalg" }{TEXT -1 62 " but it has add
itional commands for creating special types of " }{TEXT 269 8 "Matrice
s" }{TEXT -1 101 ", and improved Matrix algebra. Also, especially when
 calculating with large numeric (floating point) " }{TEXT 270 8 "Matri
ces" }{TEXT -1 150 ", it is much more powerful and efficient.  Note th
at the new commands in LinearAlgebra are genrally capitalized unlike i
n the package linalg. So that " }{TEXT 271 6 "matrix" }{TEXT -1 5 " an
d " }{TEXT 272 6 "Matrix" }{TEXT -1 63 " are different data type.  To \+
simplify matters I will stick to " }{TEXT 273 6 "linalg" }{TEXT -1 17 
" in this course. " }{TEXT 345 92 "If you need to do calculations with
 large numeric matrices you should look into the package " }{TEXT 274 
13 "LinearAlgebra" }{TEXT 346 1 "." }{TEXT -1 1 "]" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 14 "?index,package" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 20 "with(LinearAlgebra);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 " Arrays and
 Matrices" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 
18 "Before discussing " }{TEXT 277 6 "matrix" }{TEXT -1 40 " let's dis
cuss the more general concept " }{TEXT 278 5 "array" }{TEXT -1 40 ": M
ore details can be obtained by use of" }{TEXT 279 9 " ?array. " }
{TEXT -1 136 "They are best understood by looking at examples: Arrays \+
are one type of indexed variable. The subscripts each must run thru a \+
specified " }{TEXT 280 5 "range" }{TEXT 398 1 " " }{TEXT 399 12 " of i
ntegers" }{TEXT -1 107 ": Arrays come in different dimensions: 1-dimen
sional, 2-dimensional, 3-dimensional, etc. Here are examples:" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "A:=array(-3..4);\n" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "The array A has indices -3, -2, -
 1, 0, 1, 2, 3, 4.  A is a 1-dimensional array. We may assign values a
s follows:\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "for i from \+
-3 to 4 do A[i]:=i^2; end do;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "
Note that you cannot use other values for the index of A:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "A[5]:=2;" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 77 "Now we define a 2-dimentional array B. Again we specify t
he range of indices." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "B:=
array(-1..2,0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Here we ass
ign values to the array:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 
"for i from -1 to 2 do \nfor j from 0 to 1 do \n B[i,j]:=i + j*x;\nend
 do;\nend do;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "B;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(B);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 9 "B[1,-1]; " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "B[-1,0];" }}}{PARA 0 "" 0 "" {TEXT -1 2 "A " }{TEXT 
281 6 "matrix" }{TEXT -1 153 " is a 2-dimensional array where the row \+
and column indices start with 1. One can define matrices directly or c
onvert a 2-dimensional array into a matirx." }}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 22 "MB:=convert(B,matrix);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 34 "B[-1,0],B[-1,1]; \nMB[1,1],MB[1,2];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 3 "MB;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "evalm(MB);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "C:=array(1..4,1..2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "for i from 1 to 4 do\nfor j \+
from 1 to 2 do\n  C[i,j]:=(i-2)+(j-1)*x; \nod;\nod;\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "C;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "eval(C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 
"type(B,matrix);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "type(
B,array);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "type(MA2, ma
trix);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "type(C,matrix);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "L:=[ [1,2],[0,1],[1,2],[4
,5]];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Note that the following \+
all give the same result, namely, a 4 by 2 matrix." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 84 "X:=array(1..4,1..2, L);\nY:=array(L);\nZ:=m
atrix(L);\nW:=matrix(4,2,[1,2,0,1,1,2,4,5]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "ty
pe(X,matrix), type(Y,matrix), type(Z,matrix), type(W,matrix);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "If you just specify the dimensions
 of the matrix you may treat the entries as variables." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "H:=matrix(2,2);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 2 "H;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 9 "evalm(H);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "To change one o
f the entries uses an assignment statement as in the following example
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "H[1,2]:=10000;\nevalm(
H);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 
3 "" 0 "" {TEXT -1 7 " Tables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "We
 now discuss the datatype " }{TEXT 282 5 "table" }{TEXT -1 36 ". Table
s are similar to arrays, but " }{TEXT 287 93 "the indices need not be \+
integers and one need not declare that a variable represents a table.
