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Transcript of NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È...

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Page 5: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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í)î�ïBð�ñ¥ò�óbï�ôRõXWO N�Z^K�J*�y�I�UT,p ÿ�ù � ��ZY p � LON ? ø J C+F<>= Z = T ø T Z ø T�sRJ+�����©s = Z^NO�Id(L�NO� = NOs ? J C ú$[ ? ��K � p ��\8�] LONO� = NOs ? J C+F÷ Z r7ø[ù+ú = Z ù J+K�J+�«p ÿ 7 û 7 �� � ø[ù±r7ø[ù+ú = Z ù J+K�J+� ÿ 7 � 7�p ����^'�û ;�p ÿ_)÷ Z ù�" J*Z[Ja` ùEù NOs ? J ù�" J*� ø[ù±r7ø[ùb� = C �aJ r �IJ C JbZ ù Jb] sBdfN�L�JbH ù ��� � �-'c�1de��� � ��� 0 ��� 2 ) P^k = ù�"<� 6 ü¡ýMÿ�� p�� Ff gEh gig gkj gEl gEm gkn gEo gkp gEq gEr jch jUg jij jcl jcm jins hihihih hihihtg hihtgEh hihtgig htgEhih htgEhtg htgigEh htgigig gEhihih gEhihtg gEhtgEh gEhtgig gigEhih gigEhtg gigigEh gigigig

Page 7: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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][JbH = T(N ? ] = K = ù wvx evUwv pÿ ÿ�ÿ�ÿ�ÿp ÿ�ÿ�ÿ p ÿ�ÿ p ÿp ÿ�ÿ�ÿ� ÿ�ÿ p�pp ÿ�ÿ p ÿ p ÿ�ÿp ÿ p ÿ� ÿ p ÿ pp ÿ p�p� ÿ p�p ÿp�p ÿ�ÿ# ÿ p�p�pp�p ÿ p� p�p�p ÿq p�p�p�p

g = Z7H/J ù�" J+�aJ�NO�aJ C = `\LON ? ø J C ù�" N ùR" NbL�J ù k¼� r � CaC = s ? JYJbZ7H/�u] = Z[K C P ù�" J+�aJ�NO�IJ H ] =zy J+�aJbZ ù H/�u][J CbF

Page 8: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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q�q*q�q�q�; �� $����#$�&���8 _����-'Àÿ p�p ÿ"ÿ p ÿ p ÿ p ÿ�ÿ«ÿ�ÿ p�p ÿ�ÿ p ÿ_)|{~}*�� _ p�H/�i][J " N C ù�" J(H/�UT r ? JbT(JbZ ù NO�ad r �a� r J+� ù d�P C ø H "�ù�" N ù�ù�" J C ø s ù �IN�H ù = �UZ = C ][�UZ[J�sud>H/�UT r ? JbT�JbZ ù = Z[KJbN�H " s = ù+ú �� $����# � 'Àÿ�ÿ p�p ÿ p ÿ�ÿ p ÿ p�p�p�p ÿ�ÿ p�p ÿ p ) �|��� 0 v H/�i][Jq�q*q�q�qK; �� $����# � ' p�p ÿ�ÿ p ÿ p�p ÿ p ÿ�ÿ�ÿ�ÿ p�p ÿ�ÿ p ÿ_) �|��� 0 v H/�i][J

Page 9: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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Page 10: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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Page 11: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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ÿ ÿ ÿ Nÿ ÿ p sÿ p ÿ sÿ p p Hp ÿ ÿ sp ÿ p Hp p ÿ Hp p p ]

