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{SECT 0 {SECT 0 {PARA 3 "" 0 "" {TEXT -1 34 "Soluciones del examen B d
e Maple. " }}{PARA 3 "" 0 "" {TEXT -1 35 "Matem\341tica Discreta. Febr
ero 2005. " }{TEXT 270 0 "" }}{PARA 0 "" 0 "" {TEXT 261 67 "Ingenier
\355a T\351cnica en Inform\341tica de Sistemas (tarde) y de Gesti\363n
." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT 262 13 "Ejercicio 1: " }{TEXT 265 110 "Calcula e
l porcentaje de n\372meros entre 20.000 y 40.000 (ambos inclusive) que
 son relativamente primos con 24. " }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "with(numtheory):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 89 "c:=0: for i from 20000 to 40000 do\nif igcd(i,24)=1 t
hen c:=c+1 fi od: evalf(c*100/20001);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 263 13 "Ejercicio 2: " }
{TEXT 266 191 "Tomamos una urna con 100 bolas y extraemos 7 bolas. Det
ermina el n\372mero m\355nimo de bolas negras en la urna para que la p
robabilidad que las 7 olas extra\355das sean todas negras sea al menos
 1/4." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "with(combinat):" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "p:=n->binomial(n,7)/binomia
l(100,7);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "c:=0: for i fr
om 0 to 100 do\nif p(i)<1/4 then c:=c+1;fi;od; c;" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 34 "evalf(p(82)-1/4);evalf(p(83)-1/4);" }}}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
256 12 "Ejercicio 3:" }{TEXT -1 6 "  Sea " }{TEXT 257 7 "G=(V,E)" }
{TEXT -1 82 " el grafo no dirigido definido por la matriz de adyacenci
as B (ver fichero B.mws)." }}{PARA 256 "" 0 "" {TEXT 260 16 "1) Verifi
ca que " }{TEXT 259 2 "G " }{TEXT 267 9 "es simple" }{TEXT 258 2 ". " 
}}{PARA 0 "" 0 "" {TEXT -1 78 "2) \277Cu\341ntos caminos de longitud 1
2 hay entre el primer y el quinto v\351rtice de " }{TEXT 268 1 "G" }
{TEXT -1 1 "?" }}{PARA 0 "" 0 "" {TEXT -1 7 "3) \277Es " }{TEXT 264 2 
"G " }{TEXT 269 10 "euleriano?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 915 "B := matrix([[0, 0, 1, 1, 0
, 0, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 0, 1, 1, 1, 0, 1, 0, 1, 1, \+
1, 0, 1, 0, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0,
 1], [1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1], [0, 1, 0, 0,
 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0
, 0, 0, 1, 1, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, \+
0, 1], [0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1], [0, 1, 0, \+
1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 0, 1, 0, 1,
 0, 1, 1, 0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0
, 1, 1], [0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0], [0, 1, 1
, 1, 1, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 0, 1, 0, 1, \+
1, 0, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0,
 0, 1, 0], [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0], [0, 0,
 1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0]]):" }}}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 131 "1) El grafo es simple ya
 que B es sim\351trica, los elementos de la diagonal son todos nulos y
 sus coeficientes son iguales a 0 o a 1:" }}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 930 "with(linalg):\nB := matrix([[0, 0, 1, 1, 0, 0, 1, 0,
 0, 0, 0, 0, 0, 1, 0, 1, 0], [0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0
, 0, 0, 0], [1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 0, 0, 1], [1, 1
, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1], [0, 1, 0, 0, 0, 0, 0, \+
1, 1, 0, 1, 0, 1, 0, 0, 1, 1], [0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1,
 1, 0, 0, 0], [1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1], [0,
 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 1, 1, 0, 1], [0, 1, 0, 1, 1, 0, 0
, 1, 0, 1, 1, 1, 1, 1, 1, 0, 0], [0, 1, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, \+
0, 0, 0, 0, 0], [0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1], [
0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0], [0, 1, 1, 1, 1, 1,
 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 0], [1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0
, 1, 0, 0, 1, 1], [0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 1, 1, 0, 0, 1, 0],
 [1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0], [0, 0, 1, 1, 1, \+
0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0]]):\n" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "rowdim(B);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
114 "c:=0:\nfor i from 1 to 17 do\n     for j from 1 to 17 do\n       \+
  if B[i,i]<>0 or B[i,j]>1 then c:=c+1; fi; od;od;c;" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 22 "equal(B,transpose(B));" }}}{PARA 0 "" 0 "
" {TEXT -1 2 "2)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "evalm(B^
12)[1,5];" }}}{PARA 0 "" 0 "" {TEXT -1 55 "3) No es euleriano ya que h
ay v\351rtices de grados impar:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 82 "g:=0:for i from 1 to 17 do\n    for j from 1 to 17 do\n       \+
g:= g+B[i,j]; od:g;od;" }}}{PARA 0 "" 0 "" {TEXT 271 12 "Ejercicio 4:
" }{TEXT -1 57 " Calcula la matriz de la relaci\363n de equivalencia s
obre N" }{TEXT 272 1 "5" }{TEXT -1 17 "=\{1,2,...,5\} que " }{TEXT 
273 0 "" }{TEXT -1 119 "se obtiene calculando las clausuras reflexiva,
 sim\351trica y transitiva de la relaci\363n definida por la  siguient
e matriz:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "A:=matrix([[0, \+
1, 1, 1, 1], [1, 0, 0, 1, 0], [1, 0, 0, 1, 1], [1, 1, 1, 0, 0], [1, 0,
 1, 0, 0]]);" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 100 "Para hacer la clausura reflexiva de la relaci\363n \+
tenemos que a\361adir unos en la diagonal de la matriz:" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 " AR:=matrix(5,5,(i,j)-> if i=j then
 1 else A[i,j];fi);" }}}{PARA 257 "" 1 "" {TEXT -1 15 "A es sim\351tri
ca:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "equal(A,transpose(A))
;" }}}{PARA 257 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 50 
"Necesitamos calcular la clausura transitiva de AR:" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Matrizdeun
osyceros:=proc(M::matrix)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "  loca
l i,j,Mat;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "  Mat:=matrix(rowdim(
M),coldim(M),0);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "  for i from 1 \+
to rowdim(M) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "   for j from 1 \+
to coldim(M) do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "    if(M[i,j]<>0
) then Mat[i,j]:=1;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "    fi; " }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "   od;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 5 "  od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "RETURN(eva
l(Mat));" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "end:\n" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 25 "Warshall:=proc(M::matrix)" }}{PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 18 "  local i,j,k,W,n;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 83 "  W:=matrix(rowdim(M),coldim(M),[seq(seq(M[i,j],j=1..
.coldim(M)),i=1..coldim(M))]):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " \+
 n:=coldim(M);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "  for k from 1 to
 n do" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "   for i from 1 to n do" }
}{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "    for j from 1 to n do" }}{PARA 
0 "> " 0 "" {MPLTEXT 1 0 36 "     W[i,j]:=W[i,j]+(W[i,k]*W[k,j]);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "    od;" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 6 "   od;" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "  od;" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 " Matrizdeunosyceros(W);" }}{PARA 0 
"> " 0 "" {MPLTEXT 1 0 4 "end:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 13 "Warshall(AR);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 3 "" 0 "" {TEXT -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
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