" }{TEXT -1 171 " If you make an assignment of the form G[z]:=w, then \+
Maple creates a table with the entry  z=w. Then you may add to this ta
ble at will.  You may also use tables to create " }{TEXT 288 17 "index
ed variables" }{TEXT -1 88 ".  In fact indexed variables are just tabl
es. We illustrate by creating a table named G:" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 63 "G[z]:=w;\nG[box]:=red;\nfor i from 1 to 5 do G[i]:=\{0,i,i+1\}
; od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "G[z]; G[box];G[1];
 G[2]; G[3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 2 "G;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(G);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 29 "type(G,table);\ntype(G,array);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "cat(E,d,w,'i',n); " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "for i in \{p,q,r\} do G[i]:=cat(i,s
tuff); od;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(G);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "G[r];\nG[box];" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 39 "Here's how you can  get a list of the i" 
}{TEXT 283 7 "ndices " }{TEXT -1 29 "of a table and a list of the " }
{TEXT 284 7 "entries" }{TEXT -1 84 " in the table. Note that the indic
es and entries are exhibited as one element lists." }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 11 "indices(G);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 11 "entries(G);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 8 "On
e can " }{TEXT 286 6 "change" }{TEXT -1 25 " the entries in an array:
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "G[1]:=G[1] union \{infinity\};" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 8 "One can " }{TEXT 285 6 "delete" }{TEXT -1 54 " indice
s and entries as in the following two examples." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 30 "G[1];\nG[1]:=evaln(G[1]);\nG[1];" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "indices(G);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "G[2];\nG[2]:='G[2]';\nG[2];" }}{PARA 0 "" 0 "
" {TEXT -1 42 "                                          " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "indices(G);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 176 "Note that we do not have to declare before hand tha
t a variable is going to represent a table. However, we must do so for
 arrays and matrices: Consider the following examples:\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 62 "for i from 1 to 2 do\nfor j from 1 to 2 do\nZ[i,j]:
=i+j;\nod;\nod;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "type(Z,t
able), type(Z,array);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eva
l(Z);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "But look what happens if
 we first say that Z is an array:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 92 "restart:\nZ:=array(1..2,1..2);\nfor i from 1 to 2 do
\nfor j from 1 to 2 do\nZ[i,j]:=i+j;\nod;\nod;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 44 "type(Z,table),type(Z,array), type(Z,matrix);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "eval(Z);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 
" Functions vs Tables" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 289 10 "Functions:" }
{TEXT -1 94 " We have already seen three ways to make functions from e
xpressions.\nHere we show another way:" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 256 "" 0 "" {TEXT 293 13 "First we can " }{TEXT -1 22 "assig
n values at will " }{TEXT 294 60 "as follows: We make up a couple of f
unctions called f and g." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "
restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "f(1):=2; f(2):=
3; f(3):=1;\ng(1):=2; g(2):=1; g(3):=3;" }}}{EXCHG {PARA 257 "" 0 "" 
{TEXT -1 82 "This explains why f(x):=x^2+1; does not define f. Actuall
y it defines the function" }}{PARA 0 "" 0 "" {TEXT 301 64 "f at the va
riable x only, but not for anything different from x." }{TEXT -1 13 " \+
For example:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "h(x):=x^2+x
+1;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(x);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "h(y);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 62 "This indicates that you have not yet defined h at the val
ue y." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT 290 24 "Composition of functions" }{TEXT -1 53 " in Maple is \+
given by @: For example, the composition" }}{PARA 0 "" 0 "" {TEXT -1 
81 "of f and g as defined above is f@g. Also the composition of f with
 itself n times" }}{PARA 0 "" 0 "" {TEXT -1 17 "is given by f@@n:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "(f@g)(1), f(g(1)); " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "(f@@3)(1), f(f(f(1)));" }}}
{EXCHG {PARA 256 "" 0 "" {TEXT 291 15 "Second, we can " }{TEXT -1 33 "
change the value of any function " }{TEXT 292 34 "whether we made it, \+
or it is built" }}{PARA 0 "" 0 "" {TEXT -1 42 "into Maple: (Not someth
ing to do lightly.)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "sin(
0):=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sin(0);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "restart;\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 7 "sin(0);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