k " J+�IJ ¹ ��ºw��»�� N�Z^]!J NO�aJ s = Z^NO�ad�LONO� = NOs ? J CbF = �a�UT ù�" = C ][J C H/� = r^ù = �UZWN�ZBd r NO� ù = H ø ? NO�(Ja`[N�T r ? J>H+N�Z�sRJK�JbZ[J+�IN ù Jb]�sBdtN CaC = KUZ = Z[K-LON ? ø J C ù �Z¹ ��ºw��» P¿N�Z^]¼J Fs ) g = Z^H/JR¹ ��ºw��»w� N�Z^]½J"H+N�Z�JbN�H "8ù NOo�J ù k¼�«LON ? ø J C 'Àÿ ���Ãp ) P ù�" J)Z ø T�sRJ+�n��� C diT(T�J ù � = H�� ø Z^H ù = �UZ C ��� ù�" �aJ+JLONO� = NOs ? J C = C � � p � FH ) V5C diT(T�J ù � = H C k = ù H " = Z[K�� ø Z^H ù = �UZ " N C ù�" J C N�T�J�LON ? ø J������YN ? ? NO�aK ø T�JbZ ù LON ? ø J C ù�" N ùR" NML�J ù�" J C N�T�JZ ø T�sRJ+�Y���;pe¾ C P C = Z^H/J"N ?@? ù�" J C J«LON ? ø J C H+N�Z>sRJ���s ù N = Z^Jb]�sud r J+�IT ø^ù = Z[K ù�" J"NO�aK ø T�JbZ ù C+F � �UZ C J4Q ø JbZ ù ? d�Pù�" J)L�N ? ø J C ��� ù�" J)� ø Z7H ù = �UZ�����9>NO�aK ø T�JbZ ù C NO�aJÃ][JbH/�UT r � C Jb] = Z ù ��9 M p)H ? N CaC J C ][Ja�¿Z^Jb]�sud ù�" JÃZ ø T�sRJ+����Ype¾ C = Z ù�" J�NO�aK ø T�JbZ ù L�JbH ù ��� ' ��� ø �*H ? N CIC J C = Z r NO� ù N ) F Q " J+�aJ+�����aJ ù�" J C J ùK¿ H/�UT r ? J ù J ? d�]^Ja�¿Z[J C ù�" J� ø Z^H ù = �UZ F] ) Q " J�� ø Z7H ù = �UZ " N C LON ? ø J(p�k " JbZ[J+L�J+� ÿ��� ��� � NO�aK ø T(JbZ ù C " NbL�J�LON ? ø J(p F Q " J ù NOs ? J = C

k ` d ¸ %ÿ ÿ ÿ ÿ pÿ ÿ ÿ p ÿÿ ÿ p ÿ ÿÿ ÿ p p pÿ p ÿ ÿ ÿÿ p p p pÿ p p ÿ pÿ p p p pp ÿ ÿ ÿ ÿp ÿ ÿ p pp ÿ p ÿ pp ÿ p p pp p ÿ ÿ pp p ÿ p pp p p ÿ pp p p p ÿ

J ) ÷ Z r NO� ù$' H ) k¼J C NMk ù�" N ù;ù�" J«NO�aK ø T�JbZ ù L�N ? ø J C ��� C diT(T�J ù � = H C k = ù H " = Z[K-� ø Z^H ù = �UZt���À9¡LONO� = NOs ? J C NO�aJ] = L = ][Jb] = Z ù �C9 M p2H ? N CaC J C+F = ���­JbN�H " ��� ù�" J C J�H ? N CIC J C ù�" JÒ� ø Z7H ù = �UZ«H+N�Z " NbL�JÒL�N ? ø J)p¼��� ÿ F � �UZ C J4Q ø JbZ ù ? d�Pù�" J�Z ø T«sRJ+����� C diT(T�J ù � = H C k = ù H " = Z[K-� ø Z^H ù = �UZ C ���N9�LONO� = NOs ? J C = C .�ÁN0 F� ) j;�[P ù�" J�H/�UT r � C = ù = �UZ>��� C duT�T�J ù � = H C k = ù H " = Z[K8� ø Z^H ù = �UZ C = C Z[� ù Z[JbH/J CIC NO� =@? d N C duT�T�J ù � = H«� ø Z^H ù = �UZ F� �UZ C = ][J+�EN C H/� ø Z ù J+�aJa`[N�T r ? J %('�û���¶�����)�&%eÂÄÃ��8'i%�Å~Æ�'�û���¶Ç)5����)

Page 12: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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N�Z^] = Z ù J+�IH " N�Z[K�J ù�" J�L�NO� = NOs ? J C û N�Z^] � FK ) Q " J ù NOs ? J = C

N s H % 0 % � %ÿ ÿ ÿ p p ÿÿ ÿ p p ÿ ÿÿ p ÿ p ÿ pÿ p p ÿ ÿ pp ÿ ÿ p ÿ pp ÿ p ÿ ÿ pp p ÿ ÿ ÿ pp p p ÿ p ÿ"3) X J ù½È sRJ ù�" J C J ù\ýMÿ�� p �������w� 9À�tk " J+�IJZ9 = C ù�" JfZ ø T�sRJ+�"����L�NO� = NOs ? J C ��� ù�" J C diT(T�J ù � = H�� ø Z^H ù = �UZ£PN�Z^] ? J ù�¿ sRJ ù�" J C J ù ][J C H/� = s = Z[K ù�" J(� ø Z^H ù = �UZ % F g = Z^H/J ù�" J�H/�UT r ? JbT�JbZ ù ��� % = C ÿ k " JbZ % = C p(N�Z^]

L = H/J+L�J+� C NiP = ù = C �aJ r �IJ C JbZ ù Jb] sBd ù�" J C J ùC¿C� C ø H "tù�" N ù¿´�P�ÉÈ ; ¿= ���2Ja`iN�T r ? J�P¥H/�UZ C = ][J+� = Z[KÃN ev LONO� = NOs ? J C duT(T(J ù � = H+N ? � ø Z^H ù = �UZ"k = ù�"L¿&��ýMÿ�� p��UPik¼J " NbL�J ¿ � ��ýw������� � F

