4 l .: I, 1. .; L r It, ('3 l. 'l ' mn ' 1' Up," "'Til ( II 1" l c x ' lL'A" it v (I) l) Hu 'ln xx "rt-mu "ul .. uv (' UH 'nm H ' lIIt-t vu l "nm ut ' gr "MN "'an 'X'H'lflu'r'h" Nn'u ' null ' V'l . >, ' , 4..->*'.n' " mm" '1 "Hard (NL ' ml "'" (H hu'ulls "varflnuh' "'WT' '" Ul ' ok I l and llu' Hill" "'T'u" " '1 i , ' H' In < x ni ir me lunnn si . 'L', Jr, fl, "' "'lxtrnu. "(VU)" . ' >.l lib.x ' . it 1. (Mann) ami (J' likes cake)" l . ' Very * ' r. , ui . Ill-'ixnifihr'xklil funny" "grimm" ("My - A. "prv'r. "to (N) I ' ' ' l'(:r " Which '1 ln) 'l(;l"'H(l nu qi > xu "mud '0 "mYH-N" mum"? LN. , .' w vu" MV II "He ) l'" " I. , l hl-'irutvs t} ' . '" "(l' Ui lhfl'N' words is used. in "HF" "xp" ht.' ' . te. . Drum" Un.- ( l ns p(""'. Lot the universe of (YXSCU'H'5P he the su ni Hm" " ' jlfllrxp" , . , al v .. * ' ni ltt C be we ll celebrate? . Lll -, - ' ( scope: of UN. fm," ti (VUDU) = C, U) ' r.lll(:r('1;c)is' 5 Just I'(I) in the fi . . entente ' (VI I' ' the scope of the " LE) :> CL ri" ovefb'bo(b{s (wantif' 7 mr (VI ' is ' > ) I (1) = C. One of these two sentences "102an 7 VVC, C' 1 .: (ibrato " z- The other means "if anybody passes the course, WC (Ilint' Th 7 *" comma":- stntcnce (VI)P(;1;) :> C umm mm be wrum" as Kenyan" :;v C ), x . l . aliases the course lnch is which'; (b) In the sentence mp am i (Ex)P($) : C, tm , mo'thech " ' lnnufiu (ax) rs just P(m). The sentence KS) means it" somohody puSCs "Wt and m var (:( r . "Irma we'll celebrate" and it ' fl 1 . . is ogically equivalent to one Of the two sentences (3) (it). Which one? "Bob do" Marie SMS entenccS him: she Ptm) he the sentence (am? Suppose 'Nhich oi the two s Exerc' me 3 Let t. x ' . not, km)", It." Let "IS I'lmVCrse of discourse be the set of French words. Let "Bob does not kno go the sentence (V$)P($) and let R be the sentence W a word of French" and Jacques replies "XVh'rch word?" Q 'LIKI R C . do or ' ' y 1 thlnk lxlarle meant by what She said and Which of these sentences did Jacques t ' X about how everyday language compares WWh t' le meant? ' , . . lan nd Cofmmcnt briefly on what this example illustrates . , . , . , 8 ga" O loglc? Wlth respect to precise expression of meaning. is true and the other is {al is true.) 'e. Which one is iaslse'l Y'vrove that W Exercise 4. One of the following sentences IS false. (You need not prove that the true one (a) (VJ: E N) (at is even or 21; is odd). (Va: 6 N)(m is even) or (VC!) 6 N)(CE is odd)- . 