298 9 "Functions" }{TEXT -1 8 " versus " }{TEXT 299 6 "tables" }{TEXT 
-1 25 ": Expression of the form " }{TEXT 300 14 "name(variable)" }
{TEXT -1 13 " are of type " }{TEXT 295 8 "function" }{TEXT -1 50 ". On
 the other hand if your execute the statement " }{TEXT 296 22 "name[va
riable]:=value;" }{TEXT -1 17 " then Maple give " }{TEXT 347 4 "name" 
}{TEXT -1 10 " the type " }{TEXT 297 5 "table" }{TEXT -1 41 ". This gi
ves us the peculiar result that " }{TEXT 302 6 "sin(x)" }{TEXT -1 6 " \+
is a " }{TEXT 303 17 "function in Maple" }{TEXT -1 6 ", but " }{TEXT 
304 3 "sin" }{TEXT -1 4 " is " }{TEXT 305 23 "not a function in Maple
" }{TEXT -1 2 ". " }{TEXT 306 9 "NOTE WELL" }{TEXT -1 107 ": This usag
e deviates from the usual mathematical meaning of function. In ordinar
y mathematics we say that " }{TEXT 307 3 "sin" }{TEXT -1 19 " is a fun
ction and " }{TEXT 308 6 "sin(x)" }{TEXT -1 17 " is the value of " }
{TEXT 309 3 "sin" }{TEXT -1 14 " at the value " }{TEXT 310 1 "x" }
{TEXT -1 234 ". In certain constructions, for example when solving rec
urrences and differential equations one is required to use the Maple n
otion of a function. The reason is that in these cases Maple needs to \+
know what the independent variable is." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 31 "restart;\ntype(f(x),function);\n\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 19 "type(sin,function);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 22 "type(sin(x),function);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 20 "type(sin,procedure);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 38 "type(g,table);\ng[x]:=y;\ntype(g,table);" }}
{PARA 11 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 348 8 "
whattype" }{TEXT -1 89 " will give the type, but sometimes one needs t
o use eval first, as in the examples below:" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 14 "whattype(sin);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 20 "whattype(eval(sin));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "g[3]:=3: \nwhattype(g);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "whattype(eval(g));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 15 "whattype(f(x));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "whattype(.34);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "whattype(3/4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "w
hattype(-3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "whattype(3)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "type(3,posint);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "whattype((x+z)*(y+3));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 35 " Introduction to the linalg package" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "There are lots of \+
things one can do with matrices after the " }{TEXT 361 6 "linalg" }
{TEXT -1 21 " package is loaded as" }}{PARA 0 "" 0 "" {TEXT -1 73 "we \+
did above.  If the matrices are not too big maple will easily compute \+
" }{TEXT 365 12 "determinants" }{TEXT -1 1 "," }}{PARA 0 "" 0 "" 
{TEXT -1 5 "find " }{TEXT 362 11 "eigenvalues" }{TEXT -1 2 ", " }
{TEXT 363 12 "eigenvectors" }{TEXT -1 2 ", " }{TEXT 364 22 "Jordan Can
onical Forms" }{TEXT -1 41 ", etc.... Here are a few simple examples:
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 50 "M:=randmatrix(2,2);\nA:=randmatrix(2,2,symmet
ric);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "B:=randmatrix(2,
2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "multiply(A,B);\n" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "inverse(A);\n" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "det(A);\n" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 14 "eigenvals(A);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "EV:=eigenvects(A);\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "EV[1];" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 27 "a1 is the first eigenvalue:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "a1:=EV[1][1];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 9 "EV[1][2];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "EV[1][3]
;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "v1 is an  eigenvector corres
ponding to the eigenvalue  a1." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "v1:=EV[1][3][1];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "We now check that Av1 = a1*v1:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "multiply(A,v1); \nmap(expan
d, evalm(a1*v1));" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "A+B^2;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(%);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "Id:=array(identity, 1..4,1..4);\nev
alm(%);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT 
-1 9 " Vectors " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 2 "A " }{TEXT 323 7 "vector " }{TEXT -1 10 "is just a " 
}{TEXT 400 46 "1-dimensional arrary with index beginning at 1" }{TEXT 
-1 36 ". It may be formed with the command " }{TEXT 326 15 "vector(lis
t) , " }{TEXT -1 12 "the command " }{TEXT 324 11 "array(list)" }{TEXT 
-1 17 ", or the command " }{TEXT 325 26 "array(1..n,list).  Vectors" }
{TEXT -1 73 " have some seemingly strange properties. When printed out
 they look like " }{TEXT 327 4 "list" }{TEXT -1 4 " or " }{TEXT 328 8 
"matrices" }{TEXT -1 222 ", but they behave differently in some cases.