Page 13: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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í)î�ïBð�ñ¥ò�óbï ôRõÎö�u Q " J = Z r7ø[ù ��� ù�" J C d C ù JbT = C û �Ê'�û3de�±û � �±û 0 �±û 2 ) P2N�Z^] ù�" J8� ø^ù±r7ø[ù = C ù�" J\s = ù|v L�JbH ù ���� �^'c� 0 ��� 2 ) F Q " J�� ø[ù±r7ø^ù JbZ^H/�u] = Z[K�N�Z^] ù�" J��IJ C r JbH ù = L�J*� ø Z^H ù = �UZ ù NOs ? J*NO�aJ ú

¸ ��0��12p ÿ p p ÿ� p�p

û�d0û � û 0 û 2 � 0 � 2ÿ ÿ�ÿ�ÿ�ÿ v vp ÿ�ÿ�ÿ p v v ÿ�ÿ p ÿ v v� ÿ�ÿ p�p ÿ p ÿ p ÿ�ÿ v v� ÿ p ÿ p p ÿ� ÿ p�p ÿ ÿ p# ÿ p�p�p v v� p ÿ�ÿ�ÿ v vq p ÿ�ÿ p p p

p ÿ p ÿ p ÿ p ÿp�p p ÿ p�p v vp p�p ÿ�ÿ ÿ pp � p�p ÿ p v vp p�p�p ÿ v vp � p�p�p�p v v� 0 ú �UZ[J v C J ù1'i� P quP@p ÿ_) Pĸ+J+�a� v C J ù1'i� P � PSp �) P7]7H v C J ù1'Àÿ P@pOP P P # P � P@p�p�P@p � P@p P@p ��)�12Ãú �UZ[J v C J ù1'i� P � P quP@p �) P*¸+J+�a� v C J ù1'i� P©p ÿ_) P¿]7H v C J ù1'Àÿ P@pOP P P # P � P@p�pOP p � P@p P@p ��)

Page 14: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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í)î�ïBð�ñ¥ò�óbï�ôRõ ô£ö ú �)C = Z[K G � � G g^g vU� H/�u]^J�P ù�" J*� ø Z^H ù = �UZ C r JbH = �¿Jb] = Z G `iJ+�IH = C J F p = C ][J C H/� = sRJb]8N C úû � T � H �wË � � �1d � � � 0 � 2ÿ v v v v v v v vp v v v v v v v v v v v v v v v v� ÿ ÿ p p ÿ ÿ p p ÿ ÿ p p ÿ p ÿ p� ÿ ÿ p p ÿ p p p� ÿ ÿ p p p ÿ ÿ p# ÿ ÿ p p p ÿ p p� ÿ p ÿ p p ÿ ÿ ÿq ÿ p p ÿ p ÿ ÿ pp ÿ ÿ p p p p p ÿ ÿp�p p ÿ ÿ p ÿ p p pp p ÿ p p ÿ p ÿ ÿp � v v v v v v v vp v v v v v v v vp � v v v v v v v v�)C = Z[K(N ù k2� v ] = T�JbZ C = �UZ7N ? ù NOs ? J ú

û 0 û 2û3d û � ÿ�ÿ ÿ p p ÿ p�pÿ�ÿ v v v ÿ�ÿ p�p ÿ�ÿ p�pÿ p ÿ�ÿ p�p ÿ p ÿ p ÿ�ÿ p�p ÿ p�p�p ÿ�ÿ p�p�p ÿ�ÿ p ÿ�ÿ p�p�p ÿ p�pp ÿ ÿ p ÿ p�p ÿ�ÿ�ÿ ÿ p�p ÿ p ÿ�ÿ p ÿ p�p�p�p�p ÿ�ÿ p ÿ�ÿ p ÿ p�p�pp�p p ÿ p�p ÿ p ÿ�ÿ v v v�)C = Z[K��UZ[J v C J ù N�Z^] ]^H v C J ù �aJ r �aJ C JbZ ù N ù = �UZ úÌ���Tb�ÎÍ 9�Ï ÐwÏ4Ñ ' p�p � p �)5� J » ÐwÏ4Ñ ' p ÿ�� p�p � p �� p ��� p �� p ��)Ì�� H �ÎÍ 9�Ï ÐwÏ4Ñ 'i��� q � p ÿ_)5� J » Ð�Ï�Ñ ' p ÿ�� p�p � p �� p ��� p �� p ��)Ì�� Ë �ÎÍ 9�Ï ÐwÏ4Ñ 'i���� ����������5#�� q � p ÿ�� p �)°� J » Ð�Ï�Ñ ' p ÿ�� p�p � p �� p ��� p �� p ��)Ì�� � �ÎÍ 9�Ï ÐwÏ4Ñ 'i���� ����������5#������ p ÿ�� p¥p � p _)5� J » Ð�Ï�Ñ ' p ÿ�� p�p � p �� p ��� p �� p ��)Ì�� d �ÎÍ 9�Ï ÐwÏ4Ñ 'i���5#������ q � p ÿ_)5� J » Ð�Ï�Ñ ' p ÿ�� p�p � p �� p ��� p Ç� p ��)Ì�� � �ÎÍ 9�Ï ÐwÏ4Ñ 'X ������ p ÿ�� p�p � p �)5� J » Ð�Ï�Ñ ' p ÿ�� p�p � p �� p ��� p Ç� p ��)Ì�� 0 �ÎÍ 9�Ï ÐwÏ4Ñ 'i�������5#�� p�p )5� J » Ð�Ï�Ñ ' p ÿ�� p�p � p �� p ��� p �� p ��)Ì�� 2 �ÎÍ 9�Ï ÐwÏ4Ñ 'i���� ����������5#�� q � p�p )5� J » ÐwÏ4Ñ ' p ÿ�� p�p � p �� p ��� p Ç� p ��)