'x is true and the other rs mw. E'W em X iaise. Whireh (me '3 he following sentences rove that the true ore is true.) if cc < y, then (Ba '; + a = y)). then (an e mate + a I "93l- One of t is false. You need not p (a) (Vb' Z)"(Vy E Z)lIN] ( I'Jir,'ll IA I'Wi' (us) If, [WT Ilun,' [Hi'lybry ir L' 1 MJ: "walt/ul Hu Xlaxy'y{:u11'k[ (If; 'I'Ill'urr'nx. 1 "fi: i,:,u':.'x/ mf" is; un! wl '1'1 III II' ( Jr 13/ /4' (VT '/ ' ' IT, 4 fm "; ' "1 v. "I' JHWI) n [warn/IL" {un/Iv N'Y7b ii "1 ' 1'('1 r') is [my (wi/ir; Irt'ull'h' u .H/'(.r) i< mu" ' I' IN: Iv ' 'v" V; vr . ' . .. >w / IA 1 k ' 'v '(,t'.',. in I' 3" [Inn-mnuut '. A ," , . n . ' " 3 r' !' il" ;' (r'u' ' ' vu" " I ' AM ' HUM) wtrnwlu tlu Nnn ""' -.A,A','4 v v ' ,( iii' rlw wntvnu- "(BU) x /' nm" ar 1' (h) h'u/u/MH. (a) By I< . v w , am (1)) For (M'h I'm] number ;1', we ham! ;1;" # Ill 1 0 Exewcim 6. Let I' be the sontonce ,-.--------"-" .2 ./' W > O . (Vr 6 RW ' "J' + b " ) . , {dent to a 1 v Alt .'. Morgan's laws to show (ha (Ex 5 am? - Gx+ 8 < 0). (ac) Usc (me of the generalized De fin: not be eddy?) any fi) 6 f' twain: (ar Mae? maui'baammmme (3@BM2@m/x+25sx4r.'w5M;v Uh? Hi: up";. Ingu' I "uf'vllt! IN" . ' Ill. III "" 'Q'II I'llv' In" Al I'. " b br- Hitrlt' Hum "nr wu h xn V(Y7 O' 4 xp]: 1 ' t'n'hi'v I rim! fI11'1('l'< ul HIrI-1 """ "I d I 4 'I." ' It nc HIM" [Pm I'I'VAHWIIH' Hz: "'1' n [In ' ul" xl I Nnrlh (mv x vhh Ut ,. HU" 7 "MW" "IN" walt w)' Him". :( 3 ' ' ! , n '" Hm I ' anybr : iq 1 C'o'v ' V SJ:I,N!,M.'1;:' "uh. * k ., IHI/'l'<'.""'l[=f? wm-u L um I'u'lru;*':'--V:'iw , I'll" fmrl'k 4' "A 1 IN ' u.'.i '1'?*' Illuiwxgq ' 11' v-', v. '. nu , ' . ' v v 11' ' x ' - .. Itu-Hr "my." wnr'wwx UH xin lmltl , 'a Run! nr fr H; k' .. . ... X' I' l ' i h, ml" .5 1{'- 1'1'" ' " 'u 4 ) fi- x ,. RI ' , IIIIquzIJc-iugg; . l, H 'l , x , n."1m.wl)'"f "ul H'" " ' ' ' v x " uni yu v' ' ' "Hull"; ' H'I nhl! I:,X.'11n"y)p. '(3', Au ' P p I ( PI 4' ' ' ' . Il'r u ' ' 4, I ( "! 4 . ' 1 l s I . u '1 'i'J/ IS ('llHJlfU )" Y' ), H 'I lz' "" 'l ., A A am" .;,(n, ix ;' H. , I "HI '; ;,lu "'"Hht'r and; [12'1 . r In 2 ' ""'7 n y/ 7 h .1' 1 I ' ' l v' l"""' n "" "UV 3.18 1' 'k ) "am pu: UH Ill 41 m ' LXduulmu '1 IN "'"anu' (VJ: .' , YI": 1' HI. I r 1'" l ' , ' "Hd " 'k/ ' In no ron ' , ' ' " ' ' *' n nw ' ga lu ' I'u FI: r" minh" "/ "UP?! that .1* -.' J' H ("HIM 5" ' "Il ) IN?" '" 1 ' 5 9 " 'I' 1' """ ' 3 ., i - -' ' r , 5 fi" 21 mal mulfln (Jay "Hmv "521/ mi 1) is true. Mruuw' ' >"" ' s l' ' q ' Il um ' ' ' ' I ' I ' ' JJOI' sm; : . ' v x;. '5'" "" (VJ: GENE"; CR "Mt 'l" %'f/ 1: "WI ;/ ru 1 > .m- "' l u ' "l w )("'+.'