{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

07Ffs - ti 1" Wm(wt w— M We 3 xéudtwuk 26132...

Info iconThis preview shows pages 1–17. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 14
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 16
Background image of page 17
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: ti 1:" Wm (wt w— M We. 3 _ xéudtwuk 26132; pron/3‘7)?” :Pmig 3% W I (aim; Mam. Jams.” .15- 99.66:- _ @SOMJ‘CL Inspr¢+1MW m réL’ZMMS_aP€. W E l __ _ ..... ________ ._ ._ . __ '_ J PM?! This Wblem has numLJLfg .1. +3 25: (1’51) 1011*} I“? 1F?! In blanks. Wfl‘l‘lA S‘f'nmbaloud ‘11: Mike, L—lvawlraion, .. “5-12.. Pack. _Jchr my and 1m .mfifii; 1 [(?=>Q)A(Q=1>RJJ=§ J I :2 E~JP=PPJ1I4¢P _E_ 2.11% (P=;>@J] => B :3 111111) JPR] PP __g__ a.‘ {~12} (a 11.19)] <=> A 11. [(PAQJ%RIJ <:==> ”L “I UPPER] , ©.__E’_ PJEPPLPPRJZJ 1"“ __[’L_ 5. .11 PMPPRJJ <=> __L _ .15.: [P31911211 «P _N_ . ,LPV (GARY—J <~e> _G:___ H [1111111)ij <==P R . E PV (9.11%)} E19 i 11. 111-12111) 11le PP .JEIPA..(Q.!\ RJIJ PP... _tL PP_-_1131111m111==> . E FWD. 3 _ _._<--+>_.. 5;; x1 it “’ (?Aa)3I (=5 D... WIDEHarfian__ 3 Law __ :1: [~(PN9JJ ¢%> E1 WW 11° AND“ mfiIfi‘Q‘fi—‘f‘H 8 1 '1 ._._— ....203 (311: JIPfoI E (an 1th 1—_me. __11, (stPtxJ 5113 L11) 1’fo £1151 (Mm IOA‘H‘QPZ.IEI3-3 FEHM Hanks w‘H/l II__S_I.. PT :5 NOIITI—I 2211;»"62 15: not __ GLLLJIW 1,1, (SJ-b. P 7.15.5}: “—93_ L5 fitémnrriwx .130 (F91) TSP (”PIDI (idJ EOnPPIJHPPEIanEiJIM 11111.1 11.11. : COPPPPsE I11 :QP_PPPP0$_UIP.E. 111 We .. .C:€Jfl+1’m2§§|I‘HV€.. P'f' .J’PPQ «is (“6!) 73’0”?) 251m. converse; _ 196E PE _. 61-- =>_P_._.. [For (1.1); 1 -11 :_ _ W13’,(21JPPJ1(V_~1J1PPJ(P(11)A P1219142.) 1 P WP (WQJ ACPIJ/RJ 1111M QJP>R 3162 _ J1, _(PAPJAP _(P P=> 6.0M P—RJ 1 12/1 («PM (1» 61). 1/..C..P\(.GI..J..JI R .PPRJMQPRJ . ___. P (PP) P (PPJ P. =>R Pi (.PPQJMar-JPJ Z} P ~61 P254431 61’. (HJEPPJPJJ _ Z. LPAQJ J! (PAP) CPA PP. 91,11 E‘é_____.£3'1J_EPPPJJ TITI f 1111136111111 1111. 111. ’ . . ._ in _.. . .. ., .. . ...... _... .. ..M....._.__. _ ..—.. .. ._ .. _... .. ._ _... _.... mg. I 1F=>Q<=1>P A’VQJ. 111101111 TE... 60111111011155 huddle w .. . Elfploufl 1’96? 4:) 'PA “9 MLC‘KvW1‘Y9AAvC-1‘Dmn 12mm tat. M... £1.11." 1.. ....1.-JJ- 1.3.1161- amowwvi .1%-1111113 Make/L751 . 