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Page 16: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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Page 17: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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Page 18: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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Q NOs ? J F p ú�� �I�B���®���®] = C ù � = s ø[ù = L = ù d������ C d C ù JbT = Z G `iJ+�IH = C J F qí)î�ïBð�ñ¥ò�óbï�ô£õ ôÇu Q �\sRJ«N\sR�B� ? JbN�Z N ? K�J+s7�IN ù�" J C d C ù JbT T ø C ù C N ù = C ��d ù�" J r � C ùaø ? N ù J C ���CÜ5ÝeÞ�Þ�ß�àxá�àcâkãwâXàcä�åæ âkç°àcè°âié�ß�àcâkãwâXàcä�å�Ü�ÝeÞCê3ë,ìaÞZì�í�à�áeí æ á æ�æ âcàcâkã�ìïîIÞ�ß�ëzàcâðê3ëñâiÜ�á�àcâkã�ìtòAâ æ ì�í�àcâXàcä FþX J ù C H/�UZ C = ][J+��JbN�H " �UZ[J\��� ù�" J C JH+N C J C úÌ&' N )ó' Û )5�1' e ) NO�aJtH/�UT(T ø^ù N ù = L�JtsRJbH+N ø C J ù�" J ù NOs ? J = C"C duT(T(J ù � = H�NOsR� ø[ù"ù�" JtT(N = Z�] = NOK��UZ7N ? P C �

r � C ùaø ? N ù J(p 'cô p ) = CEC N ù = C �¿Jb] FÌ&' s ) = ����] = C ù � = s ø[ù = L = ù d�P7k¼J(T ø C ù C " �Mk ù�" N ù ¹�Û 'iº e »4)C�õ' ¹�Û º�) e ' ¹�Û »�) F�X J ù*ø C r �a�ML�J ù�" N ù�ù�" = Cr � C ùaø ? N ù J = C ù � ø J��y��� ù�" J C d C ù JbT sud r J+�a��JbH ù = Z^] ø H ù = �UZ£PiN C;C " �Mk;Z = ZtQ®NOs ? J�ö�ö FÌ&' H ) Q " JtN�]^] = ù = L�J = ][JbZ ù = ù d³J ? JbT�JbZ ù = C ÿ C = Z^H/J¼¹ie ÿ÷� ÿ e�¹ � ¹ F Q " JfT ø ? ù = r ?@= H+N ù = L�J = ][JbZ ù = ù dJ ? JbT�JbZ ù = C C = Z7H/J�¹�Û $�& Û�¹ � ¹ FÌ&' ] ) = ���-p ù � " NbL�J-N�H/�UT r ? JbT�JbZ ù p Ô k¼J«Z[J+Jb]�pbe�p Ô �É N�Z7]³p1Ûfp Ô ��ÿ F � �UZ C J4Q ø JbZ ù ? d�P¿�y�a�UT ù�" Jù NOs ? J*k¼J C J+J ù�" N ù p " N C Z[�(H/�UT r ? JbT�JbZ ù F

Page 19: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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jY� ù J ù�" N ù = ��� 0 = C Z[� ù J4Q ø = LON ? JbZ ù©ù ��� � = ù = C®C ù =@?@? r � CaC = s ? J ù�" N ù­ù�" J���� = K = Z^N ? Ja` r �aJ CaC = �UZ C NO�IJ�J4Q ø = L�N ? JbZ ù FV ZtJa`iN�T r ? J���� ù�" = C ù d r J���� C = ùaø N ù = �UZ = C ¹ º Ô MÚº¯� ¹ MÚº P = Zfk " = H " ¹ º Ô N�Z7]¼¹\NO�IJ�Z[� ù J4Q ø = LON ? JbZ ù F