-,'/=:1)istruv. ) . hu ' "'h H'i" " 31'9Ex' ' 411 hi " vr - ,an1p}0 IT) ' v mmp , ,' (( . IC bClltmwo ( w v r) . wv ('Hmhu - l, hi Al . u; p . r -I G R I" s 1 manu " , > 1 s 'I ' 611111] to 4." This scutcnce is falw bcegfwv '7 'n)1d " '110 iwn (hilk'H'l'i' " '!l we 1 He." ' w uu' "x-* ' .1'xl?.1l'l I' , .,.I IleI. M . 4 H.' HIII'I'" " . A , L" G RXJ'Z : MJ) menus "'lll('1"""*"l' ', ' Vlu'u u x "v """"" "' "on" re ia" no ("ll 'nllw "1 " "'l X ' U'hi""l'l l" W "#*W 3320 E ' ' r ' "ample. 11-10 sentence (3 This tiL'IHCIICC is rho CMW be mega five. 0 . .1." m worm] tea -4 n ' {also became maw rein maw] g V FM such of am Mowm fiflmwa' g 53" ' EXEwL 7>== WI "Hawker"; Be) (it c W ' m " v vvv"""""""""m Ku hun il Qunn'ifwh 29 "u- "'Npr l "1 7ln- 1 l ' l-I-Ht'mc-l' I"v ' ' '"' l" *' hk" "" 'EWHII 'chl x Illn's ull'p x ' I 'Hr'hlufl). 'I'hh'mu- 12'. "walt mm hf "kvx " 'I'k" " I ll *( "Ir > ' ' lf- If" . in" 1 li "" n A . ' IF? .I " ' ' fi. uh "Ht * I'l NHnlul nu mu'nuxs ul 1 llll' Imuml uml l Hm! . .- ( m't'mrcmr ul 1 hk" 144'ulun'" 1 i ' II I l -I u nun" h "mim- Jl wnlruu llI mr'uh "l' ' . .Xl'l'"llul'v "If" N" '"u ' - , . v m um. I 'll ul" ' Am- lln- sum!" Hum: Am llu M'mvlu'r' ., ., .. [3 ml 1/ lll.' 5 II I' H'ulnl nlul IJ lllu's ('Jlliv'l " ' xu " AM "null: null xl . npc. Hf Mn. ' In". El}le11!u((.lnlrli" I 'l' "l't'n" Alul ' wxmmluu.' qmmm" l ' ""' "hlmullx unrmpun-I lu muvu-ul qummllv in "mud" v'm am "run". xl. . fl Ibn llc'ln'ml uh uh"), "1 Hu," "mil" F IN." ln Hm ' WY i . . Ill ' ' ' ' " Ill 1' ' r' ' ' ' l > ' It ' q . "nj Un. mu" ,!'."5 If: [MJ] En. h ' . . UNI llt x HIM pmm lyl lln "mu-pd. "fdlfiwunwl 1". HT ., ' . (fl) I c ' lim-"tx 'h" "Illrhl'. HIM! ll'l (' l ' " ' ' " l N I Mr "LHWIH' ll -lll' Sl'lllu'le - H "P (-"'I"r;". l''.'I/'I.II -.> (' ') lln Stu v ' . MU in III II" WHHH'IHPI Mtr} ix Uh, [J' 'I In 'I t . 1 H' wu l'lIH' li'r liPlll -.- (" . ty Ilu Nt'upr ml" 01 " I" tm lullfw ' ' . ' . l n l' is IN) N- ( (lnp .,1 (hm... Hu, whtpwn mwms -" ., fl : ' - At ' runv [ML-HP" llH' (wulllm- wu] w A " ' "'hH-h l.' wim h. Ii" lullflhl'm' "xv ullwr IllO'llllH "If unvlmclx guy-wa llw rmuxa- x'-"'ll I r'l h"!' ' . - v uh lr' wnlunn- I'Lr 'I -' ' ' - ' ' ' ' M Q. (h) 1" lb" "whim 'l JI Il : ( uullll] AIM; lw wnnml As. [(z'nl'l 1. z, (' l 1' i.l'l["'.l'l r: ('. 951 lhc' sin ;0 ul Ii" ' ' fi ' ' . J ( n; . - ' I ' ' "uh" in H I R l l 'lllllllr ...r) IF. JllHl I'll) lln' wnh'lu'r ln) llll'nllb 'U' Hullll'h'lll's' puwk' 'l . Il'"; . , - ' . . 'lv ;r n .nul It Is luguall) wunvulmn t m- nl thr nw arun-nun I'i) and I lp u 1 t . .. .. 1 nt in l. "'lllt ll uhv.' [vart'is'v 'l lu nl ' - - . . 