111.14.111.11; _.....gle. f... “Maw/112'. 1L... J1_M.O11€%W1F)_ 1141M _ .. (21) T Ill— 1‘ T l T' ---...1 1E? _=_> (617111)] "> I; R. (1" "-5 am)“ jm (7111) _ .m-Imm 1’_ 01112 71111—7 131.719fl --|-_-— T F.-I-_-_ Tl-I-_-— -.-|—-_ ..II---- T ll- 1‘ T T Curde. .0011... '1 I ..= .11 ., . " $1.101?» ' T116 Mrwhmaj 151-111. (111.111 11.... 111.111.1101» 11501111111143» .11L M11114.- .. “ml-Fog .2001- -.<31....$:: 1 31» .2111: .1%er..1’11 3750.11.11. 111.111.111.111me 411271719 (17711111111113.11131111151111 Illl I r L35) Whéjmng @923me i EMIJCT _QMNrIFiERSM . _ U55. 14a. _L It?" Widen-c .5: 0-,“ classes. .._.!___.M...£1t).. =.. 1-25,. 5.5%?»va 51.555 L Fix} .5 x 55.5.2225. £5555"... '_ 2.2 ML m+h 0&5555 are £251 LLV 2} I . .L .. __ . . . _ ........ 2,:3m1, med-21 5.3555933 are. lem (3 ’X 5110': Mbfi) A F0301 ; (mm) agsmflm—g 0) . .. h .. of... " FALSE U 7, (cynic. WWPWOJtQ-VEWB 5| 5555;4'15 £55;an '. LL (:9 ' @59551553” 2.2;; 35qu w ___.“ ‘ (Brut. MSW) L" I 2 ____.__L5m,n{( we) Law-5201 I | I ' __ 5 [Is 22;; W '2 . ma QL‘FALsgr-t)? [c5515. 555—55 war} IITI f i ' (5 til-'7' #1 12. 3 . I. ' it. mm:- M.)_ ML LIL_..(M+..M) m......‘._rsroam~de ' i .. . . . .. _ Problems with this title throughout this book ask you to analyze an alleged proof of a claim and to give one of three grades Assign a grade of ‘ A (correct), if the claim and proof are correct, even if the proof is not the simplest or the proof you would have given. C (partially correct), if the claim 1s correct and the proof is largely a correct ""'""‘ ' claim but contains one or two incorrect statements or justifications F.(failure),' if the claim is incorrect. the main idea of the proof is inoor» __ __ . . _____ __ .. _ _ __ __ rest, or most of the statements in the proof are incorrect. ' 'h " . ' Justify assignments of grades other than A. Whenever a proof is incorrect, _ f explain your grade. Tell what is incorrect and why. . ._ _ . . S'Jli'Pfi‘S'3 x 311d)? are integers. "..____ .. ..._-_.__.._ _...._._ .. . . .. .. . -Claim.1fxandy are even thenx + y is even. . ’ . “Proof” Suppose x and y are even but at + y is odd. Then, fOr some 1 integerk, x + y: 21: + 1. Therefore,x+ y + (—2)k ='1.Theleft side ofthe' éfiuatio is evenbeca iti e sum of even numbers However, the right side, 1. is odd. Since an even cannot equal as odd, we have a _ ._ . . . _ .. .