CSE Discrete Structures

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Roy Little
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1 CSE - Dirt Strutur Em : Comprhniv Nm: ID#: PIN: CSE - Dirt Strutur Fin Em - Soution - F 999 Cod boo, od not Dt: D. 999, :0 pm - 8:00 pm. Giv th orm dinition or th oowing dirtd or undirtd grph. Ao indit i th r oop-r, imp, onntd, or i. ( point 7! "# $&% ( *+ $&% *, $-%. */0 $-% 0 $-%!. / $-% "( / $-% #. 7 Thi grph i oop-r nd onntd.! $&% % ( $&% % 0 $&% % $&% % $&%! % Thi grph h non o th proprti.. Drw th dirtd or undirtd grph orrponding to th oowing orm dinition. Lit in th grph. ( point 89:8;<=>! $&% C:DB=0 $&% 8 GH> $&%!( I:JH; 0 8! $&% % $-% % 0 $&% % 0 $-% % 0 $&%! % d b 9;<:8=0 Th on in thi grph i Thi grph i i. Mnrd Hubr Pg

2 ! CSE - Dirt Strutur Em : Comprhniv. For th oowing pir o dirtd/undirtd grph indit i th r iomorphi. I th r iomorphi giv th orrponding iomorphim. ( point K M L O N K M N L O Th grph r iomorphi. P % Q0 P % Q P % 0 P % P % R P % >8 P % >8" P % >8! P % >8 P %! 7> P % " > Th grph r iomorphi. P % S P % T P % U P % P % I P % ( > " P % > P % 8 > P % 8 >! P %! 7> P % ". >. For th oowing grph indit i th r pnr. I th r drw n quivnt vrion without roing r. Othrwi juti wh th r not pnr. ( point Thi i pnr grph. Thi grph i not pnr (it i V.. Giv th prordr, potordr nd inordr trvr or th oowing tr. (8 point b d g h i m b d g h i m n o p Prordr: 9:<;<=0> P $ WXYZ&[\] Potordr: =0> P ;:8YW^[\]_Z- $ Inordr: =;<> P :89YWX $ [Z-] Prordr: 9:8> P [\;< $ ]HWXàb=YZ-d Potordr: > [ P :<]H $ àbw^;<ydxz&= Inordr: > :<[\ P 9]H $ ;àwxby=xz Mnrd Hubr Pg

3 CSE - Dirt Strutur Em : Comprhniv. For th oowing grph giv th djn mtri nd dtrmin th rhbiit mtri uing Wrh gorithm. Lit th intrmdit mtri /g. (8 point h m mn q(r o mn on p rrrrrr o m pon p n o q(r n n o n n o rrrrrr!vi n n o o n o p po n po no m n p o po n q(r rrrrrr q(r rrrrrr p n t n n t no t n n u o n t q(r rrrrrr 7. Dtrmin i th oowing grph ontin n Eur pth. I uh pth it it it. ( point 7 Thi grph ontin n Eur pth:. 0 w0w 0 7 Thi grph do not ontin n Eur pth (thr r odd nod. 8. U Dijtr gorithm to = dtrmin th hortt pth btwn nod nd. Lit th intrmdit ditn nd prdor untion g nd g, w th in pth nd it ngth. (8 point 00 } q(r 9 rrrrrr Mnrd Hubr Pg

4 ˆ Œ CSE - Dirt Strutur Em : Comprhniv ~ = } ~ = } ~ = ~ 0 = ~! 0 =!! Th hortt pth rom to i 0 nd h ngth. 9. Giv minimum pnning tr or th oowing grph. ( point Š ƒ 9 0. Prov th oowing quiti or Boon gbr. Juti our tp uing th bi proprti itd in th tb t th nd o th m. ( point % Ž %d + Ž % G % *Ž % Ž % + Ž % % / Ž % % + Ž % = Y % / Ž % % Ž % ;(b8] ] % / Ž %% / Ž % 8 % / Ž % Ž % ~ = >.]b8 \>`& % / Ž % ~ = >.]b8 \>`& % Ž = Y Mnrd Hubr Pg