4' Ir umu'h-r ul llw'nur. r-I -l ' -. :. . l . . ""I kill'" 1' " LN U lw llxv an . . ': lu xu Hf [ U "('h "Md" L'" I'll] lw llw Wim-mr "[Mlhdtws "H' , l ' l a Hm mr (.1)I l.r.) uml ln! If lw llw senlt'um' IIIPUl qupptru "am o'nj'lt' - . ' -*. . - 'f. D." l l" i"" """' " "UNI Ul III-url: and Jacques rvplivs "Which word" "hull of UK No t "my: nm -. ' ' . ' , ' . ' - f ' Stan cuccs (J (( wu Ilnnlx Maru un am In uh." hhp saul amd wlm-h .,[ Hug. wmpm." dm lawn"... nm P 1 ' ' lu 5 IF " hi! 1" (' " . ' .. ' l n unmu n! lnu ll} un ulm! llm c-MIHpIn llluslrzuvs nlmul how (n'm'diu' lnuguauv compares mil tl - . . v ' I it luummgv ul lngu'. with rt'spu't m prm-iw vzqnusion ul' muming. Excrcnio 4. Univ of the fullowmg 5("I'II(u('m lb "'Ulr illltl the other is fable. Which mu. '15 (me? pmve am ' Is fills-'v (Mm need not prm'v that llw mw mm is mw.) ' (il) ("if i NMJ' in" vw" nr .1 is mid). (b) (vlr v'i N)(.r is won) or {VI 6 N)': in odrl). Exorclse 5. One of the folllm'ing soulmwes is mm and the mlmr is false. W IS also. [You mood not prove llmt. the true one is true.) (a) (VI 6 ZHVy E Zil[il.r < y, then (an E NHJ' +u = ml. (b) [VI 6 FUN); E R)[if .r < y. then (30 E N)(;r +a = y)] The Generalized De Morgan's uplivalem. to ("P1) V ("PQJ and that, -*(Q1VQ2) is logically equi. . De Morgun's laws are generalizations of these. logical equi existential sent ences are generalizations of conjunctive and disj , > .; ' Here are a couple of exampla; in ordinary English to illmtrm. Similarly. to say "It is not . dislikes spinach." " Now here is the precise statement and proof of the ;t' ' ' ' .) hl "ml i("(.l , r) "Ul ul ! 4'-'/""" ' "wu v l-u's) It! ,( ' .. n 5 v' 4 . 'lx' 'lulkJ 'I' uh. 'I' Jv) fl"... Imf) ' v' """"""" " > .... .r l ' -v . l' * .> "sun" 3'5 Inn". ln. : . r'HH'" 'l! mu" . H .'.J.' I" ( fl ' U '()"r)' 's' "IIS" '" Hu ' in "IV Mn v-""" " " ' I . cs'""'"' v.' k ' ' ' ' .r) " .. ul f ' un. i'. v) "HU": : uh, ny". lvt il) ( "whlhl fi" fi." ' : " '1 u ix'mhw'L 'th'" ' wmvnn' l'" wi?" ' ' .. , v " l: - ' [1 u' . "fl 1 *' ' ' .. It . h" 'u ' um .['(.:') ' ' ' '0 (NJ') . - [UP It . .. Iv hvn[( lu ' 1I'J'L" ' ' . ' ' '11 l' ' ' ' "H'Y ' - I'll?" . .. . -. I (-( nw. Fl) vwmm." ' !r n lat! xhpmz' HUM) ul . . in .I IN Hu MN" 11 . I as Mm- Mr M hum uxu' HIM: ' ! "IN"? l mw UI J , . . '.'. ri 15 !:m' ' fl,' w w ' ' sulh Eh" 'I 1 t ' If" 't ;.. t "(J' 5') is "In l 't"(.(.) Lg tn" (IN IN!" N'Hlflu ' Rr c - " fur ("'t'r.' " ma' 0 n' N nh ' zw nw" Lahu' of r m 4 L? H " "UN' . Hr ' I' 13." in mm I ' ' "' 't'nh'lh't' SI; qumplv. IN I bc lm > 1' N '3 '+ x' Hare R'H' ' " - ' " tn ' k [) .' INLER'HHV ('qlu'dh'nt ' L' l ' ' "" "li" ' ' t ".'L Q ) l U' ' 5!' [Un q'rlwflha i Ix' NIUH'NH h' It (" "" K x n . '( v 9 'L 'r c' "Hr" - II + x' it ( ' ' Rr nm'm.' nu! to skip '*"" ""N' (M ls l' mw or "In" ' ' n ' "'Q' ' ' r' ' J ' 'I' Hi" . Id"- '-'1451-'?1. (u' H) [hr fin! gt'llt'hhl.t't IN MI .k ' "(vr 6 RNA": - Jr + 5 > U) ' " '.