__.. ......H .. mlconn'adiction. Therefore, 1: + y is even. - I wit. 1” -.__._..- ' .... __ -___ i- --.: _ imiififfii'f;wig?!)Hiram/1%MM) =>Lx+3MMM>J _ - __ _- T’Eé..'..;ttr{.-.ot'_a ......MQJLM. g. __M —liéiffi’]""."f.:_7"f_'ff: " ("so M is - trims/ism ____+_.____.,.__.. ___...__..., __.f. _ ___._ __. __._..___ _ _ J I. 91' tiara}. %.._.Attm_f ,_ if.“ LIA-MS. 20‘s.. .Mf/Wr- 63W commas-.7! mg, wro- 5: est-Md Ina/Md" .. . __ _ _. _EW _ -'.-‘ i ‘ __ Mastiff-...... ..-; 010.“ £32.12M dip-ta»; Stair-.213 JfiM-‘u ..__._ wad. s Lmtfi’ow'f AH. . . ___---' g C . \J _ ._ , {:1 I. L1H}? :QJ'JJET', ”a 31.2% &J.._..__ _.‘fflff _._-5... fair. .. __......... .. ....._.__... ..__.__.._... - ... .s-.-.._.- sf::t:<-_2M-.E-. song's--.§_-_._-__ -_ --___.__.-_ ._1 19) 1‘ Claim. There' Is aunique set of three consecutive odd numbers that are Iall prime. , “Proof. ” The consecutive odd numbers 3, 5, and 7 are all prime .r Supposex,y, andz areconsecutive odd numbers, all prime andx ye 3. " "' " " " " "" ""'" -—.- E Theny— _ x+ -—23ndz - x +4. Sincexisprime, whenxisdividedby 3, I theremainderis 1 01-2 Incaseflremmamderis 1thenx= 3k+1fer -J-- some integer k >1. But then y— = 3: +2: 31': + 3— - 30‘: +1), so 3,) is —--- - .‘ not _prirne. in case the remainder 1s 2, then x— _- 3k + 2 for some integer k >1. Butthenz =3: +4= 3k+2+4= 3(k-l-2), sozisnotprime In either case we reach the contradiction that y or z is not prime. Thus . _ x = 3 and y_= 5, z = 7. Therefore, the only three consecutive odd, _ primes are 3, 5, and 7. I 3.32_3.j;:...,r; 3 3 3.... "W... .3 .33.... 7‘? .13 5,3,}- -33 {I113 243.. ML MI 113 IIsi MI..." 1‘15 33.313. 9... . ..2.1...I£_._...p-gsa.1mm£— number“ .3thQ..I’I.. p>i __ ___. 3-3-3331! fine, CliVlSIm. mlgmihm :Fgr_Z (dwdmfl )2 A: 3) I I I3... 333) ..-..-3II33»... 3.2.) I2 *3] 51.5 zit—la 71/ ’5 (J (.{agl’} 95% mwfltm ntmk 1 . 5.. CWsialer problems 15'!) and. (—5 21 Br m1 flambm rxanri 3 AF. “3.21:1.Irm11m1 . Ham ei‘Hmr .12-. an”. '18.- brad-1M __.. ‘ 313117314171 tum um. no I i _jizg {LSW rad-smol (5% _._fo "E; Q a) (3P£Z>(3 léZDM £1,710) IQt {gs mpg-.5331 5’s) _x_d “Q. .... ..., . . (50L) {WINK Weala 145m? Wet/{1 (5‘24) :35 5118’“? Mt?» WIRCJ 'l_,1_._. cm. 1 1.. . ‘f 1 ! '3 " a. \ .. ...g ....I..E2.. ....Q. .s.