CSE_20_exam forumulas

CSE_20_exam forumulas - ‘ Theorem l.l.l Logical...

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Unformatted text preview: ‘ Theorem l.l.l Logical l‘lquimlulccs \ ‘ Gwen any slutcmcnl variables In}. und 1'. u tzlulnlugy t and u cnnlrudiclinn c. the t'ulluwing Ingicui cquivatenccs ‘ hold. ‘ . I nus nl‘ ‘ I. (ummnfurrw funny.- ;1 A q E q n p p v 11 E if V p b‘plmflm ‘ . ‘ ‘ 2. rh's‘urimr‘w hnrx: lp A q) '\ r E p A u] A r) 1;: v q) v r E p v” 1:; v r) ‘ U : I ‘ 3. Dr'xrribmiw Iem‘x: p A (1; v r) E [p A :11 v {p -’\ H p v (a; A H E (p \x q) A (p u H j \ . 1. E 4. Monti” fun-5': p A t E p 1; V c E p "" ‘ {I' fr : h' ‘ 5. Nz'gunrm l'uu's; p v w: E t p r'\ *—p E c _ . 77 1'1" ‘ (a. Durrhh’ HI’L'UHH' law: "1 mp) E p In If)" I E I!" 7. ldmrpnftlur Imus: p A p E p p v p E p ‘ m. a" 7 Jr‘ ‘ . ‘ 8. Umrers‘ul hmmd 1mm: {,1 v t E t p A c E c 1‘. V is H l I \ \ , , ‘ 0, Do Mrnjmm x fun-x: ‘vIp A q) E ~11 V ‘—q ‘vtp V q) E «p A ‘~({ It). Amurpn'rm Ian-1w: p v [p A q) E p p A. (p v q] E p , é l l. Nt’gminm of t and 1': x1 E L' we E t ‘ E i 4: g '5 E g IL: E E ’ ’ ": 7 , '2 2 :1 I ‘ um + l I ‘. " " 3 Numul li1cl|r~1nnllcgcr~ | I _‘ + - n : — ‘ - E _— :2' 5- 3 - : j E ' .2 ' ' ' ‘ J ’ “I 7 I E 3 = .:' '7 Humnl ["1IIWL‘FH\I|I" I + r' 1- r" + m + r : I : 3 :3 = 3 I 2 Z ': .: E _ __ I '5 E .5 .1 — '— i r : _ _ a; _ I'nIIIL- l..\.l Valid Argument Fnrms : ;_ g : — '7 _ IZI' inliun _ I 'I f' 'I 7 h- I’ '1' : t 2: Hindu!» I'Imrnk ‘ I’ ' I! ‘ “" ' ‘ ' ' 3 3’ .= 3: .- 2:. I» _1 I z: '— ' .: .‘LJ ‘ [I q . __ : .‘1 z : ‘ ‘ I: rl :3 _ L I‘l-f r 7 WW ‘_ __ I” _ ' f _ I III' | E El \lmllIN'IhlIou-a ‘ p « q 'l'ransilml) ,u a q : I] ‘l' r E v _ E ”_ ‘ , E _ t 3 (Evnerulimliml u. ,u h. .,u I'mnl'hy ‘ p .y E _ j' [I I” if Din-null Inlut uM-x I, , ,. a I w ' _ , , =-.. 1‘ _‘ ‘ Hpvrializuliun u. p u [L r' ' *r 3 "' ’ g 2:" ‘- ‘ I :1, a. r H _L_ ‘ In ‘ ‘f 7 7 _ 7 7 _ 1.3 E : (-mljumfi‘m ‘ ,, [‘nnlrmlmnnn Rulv ‘ p - . .2 ._' é; , - fl H r- -|- .E ’ _ - i g 2 s L ‘ n” ‘1' g _ c_ a : -' . ’ I —— if — — _ .F : ‘_' I .- 5 E S - = E.” 2 Theorem 4.1.] h d E S E. H E = g F‘ . . -. j ' - mun ers an r' .— '- .— L. .— "‘ —-' '~ - '1 3 .“ |- mil""‘ ' ‘ U u -- 5 3. ~’ It a," H.n+|.u".+1- ---‘“ "' 1"” b If; “I‘dqt’lmu‘ flirt:er -- m' E‘ '- - — —‘ II - H ' ,' -' ‘ "man UHIH'] 3‘3 '—' ' "" Big? 5 3’ is. any real number. then th MINNIE! uIW'“ " 3’ ‘9; _ : z 3" : é" —: " r: ' J : : : : n u N -E. I A I: _‘—- -£ .— '— i:- u '3 r r: "J E H .4 7.4 ‘ I) = I”; + IN) = m . B '“ .‘J .C H .1 I H; + A ,. .. . .. . p— ._ _ 11 Azm kzm L=m ‘ t E i- ; _ " _ 4T 1" : — 3 '— H ” ‘v: l E ‘ ‘3‘.) 2f“ 1’ ': m . .‘ ; ' - JII ihulivc 1i!“ ‘ 2—: 1| — E E 2 t E m = I m. gumnllludthr ‘5‘ II ll = n (I 1.;Iu i="' .1. "3 _" I H n H 5 | E 3 t: é - .2.‘ ._ _ =4: -= I 1. nm - nhk =nm‘ b” .3 .