- ) ifl' (Hr c' RN)" - 41 + o > t) ., . < . v. - ,- - . 1 fl fl). nfl (511' t RN" "- + ' ' MU For wu'h md number I. we h.' Fu-rriw 6. l & ' ' . 7 ' . . 4 ,. (J. >)2+1>()+]:1>(). Hms! lhnllf'. . mnr' - r+o= -- - wt 1' hr Mw smmm- ("rr 5 RN!" - (ir + S 3 m. (u) l'w (mr nirhv gt'm'miznxi De Morgun's hms to show that 5P is Iugiqu)' "mimic!!! lu (ir c' R '(4'2 - (ix + S < U). Hr mn'ml no! to >k (h) Ls l' mw nr Misc? Exercise 7. Ix! l' he- thu N'HN"UC(' fi - X [if c RN: 2 Hand yJT'i-2 < grm'rui'ih'd Dr Mun: imdb" Hwixulem to ip any su'ps. (al) [SP mw U( [hr (2th fi' ' LN 1( 'I: au's kms and one of the ordin my Do Mm'gun's laws to ' k ; fix- p. r' (J'x R)(.r<00r x'j'7-22 vgr+ he mwful not (b) is p m' 3.7 exmnplo. here amur ile'r illustration . - .. u l'f ' wmo'u-"c' .r {z' .- ox 81 mr nr l' 4 him thumb" and lo! q n be " wim: uro ndmdn eu" .hf}u.u." "gps vhm-uisz" unh' (x'w'x' j ( me i (5 ill)! ia? ll 1 b -i'-u"'x'')u"' "(- i'i l n" x' oco '- fi ri" uu! {liv ("inn ul 3" in k (x n'fift'm'g' "f-"un hou", i' 1.' pr x h, "'pnllg tx. mv" du! immhoowntfi a ("hq' hm ) "l' "id." h'1'-',"!fxvd zlx (nz-"di in) il d. ('n to r tu z-'n-> , - , '-*' ' Olk'o '-" ''-"1"('hlxl' .r (kluxfidug' j' and then Wo: " t) IN hkt'cl ht" rx'vryhutiy' hvr Wurds. H ". (IN kt'ti ivy tw't'tt'hmh'." hr in nt V in thr timur-Mv YU "l'm unnu- u. u IN li "him x inuit hiu". l] l' * .. ' Ph'"'l' lln' "Hurt" l' s hht'r. lu" Ittt'nltw nunn" i/. vxt-ixlunl 'H t'Xt'lNanlx " am ntul tun n Wt' htwc tnkcn PU. y) tu tncun "Evcryhmiy likes "Fur smnc y. y -""' that "I'm ."N(DHN.',""' " "kvd l HHH'T wmtis l thu' th'ut'n'm "l'ni cmh r. . mui lht't't'hit' ithml iw M't't'yhudy. l'u. yi in ll'th'. but the scntcncc tigntVIH'Lr. y) is and t.tyttV.IH'(J'. y) umy hm'c different tiiicrs in u scntcnce. Now that lit sumnmnxu' itu- (' hhw y." in it v t hI)tVr)l't.r,y) menus ll lnlhmx that tX' MI'U . g/i hhni in Mt-nh "Stunt-inuit is W rh in") Nflrh iuH'tIlm tht- nt'th'i ni thc quan HH'HH* v-r I'lkt's UI" il. ."l H' "In nt'u'l "Unix 'Nh'. "' hk"" it) ('Xt'l-X'H'ti'." nt Ht u 'hmun. Hi this t'xamph'. xx that In igt'nt'tui. 'ht'i "i/ In thm "unix. lll ('Hlu lhc' ht'llh'lli't' lnlsv "its xhut [luv n tx 'un'nntnll' i" l'i'.' wu r't'nlt'tn't's t In . . . nh "am" th'HHt'll tn in tht> vxnmphu :' WIM h/H'UJH iht' Nt'ntmu'p tiyitX'J'H'tJ'. yi HU. y is free in the t'V-I'NBNH'U. gt. i tht' ign'ttcmi i'nm't ILH. "'hich ui thc wn'inhit's .r and 'hl' >h:tH M't' that this Is Mt it 'h nt thc inni' scntcncvs (am PU. Ui i> Hut- re 7 .' ' - ' . . nu lust' 8. t nnllhth' With thv Huluthm anwm thc witnt' Sr!" "ul". Ilkir. " ' .'