~.:‘.ugm.,...g._, con-Lmfiswfluen . _..__...... ..numbu__.o£ . _j’.°=-..._. 9.5.. ..._?ng.._._£.€25<.—.+o(6...-.. ”(3.11199 3 . . . . .b. ..._:.s usquarz ,...b."... 999+9~1fl§- 036$ 3%? huMbtK 0-5- ,7‘3 . ”“5 .?";".MQ‘ . .799‘f'fi'9r5.“ _...§‘m_e . b9: owfi-QOS. £4.11, ..¢!..e...n “UH-bet“; 71.75.... -9155... . ..rfa.f:m?-... 9;“9t9rs.)_._..—Fb?-..g.orfi79§rn§. 59m.“ 9dc}_..mmber L } l 9%...1}- . 0.».5 Vane-Fufior’é- .Bwh ...s-‘rif—t- Hub" :5 o? J ‘3 . - 1bmWSr-t .. ..cdso. _.__(_Q()4-¢.,_a_fi__u mn.....e¢.r..u"- ngmbu: .. .. _...... o-F. .. -33 -.....5‘25 riff“: ..‘L‘csp-‘tofsu flpwewerr ., .“L‘xzfa _ Cannc’r____cc)n+fi;:.f‘_._.\2%. “may- .affitm .909? 993.5%. _ WWW oifi‘smmbm. 1.115.“: *anb. k°*‘\‘é- 5 .....a.L.®..‘ffif.c-ac§19fi9!‘_' Theta-hfe Jim? 3%.“..- .lf’fif’flfi-‘u. .:......3?<‘.'.-—JN ; ' 9 now. n2} [m I «>- - m; 3.3...-“ i _ . .. Lit X. and- «3. .be, wJ.,«.’ai. mLm'F-«EJ.5.. :‘E....«..¢.«'..l.l...prwa ...:Lhai..«'1£ -X.1.f3.._15. ....ietfi-iiQflCki. . .me__ «511351 ..{T-z- iiififizflfiL. or ....l3 {3 \«_('q.'l‘_««:«i_‘«€,L.C'...... Iwr !.l..._£‘]u..1!i«.!::.>.. . .. L43Conflmdmhm ._. .._...__ ...-9.12an .....Ai. (R3. 43. 5.51.3.md. ...:La.:3...fiflh .__.§.«.r.«.«.r... u..(fl_,_.:fm. ma {Gig-”$1.5.” .....ipi'and 06. flash that. x =__I/%._pw.zswd€d. (b. i5 mat .250. ..... 61‘*—«I'i.%l.-3_ ris‘ng; ...«3 2-81,. .... .. . 1mm. _m {311333 a .f'r‘.‘ q...;.—.«..«._T~s‘.‘ ._ .s.u.ci'.=._~.tbei. ...«3 :...12.,..xer-«zv.«.id.r;gi_51.1). ..Thm -.x.1«3_... _ L r - 5" r '- I ..r....53‘.23' “P?r ...JQ.C£2_..___%_...QJE1. 5_._...§v:«e....-mLTcQQ}...._.13_;.t0_« Ti.".«.«.«_3....Xt.«3r.§&.. THE: _ 43.1,»... fogifléfldidibfilfij 5‘? J’.£.Kt..«3...is .ermiiornfl._ ‘il‘UIJ‘5.11..1...'._!'=3félig'fzfl:__..§f. ...ukt'flfisc ! - _3_ __ 3—. - _____ __ ..___ - I.” _ _ __ to; 5 i i 3 LL 1*— M E V i, -_ é. cam mm; Uri) wk (.91).... I ' (cpl) umzflmotwux-I) hot mom .. f _ ' __ I. (H) iLetaL Mdumwfifw iIE--..flaw-.wfsts...._irrl'§[email protected]’u ii m1 41.11.231.51 __ quid}! 3,- . ... Mendfidficéno" dtwde._a_.arddocs nafduwalcb . .. .I f .. 13mm Halolfam Mmhmsml woman/Ls .. a ixazx mm _.<=>C13;a/Z)_(_ex-.u.g£)__ _ 3-91va _u aoloL <=>K33&ZD Lon-234) I . -1). F3!“ ...QJA LEE—3 _ ._ _. .. _ ...... . . .. ._ .. . M chum d _fiéfili .....$?>-...(3_£.§Z)_U12...