 Maple treats them as column vectors for many purposes. I suggest avoi
ding them except when necessary in linear algebra problems. We will sp
end more time on linear algebra later." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 
"with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "L:=[1,2,3
]; " }{TEXT -1 46 "#this is a list, not a vector, matrix or array" }
{MPLTEXT 1 0 23 "\n\nv1:=vector([1,2,3]); " }{TEXT -1 55 "#this is a v
ector and an array but not a list or matrix" }{MPLTEXT 1 0 22 "\nv2:=a
rray([1,2,3]);  " }{TEXT -1 55 "#this is a vector and an array but not
 a list or matrix" }{MPLTEXT 1 0 24 "\n\na1:=array([[1,2,3]]); " }
{TEXT -1 74 "#this is an array and a 1 by 3 matrix, but is not a list \+
and not a vector," }{MPLTEXT 1 0 23 "\nm1:=matrix([[1,2,3]]);" }{TEXT 
-1 74 "#this is an arrary and a 1 by 3 matrix but is not a list and no
t a vector." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 168 "Note that the 1 b
y 3 matrices look different from vectors and lists when printed out. O
n the other hand one cannot distinguish lists from vectors when they a
re printed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "a1[1];" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "The matrices are 1 by 3 and so we \+
need 2 indices to specify elements. " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "a1[1,1],a1[1,2],a1[1,3];" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 
1 {PARA 3 "" 0 "" {TEXT -1 33 " Multiplying a vector by a matrix" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 355 "R
ecall that if A is an n by k matrix and B is an m by t matrix then the
 product AB is defined if and only if k = m. Maple uses this same rule
. However if A is an n by k matrix and v is a vector  of length k then
 Maple computes the product as if v was a k by 1 matrix -- even though
 when printed v looks more like a 1 by k matrix. Here are some example
s:  " }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 23 "restart:\nwith(linalg):\n" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 30 "A:=matrix([[1,2,3], [4,5,6]]);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 37 "B:=matrix([[1,2,3],[2,3,4],[3,4,5]]);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "multiply(A,B);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "multiply(B,A);" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 19 "v:=vector([1,2,3]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 25 "evalm(v + v), evalm(x*v);" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "v2:=multiply(A,v);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 98 "Lists are also sometimes treated like vectors. But note t
hat a matrix times a list gives a vector." }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 11 "L:=[1,2,3];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 18 "L2:=multiply(A,L);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
14 "type(L2,list);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "type(
L,list);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "type(L2,vector)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "L3:=convert(L2,list);
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 298 "But in some respects lists a
nd vectors are not interchangeable. For example, one can plot a list o
f length two as a point, but a vector of length 2 will not be accepted
 by the plot command. Consider the following attempt to plot the vecto
r L2. But after converting it to the list L3 we can plot it." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "plot([L2], style = point, sy
mbol=circle);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "plot([L3],
style = point, symbol = circle);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 40 " Assignment
 4 Due Next Monday..         " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
332 10 "Problem 1." }{TEXT -1 48 "   Generate 10 random 3 by 3 matrice
s using the " }{TEXT 366 10 "randmatrix" }{TEXT -1 18 " command from t
he " }{TEXT 367 6 "linalg" }{TEXT -1 42 " package. As each matrix is g
enerated use " }{TEXT 368 9 "eigenvals" }{TEXT -1 159 " to compute its
 eigenvalues.  Then take the product of the eigenvalues and check that
 for each matrix this product is equal to the determinant of the matri
x.  " }{TEXT 371 6 "Hint: " }{TEXT -1 79 "The product will be complica
ted algebraically and you will need to apply first " }{TEXT 369 6 "exp
and" }{TEXT -1 7 ", then " }{TEXT 370 8 "simplify" }{TEXT -1 60 " to r
educe the product of the eigenvalues to an integer.\n  \n" }{TEXT 329 
9 "Problem 2" }{TEXT -1 59 ".  There is a one-to-one correspondence be
tween subsets of " }{TEXT 379 17 "\{1, 2, . . . , n\}" }{TEXT -1 28 " \+
and binary lists of length " }{TEXT 386 1 "n" }{TEXT -1 17 ", that is,
 lists " }{TEXT 383 3 "L =" }{TEXT -1 1 " " }{TEXT 380 22 "[x1, x2 , .