. WJ'H'LI'. j/i. and t 'uit't'i'il'u. yi, it pmph' { Micn. Betty. Chuck} . To 5 int ('hnck i ms shuwn in t ave writing, we ".r hkos y. is Hit' sol t hc the some! and c mt I'LL!" Itnt'l'st' ni tiist'mu'mi hc t'uiimx'ing tnhhn .Xiit'tt. h htt' Ht'HX. is un .l' nm! 31 thH'ClM' 9. Sllppust' tin' n t' shnH n'i'itt' n in] -'f-- SHIHt'lHHt's W truth X'nhtt' nt" I'tJ'. fi"N'HN' th- (I, mwww i ['til yi ' U .1' li " l' rct'crs tn the cnhnnns. For instance. I'tc, b) is true and What does the sontencc (Vyif'ta. yi mean in ordi- thont each of the two sentences (Vyfi'tb, y) and of the sentence Wy) PtLtn depends on I. ant' that in this tnhlc. .r i'ch'h tn the rows and y PULP) is false. (21) Is the scntcncc tint)" lfnglish'.' (Vt/il'tr'. yi. Make "'hut docs llit' svntcncc (b) is the scntcncc (3.r')m;')1't.:. (C) Make at tnhic shmving hmv the the scntcncc (BI) I"(,r.yi tncan'.' (d) is the acntcncc (Vt/Mir) Pu, y) true or false (e) C'onsidcr thc tionzil "XP 01' inisv'? fistinns : ti'nth mhic tV.:1)!'trI-u) t Ansm'r thc MHIN' twu qnt it tnhh' showing hmv thc (Vg/H'LJ'. in mcnn'.' yi true or (also? What dues this, sentence meat trnth mhic ut thc sentence (31')P(.r. yi depends on y. X i in ordinary Engiish'? That does '.' What does this sentence mean in ordinary English? two ('(mdi scntcnccs t3r)tVy)1'tv y) =? (Whirfl'tlu y) and (VI/haLFW'LI'. y) => (31T)(V'y)P(x,y). f these two sentences is true and which is false in this example? f the variables I and y is free in the sentence P(:c.y)? Answer Wtwty). Bunny), and Which 0 s (Vy)Pt:ry). BMW (f) Which 0 each of the four sentence Here is an example from algebra that. is somewhat analogous to. ' tm)'H ('Imph'r 1, Logic "' ' ri :: : (I)(2')) Un lln' mhc'r Imnd. " A 13" : l .. - .:I'i A'i) : (3' [S k':"(3 1 1.17'1 In"! . an (. Jr) : (3 + Mw ' h) ' 't( whc'tlwr it is 1( nm in words. am v - - 't n ' .. nu! "hat I 3.13 Example. Fur vm-h n!' the" foil" Hm- nr ("lau mu! provo yunr sm'u'1;1( . (il) MY] r RHJ' R)(.r yi. (b) (91'5RM'UE RMJ'Eg/i. (6) {31'5 NN'JJ/ 5 NJ{.!' 3 y . - ri!(' ' ' ' "(ON u lmnng N'm" ,.(, exists it real ml mmlh"r 1" mu - consider "For (mph 1 c ,' "qu TO See (his. . I!)('(' !