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Q NOs ? J F uú G `iJ+�IH = C J F �#�' s )í)î�ïBð�ñ¥ò�óbï�ôRõ0/*W' N )÷ Z r7ø[ù C ú ÷ Z ù J+K�J+� û�ü¡ýMÿ��������1� p � �UP¿H/�UZ ù �a� ? L�NO� = NOs ? J �¼ù ? ü�ý ÷ Z^H/�aJbT(JbZ ù P D JbH/�aJbT�JbZ ù �� ø[ù±r¿ø[ù+ú�$� � '�ûLM p ) T��i]tp � = � �¼ù ? � ÷ Z7H/�aJbT�JbZ ù'�û ;¬p ) T��i]tp � = � �¼ù ? � D JbH/�aJbT�JbZ ù' s ) giJ+J*Q NOs ? J�ö�ö(�UZ r NOK�JÒö�ö F' H ) c J " NbL�J ù � H " �u� C J N¬H/�i][J��y��� û P �¼ù ? N�Z^] � F � �u] = Z[K û N C û � '�û3de�±û � �±û 0 �±û 2 ) N�Z^] � N C� � 'c�wde��� � ��� 0 ��� 2 ) P ø C = Z[K ù�" J«H/�UZBL�JbZ ù = �UZ7N ? s = Z^NO�ad8�aJ r �aJ C JbZ ù N ù = �UZ F � �u] = Z[K �¼ù ? N C » = Zts = Z^NO�Id\k = ù�"�ÿT�JbN�Z = Z[K D JbH/�aJbT(JbZ ù N�Z7]>pÃT�JbN�Z = Z[K ÷ Z^H/�IJbT�JbZ ù F Q®NOs ? J�ö�ö-�UZ r NOK�J�ö�ö C " �Mk C ù�" J ù NOs ø ? NO�¼�IJ r �aJ C JbZ ù N vù = �UZt��� ù�" J C k = ù H " = Z[K-� ø Z^H ù = �UZ C+FQ " JK¸+J+�I� C J ù C NO�IJ ú¸+J+�a� v C J ù1'c�wd�)��«ý p ��������� ����������5#������ p ��� p #�� p ��� p�q ���ÿ��5 p ������� p��¸+J+�a� v C J ù1'c� � )��«ý p ��������� �� q � p ÿ�� p�p � p �� p ��� p #�� p ���5����5� Ç�������_���5� pe�¸+J+�a� v C J ù1'c� 0 )��«ý p ������������ q � p ÿ�� p ��� p �� p ��� pbq ��Uÿ��5����5� Ç��_#���_���5� pe�¸+J+�a� v C J ù1'c� 2 )��«ý p ���������5#�� q � p�p � p ��� p ��� p #�� pbq �� p �5����5�����_#�� p�q �5� pe�

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ÿ�ÿ�ÿ p�p�p�p ÿ�ÿ�ÿ�ÿ ÿ�ÿ�ÿ p ÿ�ÿ p ÿÿ�ÿ p ÿ�ÿ p�p ÿ p ÿ�ÿ ÿ p ÿ p ÿ p�p ÿÿ p ÿ ÿ p�p�p p ÿ�ÿ�ÿ p ÿ�ÿ p p ÿ p ÿÿ p�p p ÿ p�p p�p ÿ�ÿ p�p ÿ p p�p�p ÿp ÿ�ÿ ÿ�ÿ�ÿ p ÿ�ÿ p ÿ ÿ�ÿ p�p ÿ p ÿ�ÿp ÿ p ÿ p ÿ p ÿ p�p ÿ ÿ p�p�p p ÿ�ÿ�ÿp�p ÿ p ÿ�ÿ p p ÿ p ÿ p ÿ p�p p�p ÿ�ÿp�p�p p�p ÿ p p�p�p ÿ p�p�p�p ÿ�ÿ�ÿ�ÿ'c� d ��� � ����04���121)

Q NOs ? J F �uú G `uJ+�IH = C J F �#�' H )

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Ì = ø Z^H ù = �UZ ú��$�û × ¶' s ) Q " J ù NOs ? J*����� ù�" J*NO� = ù�" T�J ù = HÃ� ø Z^H ù = �UZ = C

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' H ) H/�UZ C = ][J+� = Z^K"s = Z^NO�ad\�aJ r �aJ C JbZ ù N ù = �UZt����� = Z r7ø[ù C N�Z^]t� ø[ù±r¿ø[ù C k¼J*��s ù N = Z ù�" J��y� ?@? �Mk = Z^K ù NOs ? J ú

Page 29: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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�UZ[J v C J ù�'c�1d1)��ý p � ��UZ[J v C J ù�'c� � )��ý p ÿ�� p�p � p ��UZ[J v C J ù�'c� 0 )��ýw���5#�� q � p�p � p ��� p ��UZ[J v C J ù�'c� 2 )��ýw���5#�� p ��� p � �

Page 30: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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p ÿ p�p�p p�p�p ÿ p ÿ�ÿ�ÿ p�p ÿ p� p ÿ�ÿ p p�p ÿ�ÿ p ÿ p ÿ p ÿ p�p� p ÿ p�p p ÿ p ÿ� p�p ÿ�ÿ p ÿ�ÿ p# p�p ÿ p p ÿ�ÿ�ÿ� p�p�p ÿ ÿ p�p�pq p�p�p�p ÿ p�p ÿ�1dOú �UZ^J v C J ù1'i� P # P � P quP@p ÿ P@p�p�P@p P@p �_) P®]^H v C J ù1'Àÿ P@pOP P � P P ��)� � ú �UZ^J v C J ù1'i� P # P � P quP@p P@p �_) PS]^H v C J ù1'Àÿ P@pOP P � P P ��)� 0 ú �UZ^J v C J ù1'i� P # P@p ÿ P@p�pOP@p P p ��) PS]7H v C J ù1'Àÿ P@pOP P � P P ��)�12Oú �UZ^J v C J ù1'i� P � P@p ÿ P@p P@p �) PS]^H v C J ù1'Àÿ P@pOP P � P P ��)H�p ) gik = ù H " = Z[K"Ja` r �IJ CaC = �UZ C ú �1d � û�d û Ô � M�û�d0û � û Ô 0 M�û Ô d û � û 0� � � û Ô d û � û 0 M�û�d/û Ô � û Ô 0 M�û�d/û � û 0� 0 � û 0� 2 � û Ô 2s �) Q®NOs ø ? NO�E][J C H/� = r7ù = �UZ8��� ù�" J�� ø Z7H ù = �UZ ø C = Z^K wv � ø^ù|v ��� vU� H/�u]^J FJ�Ï » û �-'�û � �±û3de�±û � �±û 0 �±û 2 ) � �-'c� � ���wde��� � ��� 0 ��� 2 )ÿ ÿ�ÿ�ÿ p�p ÿ�ÿ p ÿ pp p�p ÿ�ÿ�ÿ ÿ p ÿ�ÿ p p ÿ p ÿ�ÿ p ÿ�ÿ�ÿ p� ÿ p�p ÿ�ÿ ÿ�ÿ p�p ÿ p ÿ�ÿ p ÿ ÿ p ÿ p ÿ� ÿ p ÿ p ÿ p ÿ�ÿ p ÿ� ÿ�ÿ p�p ÿ ÿ p�p ÿ�ÿ# p ÿ�ÿ�ÿ p p ÿ p ÿ�ÿ� ÿ p ÿ�ÿ p p�p ÿ�ÿ�ÿq ÿ�ÿ p ÿ p ÿ�ÿ�ÿ p�p

Page 31: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

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� � � û � û Ô d û � û Ô 0 û Ô 2 M�û Ô� û3d û Ô � û 0 û Ô 2 M�û � û Ô d û Ô � û Ô 0 û 2 M û Ô� û3d/û Ô � û Ô 0 û 2� d � û � û d û Ô � û Ô 0 û Ô 2 M�û � û Ô d û Ô � û�0Þû Ô 2 M�û Ô� û Ô d û � û~0±û Ô 2 M û Ô� û d û Ô � û Ô 0 û32� � � û�Ô� û�Ôd û3Ô� û 0 û 2 M�û3Ô� û3d û � û3Ô 0 û3Ô2 M�û�Ô� û3Ôd û � û 0 û�Ô2 M û � û�Ôd û3Ô� û3Ô 0 û 2��0 � û�Ô� û d û � û�Ô 0 û�Ô2 M�û � û�Ôd û�Ô� û�0Þû3Ô2 M�û�Ô� û d û�Ô� û~0±û�Ô2 M û�Ô� û�Ôd û � û3Ô 0 û32� 2 � û Ô� û Ô d û Ô � û 0 û 2 M�û � û3d û Ô � û Ô 0 û Ô 2 M�û � û Ô d û � û Ô 0 û Ô 2 M û Ô� û Ô d û � û Ô 0 û 2s ��) Q®NOs ø ? NO� ][J C H/� = r^ù = �UZÃ��� ù�" J2� ø Z7H ù = �UZ ø C = Z[K P � P P@p¼H/�i][J F g = Z^H/J ù�" = C H/�i][J2N ?@? �Mk C ù k2�)�aJ r �IJ C JbZ ù N ù = �UZ C����� C �UT(J)][JbH = T(N ? LON ? ø J C ' C ø H " N C � ��� �) k2J)N�][� r^ù Jb] ù�" J;�IJ r �aJ C JbZ ù N ù = �UZ8k = ù�" ? JbN C ù Z ø T«sRJ+�����npYs = ù C+F