“- o\ m4. m¢01___.__ __ (ea) 1 WRW thmggumfim,(m 2.4; (H) . ' '_ f " __1:! we; an W. Wm: _w. ___ __ - g- cure/among. 5Ww_we%. _ .. .. . 1H2 4“") IE. £6Z,m¢§w« 30-461 «1.19M (%IX£'.I)W7. .......i(¢"2—)-.3m (Mmfléfmfim Whig . ‘Hu. growth-n i?! “Eh E9005; +£91— ¥qu+hdafigjgfiz mixgr—l ,-. -... .. .. . IITI I K 5.555755% 555.5 :55; 5 .5 J I I ...Ui. II I}; nn_I__mW ImfljIlL_[21IwL_II_[£IIj’IIIIdpt1III_n5r_;ItII_IIcIr'L;:'CI_& (5:91)” __ JIJHLILIIII .....X....uf5::_LzL£_5. . 55L u.)~iI_dQ_flgI5§IIl5(9I_ .5151. Tfiwcciflatc. 535555 559(55555555555151‘55 _J_rI_ oflmggards .-., I__. ..._153- IL W “If .25._ .55. ..Qdd 1‘15“... .3. 33252165-.. £531.35. .LQFT.1_..§.I..’..'5._._ +1.1!!!ijqu efiamflai 5.3.12.-. JR... uraag €510 -_ .. _.._ .. . ___... . 9055559....11142111 Odd anflcfiu It“? .. K ‘-= thI .. 315:?4.I_._ ”Sax-21!.=(2.'”DI.'.I. I .3. 451551554 ....‘-‘.. FILE tilt. :7.ELEM};.....L.Lpp...l¢5::5¢_s,...._.g.i_5-5-4.5195. ...EE 515.55.915.55“ 155m_kILEIIs'DI51;.IuIIaIrI5IIIILIand a'I-f_I_IinjI9dd _+_I\e_,|"\___IL_'1_I ___r'__5 {55'1"in . 55535515553; 5515:5595 32>... “539.55%” _mIIILI +a__ {Mini}. .51 III-555mm 4“er _‘HZG‘J _fi. 5999.5 3.5521 .. 3055M, 5.50.5555 .__x}:l= .859”-.. “95%" :5? KIWI!) ___1555 '5 5551:9555 .5 55.5555:5__ ___. ... ___ {.__ .__ ..._ __ ___,_, .._ _ __ ___. _. ..._ _ __ _. _ _ 5. _ REM? mafia) ALL: ins-3‘“ gbgm ..-.T:.'L:fl§£;_1__m. LQQ..(:1): ___... :. H ...}w._).'})1... . _...._ . _ .. ...... ...... Vfiwe i—fimranémufi m M Mr: M " " " 5.3, mmw .. Maxim ______ _. i (W2) L-For‘ _ em f!”- N . 3 i i ! E L '! l ! __ m“!!! as”!!! "g! MW!!! 1mm 1cm) : LIP. .S. 11.5.. .0- $9910.35! 6!]: .IN..._vs!.I.T.fi +£95.53. W0 Ifirapu'fi‘és . . ii!) 4.55.. KT“).-. . “For. asl.l....n£.HJ,.. dings) M ”iii .8 SJ. . .. .[va .5 “"..1N..! .. .. .....- .JUTSflng +iru, FPn‘nexF/Ga Ag MULHLWWE ELM!M -!---- . . . . by“; 933 13 ; low! mib..ql’bs {F J (7'79 M (4-7).. .. .. " .Iczm swag 7!!! Elle—ml! w. " % “mfi%'b1dmw’“%mi 1...“? maxi” ‘bfi ,aMIJ' filmy]. M _ .. 2: (5E<L] 1 ”LT-1.! (fiwaMA 1Q? né—fi':+l,m HRH). (5(E-f.lj__u‘i:!> (ms). 2}.. ' ,_ . 754$» ____ . . a ..._;_.__._..._...___..__.__.__._._..._..._..-'.-.._....._.._.. ' _ ¢ . , 'n 1 i;g _mmgm mm -_._..—.._........_.._....._. _ ’L .311 ‘35£%1 W ...
View Full Document

{[ snackBarMessage ]}