- Ia ll 3 ._j '5. E. ‘2'" fin." l:m i ‘ l‘ g .i E \ .r' u d 2" ’ z "-2 '3 _ Ch I 2 E a: s I _ _ E E — 7 — * ' i ’ ' 7 ‘ 3 I g- ", z : - .T ’ ’ _ 743’ '3 5 5L ' r ; : lhcurt‘fll 3- -- _: = 3 . i , . . . , -' ‘ 11W. :7 '1 E h' H 3 \ w iwn cmhcculwe “REL—‘9'" m“ “W‘NIL p u ' E E : H H ' ' . _7 i _ i _ E — 7 - _ L. If H H I'lu'nn-m 3.4.1 “It: Qu:IlitIII-Hunuimlul‘ 'I'hmrcm (iiwn uny inIcgur H um] pmilivc inlugu‘r d. there His! unique mm.ch r; and r‘ r-IIL h that l 'I‘Iwnrt-In 5.3.3 i‘niquc l'ncturimlinn I'Iu-nrrm l'ur lhu inn-net‘s Il’umiaum-uull ‘l'hrnrvln nl' \rilhmulirl (iiwn .III_\ inlcgcr n m In. . _ . A In. mid pmliiw inuyclx r‘[. :1». .. n : dq + r‘ H = I'I It I". uml U; I" - .I'; "-lIL'i‘l .pL“ If. Ih.II I. Ihurc mm il pnxuixu iulcgcr A. lilNIil‘iCl [‘[illlL‘ quuhcrx I :uul an} nlhcrcxnrcwuu III II 11‘. :I prmlucl ul'prilnc numth i». inicniiml In Ihis L-wcni. - IIL'I'huns. l'nr li‘IL‘ order III \\ high Ihc LICIUR :nc \II‘IIIL-n. (ii\-‘cn:u1_\' rcul numhcr \. Ihu finur ul‘ x. dunulml III. I» Ilciincd m. l'nllmu: i_\ i : llIIII IInulnc illlugcr II HilL’ii lhnl II - I’H'i-i. 5y lulmlicully. ii'I ix In real numhcr and n i.» :11] hugger. Ihcn v Definition III j : II \ -- If -i- (ii\'Lfll;1l1_\‘ r'cul nnmhur \. Ihc cciliug nix. dvnnlul [I ‘ iri'I : IhIII unuluc iIIlch-I‘II \llL'ii lhul II 77 | i I '_ II. S) nlhulicully. II \ ikn I'L-ul lillllihL'l'illldH ixun il]!L‘_I1L‘1‘.iilL‘li - Definitions An intcucr H ix mun il'. um] uniy it". n out];le mic ii-l II'. and mm. If. H L'LlLllli\ In IL‘c «Inmc uncgcr plus I. Sunhnliuully. If” in am iuicggL-I. Ihrn - Definition \I'cul nunIhuI'r IH raltiunal II. .IIul unl} II. II c: H IN L'\L.‘|'I II Ix mid C? ‘1'? IIIIL‘L'CI'H \\I|h.1IIIIIIII'I'HIIL‘IIHIIIIIIMU!’ \luru lnunnll}. II I‘ H I'IIII-mul -:i- ii iuIcuL-I'a n and I! ~uch lilill :- H. c wmc inlcgcr. \n integer N ix odd Hun iIIIcguI'A such II'IIII n : I!» Eiun inIcgcrA such IhuIn : M + 1. ix LIL'l'iIIL‘Ii 3h I'Ullnwm I III hc mph-“ml In a Ilunlu-III :II' Inn \ IL'ili numth iiiill Ix nuI I.IIInII:Ii |\ irruliunul. I I ix :I I'ml Illllili‘x'f. Ihcn Ii I) .unl h T- H (iivcn II nnnncgmh c inlcgur H and LI pnxiliu.‘ inlcgcr r]. If (HI' I! : th illlcgcl' Lllll‘iiL‘i“ nhluluul when H' Ix L|i\ idud by cl. and H mm! d : Ihc iIIIcgcr I‘I‘Iuniudcr nhlnincd \I. hen II I\ tii\-‘i\iL‘Li hy If. S) mhuiicully. If” and I! urc melik-U iIIIcch's. II‘IL'II uhcrc I; IIIILI I' an: inlvgcrh IIIIII [I H (III I! : I; IIIILI II mm! If I I' «I, :27 H i II'II lrl cr 1:, th‘ uumuuy n l‘m-lnriul dcnnIcII ME. I\ liL'illlL'Li In hr Ii‘lt‘ ‘ L‘I‘~ I} In] I [H II: n c inlcu Imuluut HIKIII lhu inlc i‘ur cuch ['Imitii III: and I! an: inluucrx. Ihcn V‘ I l ! 'iH :H H! 'hle In I! II'. iViNI N de .uul uni} 13.1! : IPA I'uI \nmc IIIIL'L'L'I‘I'I. Zvru l‘m'turiul. iiL‘HUlL'ti Hi. 1a LiL'illiL'Li In he I: il\L'i\. \w m} IhuI \ilL'I H' H is. a multiple ni'u’. nr .! ia n l'uctnr III' u. “I III I! is a: (IiIimr III' II. I! (li\i(lv~. n. CI 'IIL: *3 WIN—CI In Wei \\‘.P\ii — i “'ii'iliiiii‘_ "Ii: ‘i iIIL'|| ilii: in. \ JIM I! .II'L‘ IHIL‘LEI'I'N. il I‘ "Hi ii|\ itih“ H iiiL' numnun II | n i» haul ” \xnfimduulh. "\iII .III- I III. In iilI II w -i Iliil I i I IN. i-‘III IIIIcuL-I'S ~III'II Ih.:l II 1.3 .I‘iII ...
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CSE_20_exam forumulas - ‘ Theorem l.l.l Logical...

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