J�Ï » û �-'�û3de�±û � �±û 0 �±û 2 ) � �-'c�1de��� � ��� 0 ��� 2 )ÿ ÿ�ÿ�ÿ�ÿ p�p�p ÿp ÿ�ÿ�ÿ p p�p ÿ p ÿ�ÿ p ÿ p�p ÿ�ÿ� ÿ p ÿ�ÿ p ÿ p ÿ p ÿ�ÿ�ÿ ÿ p�p ÿ� ÿ p�p ÿ p ÿ�ÿ�ÿ� p ÿ p ÿ ÿ p ÿ�ÿ# p�p ÿ�ÿ ÿ�ÿ p ÿ� p�p ÿ p ÿ�ÿ�ÿ pq p�p�p ÿ ÿ�ÿ�ÿ�ÿ

� d ú �UZ^J v C J ù1'Àÿ P@pOP P � P ��) PS]^H v C J ù1'i� P � P # P quP@p�pOP@p ��)� � ú �UZ^J v C J ù1'Àÿ P@pOP P � P@p ÿ_) PS]^H v C J ù1'i� P � P # P quP@p�pOP@p ��)� 0 ú �UZ^J v C J ù1'Àÿ P P � P@p �) PS]^H v C J ù1'i� P � P # P quP@p�pOP@p ��)� 2 ú �UZ^J v C J ù1' pOP@p ��) P£]^H v C J ù1'i� P � P # P quP@p�pOP@p ��)H ��) gik = ù H " = Z[K"Ja` r �IJ CaC = �UZ C ú

� d � û3Ôd û�Ô� û�Ô 0 M�û3Ôd û�Ô� û~0aû3Ô2 M û�Ôd û � û3Ô 0 û�Ô2 M�û d û3Ô� û�Ô 0 û32� � � û3Ôd û�Ô� û�Ô 0 M�û3Ôd û�Ô� û 0 û3Ô2 M û3d/û�Ô� û3Ô 0 û�Ô2 M�û�Ôd û � û 0 û 2� 0 � û Ô d û Ô � û Ô 0 û Ô 2 M û Ô d û � û Ô 0 û Ô 2 M�û3d û Ô � û Ô 0 û 2 M�û�d/û � û Ô 0 û Ô 2� 2 � û3Ôd û�Ô� û�Ô 0 û 2 M û3d û � û3Ô 0 û 2s �) Q®NOs ø ? NO�E][J C H/� = r7ù = �UZ8��� ù�" J�� ø Z7H ù = �UZ ø C = Z^K � P vU P P v p-H/�i][J F

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Page 33: NO aJb]tsudweb.cs.ucla.edu/~ani/classes/csm51a.10w/SolOdd/chap2-sol... · 2010-01-04 · Ç^È ÉËÊ^ÌÎÍyÈ Ï^ÐÒÑ>ÓOÏRÊUÓOÉ®Ô®Õ¦Ï ÌÀÖ È[× ÊuØMÌÎÍyÈ Ï

Ç^È�ÉËÊ^ÌÎÍyÈ�Ï^ÐÒÑ>ÓOÏRÊUÓOɮԮզÏ�ÌÀÖ È[×�ÊuØMÌÎÍyÈ�Ï�ÌËÈ-ÙÃÍÛÚ¥ÍËÌÄÓOÉ�Ù*Ü ÐaÍÛÚ¥Ï�Ô�Ç7ÜÞÝ­ÌËÜ ß8à[Ü0ÖYáiâ�ãRáuä�äMá � pí)î�ïBð�ñ¥ò�óbï�ôRõEDÇ+X J ù©ù�" J = Z r7ø[ù C s�J�¹�N�Z7] º N�Z^] ù�" JÒ� ø[ù±r7ø^ù ÿ¬N�Z^] � P¥N ?@? " Ja`[N�][JbH = T(N ?i= Z ù J+K�J+� C+FÀV " = K "�v ? J+L�J ? ][J C H/� = r^ù = �UZ= C

ÿ � � ¹ ò98 ¹;: ºº=<?>�@ ïBð�A\ò�óbï

�$� � ¹ ò98 ¹ FGººB<?>�@ ïuð�A\ò�óbïs ) ÷ Z ù�" J��IN�] = ` vU �IJ r �aJ C JbZ ù N ù = �UZ�N ? ? s = ù|v L�JbH ù ��� C " NbL�J«�y� ø ��s = ù C+F