5 ž š CSE - Dirt Strutur Em : Comprhniv Ž % + Ž % / % Ž % / Y = >.`& % % Ž+ Ž % + ;?b8] %% Ž % / Ž % / = Y % Ž %% / Ž % + 8 % Ž % + ~ = >]b8 \>`& % Ž/ = Y ;?b8] Y = >.`&. For th oowing truth untion dign ogi ntwor oniting o AND gt, OR gt, nd invrtr. Ao dign n quivnt ntwor oniting uiv o NAND gt. (8 point P % J J. U Krnugh mp to driv th minim dijuntiv norm orm or th oowing Boon prion. ( point J ž œ. U prdit ogi to prov th oowing rgumnt. (A tp in th proo qun hv to b nnottd b th ru ud to driv it. On th ru itd on th pg t th nd o thi m n b ud. (8 point % Ÿ % % X % %d %d % J U % Mnrd Hubr Pg

6 Ä Á W W W CSE - Dirt Strutur Em : Comprhniv % Ÿ % % X % % X % ªY % Y«] % X % = = % U % Y ]G %«% % % D> $ %d %d % v % ^ % Ÿ % % % %«% % v % % Ÿ % % œ % % Ÿ % X v % Y `& % Ÿ % % X v % ±E> b8² $ ` %d & v % ª9Y %d Y ] w %d œ % ª9Y ³ %d ẃ ] } %«% ³ > $ %d %. Prov or diprov th oowing onjtur nd indit whih proo thniqu ou ud. You n u n o th proo thniqu introdud in (ountrmp, hutiv proo, dirt proo, proo b ontrpoition, proo b ontrdition, proo b indution. ( point Th um o two onutiv poitiv intgr i odd. Dirt proo ` Givn two onutiv intgr odd. nd ` thir um i ` % ` % ` ` 0 D7 µ ` `H B ¹ Dº For vr poitiv intgr,. Proo b indution `» B : ³. º w ¼ ¼¹ Dº ¼ Z F Indutiv hpothi: or Indutiv tp: ¼?¹ ½ ¼¹ \¾ ¹ 7 µ ¼ µ ¼?¹ µ % ¼ ¼¹ º µ ¼ º µ ¼ 7 ¼¹. Prov th ubt rtion %,À Á ^Â % Â Á Ã Á»Ä %% ÅÀ Á &Â % Â Á %»Ä % ÅÀ Á ^ %»Ä % Â Á %»Ä ÆÄ Á ^ % ÇÄ»Ä Á %»Ä ÆÄ»Ä Á ^ % ÆÄ Á»Ä % È ÆÄ»Ä Á ^ % ÆÄ Á»Ä ÊÉF % ÆÄ Á ÆÄ Á»Ä %»Ä Á ÆÄ»Ä. ( point ÇÄ ÇÄ Á Á nd thror Mnrd Hubr Pg

7 ` h CSE - Dirt Strutur Em : Comprhniv. Sov th oowing ounting probm nd indit i th r prmuttion or ombintion probm. (You on hv to giv th ormu. Th in numbr i not rquird. ( point How mn dirnt t o ing digit numbr (0-9 n b ormd? Combintion probm with rptition Ë % F Ë % (!Ì?Ì ÍÌ I In how mn dirnt w n th ttr in th word mtr b ombind? Prmuttion probm with dupit Ì Î Ì Ì 7. Form th our rquird to orm prti ordring o th rtion ovr th t Ñ Ï Ð % 9; % :8= % ;<; % => % 9: % > > Riv nd trnitiv our: ÏÒ % ; % :8= % ;<; % =0> % : % > > P 8. Show tht th untion % Ó µ ontnt. ( point ; ; 7 Ö 9:8;<=0>. ( point % 9 % :8: % == % :8> % = % > P ÕÔ h th ordr o mgnitud % b providing pproprit Mnrd Hubr Pg 7

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