 . . , xn] " }{TEXT -1 6 "where " }{TEXT 381 18 "x1, x2, . . . , xn" }
{TEXT -1 25 " are elements of the set " }{TEXT 382 5 "\{0,1\}" }{TEXT 
-1 57 ".  The correspondence is given by associating to the set " }
{TEXT 401 1 "S" }{TEXT -1 10 " the list " }{TEXT 385 1 "L" }{TEXT -1 
7 " where " }{TEXT 384 35 "xi = 1 if i is in X and 0 if not. \n" }
{TEXT -1 23 "\n(a) Write a procedure " }{TEXT 351 12 "list_to_set " }
{TEXT -1 95 "whose input is a binary list and whose output is the corr
esponding set.\n\n(b) Write a procedure " }{TEXT 352 11 "set_to_list" 
}{TEXT -1 22 " whose input is a pair" }{TEXT 372 4 " S,n" }{TEXT -1 7 
" where " }{TEXT 373 1 "S" }{TEXT -1 16 " is a subset of " }{TEXT 374 
17 "\{1, 2, . . . , n\}" }{TEXT -1 5 " and " }{TEXT 377 1 "n" }{TEXT 
-1 69 " is a positive integer and whose output is the binary list of l
ength " }{TEXT 376 1 "n" }{TEXT -1 26 " corresponding to the set " }
{TEXT 375 1 "S" }{TEXT -1 68 ". \n\n(c)  Show by a few tests that each
 procedure works.  Then apply " }{TEXT 353 11 "set_to_list" }{TEXT -1 
32 " to each set in the powerset of " }{TEXT 378 12 "\{1, 2, 3, 4\}" }
{TEXT -1 170 " to form all binary lists of length 4. Make a program to
 print out a table of the following form\n\n   [0,0,0,0] <-->  \{  \}
\n   [1,0,0 0] <--> \{ 1 \}\n   [0,1,0,0] <--> \{ 2 \} " }}{PARA 0 "" 
0 "" {TEXT -1 12 "    ........" }}{PARA 0 "" 0 "" {TEXT -1 7 "    etc
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 40 "Some \+
extra commas in the output is okay." }}{PARA 0 "" 0 "" {TEXT -1 1 "\n
" }{TEXT 330 11 "Problem  3." }{TEXT -1 14 " Note that if " }{TEXT 
393 1 "n" }{TEXT -1 20 " is an integer then " }{TEXT 387 18 "convert(n
,base, 2)" }{TEXT -1 14 " will convert " }{TEXT 394 1 "n" }{TEXT -1 
86 " to a binary list (the binary representation of n in reverse order
). Then you may use " }{TEXT 354 11 "list_to_set" }{TEXT -1 46 " to co
nvert  this list to a set. Note that if " }{TEXT 395 1 "n" }{TEXT -1 
11 " runs from " }{TEXT 396 1 "0" }{TEXT -1 4 " to " }{XPPEDIT 18 0 "2
^n-1;" "6#,&)\"\"#%\"nG\"\"\"F'!\"\"" }{TEXT -1 6 " then " }{TEXT 358 
32 "list_to_set(convert(n,base,2) ) " }{TEXT -1 25 "will give all subs
ets of " }{TEXT 388 16 "\{1, 2, . . ., n\}" }{TEXT -1 42 ". \n\n(a) Us
e this idea to make a procedure " }{TEXT 357 9 "PowerSet " }{TEXT -1 
43 "which will given n produce the powerset of " }{TEXT 389 17 "\{1, 2
, . . . , n\}" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 93 "Show by \+
a several examples that your procedure works.\n\n(b) Use this idea to \+
make a procedure " }{TEXT 356 7 "RandSet" }{TEXT -1 43 " which given n
 produces a random subset of " }{TEXT 390 17 "\{1, 2, . . . , n\}" }
{TEXT -1 2 ".\n" }{TEXT 392 1 "[" }{TEXT -1 4 "Use " }{TEXT 402 4 "ran
d" }{TEXT -1 89 " to produce a random integer in the appropriate range
 and then convert it to a subset of " }{TEXT 391 17 "\{1, 2, . . . , n
\}" }{TEXT -1 1 "." }{TEXT 355 2 "]\n" }{TEXT -1 49 "Show by a few exa
mples that your procedure works." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
1 "\n" }{TEXT 331 9 "Problem 4" }{TEXT -1 108 ". Construct an original
 example of each of the following types of Maple expressions. Check ea
ch using  both " }{TEXT 349 4 "type" }{TEXT -1 5 " and " }{TEXT 350 
10 "whattype. " }{TEXT -1 25 "Note that in the case of " }{TEXT 359 8 
"whattype" }{TEXT -1 21 " you may need to use " }{TEXT 360 4 "eval" }
{TEXT -1 32 " as illustrated in the lecture. " }}{PARA 256 "" 0 "" 
{TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 8 "list\nset" }}{PARA 256 "
" 0 "" {TEXT -1 12 "array\nmatrix" }}{PARA 256 "" 0 "" {TEXT -1 6 "vec
tor" }}{PARA 256 "" 0 "" {TEXT -1 23 "table\ninteger\nprocedure" }}
{PARA 256 "" 0 "" {TEXT -1 8 "function" }}{PARA 256 "" 0 "" {TEXT -1 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}}}{MARK "12" 0 }{VIEWOPTS 0 0 0 1 
1 1803 1 1 1 1 }{PAGENUMBERS 1 1 2 33 1 1 }