¹ �-' ¹ de� ¹ � � ¹ 0 � ¹ 2 )5� º �-'iº°de��º � ��º 0 ��º 2 )� �-'c�wde��� � ��� 0 ��� 2 ) ÿ �û' ÿ dw� ÿ � � ÿ 0 � ÿ 2 )= ��� ù�" JEs = Z7NO�ad«][J C H/� = r^ù = �UZ"k¼J ø C J ù�" J = Z ù J+�IT�Jb] = N ù J;L�NO� = NOs ? J C 4�� Ï � N�Z^]ZÐ;��s ù N = Z^Jb]��y�I�UT N*H/�UT r NO� = C �UZ���(¹\N�Z7] º P C � ù�" N ù� 6 � ¹ 6 'C4�M Ï )�MÚº 6 Ðÿ 6Ê� º56�'C4KM Ï )~M ¹ 6 Ð

guk = ù H " = Z^K"Ja` r �aJ CIC = �UZ C ����� 4�� ÏUP¿N�Z^]¼Ð«NO�aJ

4 � ¹ d�º5Ôd M Ï d ¹ � º5Ô� M Ï d Ï � ¹ 0 º5Ô 0 M Ï d Ï � Ï 0 ¹ 2 º5Ô2Ï � Ï d Ï � Ï 0 Ï 2Ð � 4 Ô Ï Ôk " J+�IJ�Ï 6 � ¹ 6 º 6 M ¹ Ô6 º Ô6

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í)î�ïBð�ñ¥ò�óbï�ôRõED�DQ " J8H/�UT�s = Z^N ù = �UZ^N ? H/�UT r NO�IN ù ����T��i] ø ? J(H/�UT r NO�aJ C " Ja`[N�][JbH = T(N ? ] = K = ù C ¹¡N�Z^] º F Q " J(� ø[ù±r7ø^ù$� " N Cù�" �aJ+J�L�N ? ø J C H+N ?@? Jb]ED � �(P7N�Z^]GF FPV " = K "�v ? J+L�J ? C r JbH = �¿H+N ù = �UZ = C

��� HIJ IK D ò98 ¹ �Gº� ò98 ¹ �&ºF ò98 ¹ FGºV ù NOs ø ? NO�E�IJ r �aJ C JbZ ù N ù = �UZt��� ù�" = C � ø Z^H ù = �UZ = C K = L�JbZ Z^Ja` ù+úº¹ ÿ p � � � # � q p ÿ p�p p p � p p �

ÿ G g g g g g g g g g g g g g g gp L G g g g g g g g g g g g g g g L L G g g g g g g g g g g g g g� L L L G g g g g g g g g g g g g L L L L G g g g g g g g g g g g� L L L L L G g g g g g g g g g g� L L L L L L G g g g g g g g g g# L L L L L L L G g g g g g g g g� L L L L L L L L G g g g g g g gq L L L L L L L L L G g g g g g gp ÿ L L L L L L L L L L G g g g g gp�p L L L L L L L L L L L G g g g gp L L L L L L L L L L L L G g g gp � L L L L L L L L L L L L L G g gp L L L L L L L L L L L L L L G gp � L L L L L L L L L L L L L L L G= ��� ù�" J\s = Z^NO�adþ][J C H/� = r^ù = �UZ£P®] = K = ù ¹ = C �aJ r �aJ C JbZ ù Jb]¡sud ù�" J�L�JbH ù ���Ò¹ � ' ¹ de� ¹ � � ¹ 0 � ¹ 2 ) N�Z^]�] = K = ù�º = C�aJ r �IJ C JbZ ù Jb] sBd º �-'iº d ��º � ��º104��º°21) F c J«JbZ^H/�u]^J ù�" J*� ø[ù±r7ø^ù L�NO� = NOs ? J ��ø C = Z[K � s = ù C N C �y� ?@? �Mk C ú¸ � � � 0 � 2L p ÿ ÿ

G ÿ p ÿg ÿ ÿ pc JfH+N�Z�]^J+� = L�J C k = ù H " = Z[K Ja` r �aJ CaC = �UZ C ����� ù�" J�� ø[ù±r7ø^ù C H/�UZ C = ][J+� = Z^K ù�" J�J4Q ø N ?@= ù d�L�NO� = NOs ? J½Ï 6 ' s = ù C ¹ 6N�Z^] º 6 NO�aJ*J4Q ø N ? ) N�Z^] ù�" J�H/�UZ^] = ù = �UZ^N ? H/�UZ^] = ù = �UZ^N ? Ja` r �aJ CaC = �UZ C+F Q " J C k = ù H " = Z^K"Ja` r �aJ CIC = �UZ C NO�aJ ú

� � � ¹ d4º Ô d M Ï d ¹ � º Ô � M Ï d Ï � ¹ 0 º Ô 0 M Ï d Ï � Ï 0 ¹ 2 º Ô 2� 0 � Ï d Ï � Ï 0 Ï 2� 2 � º°d ¹ Ô d M Ï d4º � ¹ Ô � M Ï d Ï � º 0 ¹ Ô 0 M Ï d Ï � Ï 0 º 2 ¹ Ô 2k " J+�IJ�Ï 0 � ¹ Ô6 º Ô6 M ¹ 6 º 6 P[J4Q ø N ?@= ù d8� ø Z^H ù = �UZ F