> ' ' < I) mvzms IN. ' l - J is me]: a value of .r. No" I/I) ' - . l l . - - fhlh >< " . . . nu" thru ' "( (.I r "isla," C. " '- '3' '-'- RN. *' ll {Inn (in) The wuu'nu' (le t: RHw' & ' Hu 1. ' 2' JT vqunl In (,. numhc-r r MM"! that .r lh hm r nun ' l. . < . N " I Honw lil' rf RHJ - Kin ' V any Ill" '. lht'n flu ; A/n. ' l ' (WI fi CV: S . q,A ;> an nrhilmwplum"! n! R. "(nu .' l.' IN!" "The", ON" 1 "JJ "It" N'IHO'HH' (ELF rf RN?" 53 2 IV (l {hm "HIS SWITCH!!!" n-nl numbm u. .r Is haus than or vqunl In y. "0 ( lel' Bm Mw" in pamcular' In Irma Thl-u w'r um pic'k In E R Mlt'h ("Bill 5 R.)(J'n fi: yi). "HM Ml {alma } that for rmu'ln'd n runnmhmiun, va'o (5.1' E [LIN]; rlR'HJ ;. y' "In-le") ("fists il "mum, numb". 'J MK I . (H 'l'hc' M'HU'HH' (if E NNV'U E N)(.I' ii !/I mt'a'ns I . h i.[ "liu "mum", is "mu- ro sho" this. It I . " ' " "(immiz . . . I ." wu-h "alum! munln'l y, , xx luv Hum 01 (thl H) !/- xl: ' NH ,. r li) is "'Up. "(l ('Jmm that l 15 such a "'"IH'PS In vxhibil n mhw mf .r MH'h Ifmr lhl' wmunr'v (-'1/ x. , - - . - . . . . -. - . . ("Price " v , - . '.'l'll:11X('(111(llt"l N H x'nlm' nf- ./' To an- lhis. ('ulhidt'l' un) [qu '5 NA "lo" 1 5 Mr N"" y" 15 (Ul H H ' fl - - y ,< WI"! 'nr mstdme. 1 lb ('v',I/ ': N)" f; !/). Hm prows the whim. Hu'H'flM'P (if ": NMVU '3 IN)" -' 'l/'J' in d N suv}: n mhu' n!' .r, I l-ur mrh ul (hr fulluuing wnlm Ul him: mu! pmw' ). botany" fr.) " it WI number .1 such that. for (fig) ts a {un v. To soc this. suppose It is Is 13an < ;.n # 1 . Thub. we have nos. "I'lx' out what it mmm in words. >1'{" xx In'thm' It. is true mm M.'!H'HI4'IH. (il) (31/ -;' RHTJ' , RN 1' ; fi M it). (h) "il '5 Ruin; ' Ru! .r 3, .r) (c) r' 51/ ti RN'W' '-' RM' ' !/ "- (d) (VJ r': RN 3.1/ ' n V' 7' ' L' h) (0) (lu: RHTJ'P RMJV !' (f) (vw y: Rilej/p mi "1/ H 3.14 Example. Hu in" mu:-- suz'l m'o ix ::. n'riL'UUHHI Hm. mid. " ;-I::h Hm! .I' is the Um' way to rumlw' ri mother uf y for each y. H' M'IIH'HU' no"; !11.v:1llu')i!{llulls WUUM be 7 how (XIMN .r am h that .1' is the 1110th 1 rlu- sruwnrv ('(r) muunbi7 ' to insert Aunrhor way to n'ndv Ibiluws: le scnronw (7 mother." What 'is." c . ' l'x""ll)'l' LN H 'w n Ml'bd' ' H Inmms lh; ' un upluv lunlnrl [ul o'xlr-lx hu "uh "hu h uh uh "vp Hui humnlvd u'mu 'snh'h'm' 't'. IN hmmdul uhth HT "his ('nmplvh's H l'xvrrisv LL. A> in laxzmu (il) H 5' 15 Mar wx mf 'd" rrnl m numhvrs J' M (h) H' H is 'Hl (nr Hu'h '! C u- Mflminn. ' lhv N'! uf n" H hnmulwi Mum". me'risv hu IN x mmmt uvlwl'uhn-d lh- X le'h lhn' lu! l'lu'h M be H hHH'HUH isis (5 TP Exvrriso l3. 1. "ml lm' ran'h Hu' gu'm-mh/ul lhul f 15 HUI ('n if cl} - t' and 3.16 Exmnph'. A> "lxnlnp'u' l') l Lt'! us gix'v . Inmhmla m'mvm'vs. Him svnli'm'v wv dcducv the cum: (VI 6 AMEN 7: '2. LN ht' ;: [HUN H. 'hl'H' 0 Uv Mnl'uul ui [n'm'i shuuhl mmparo wish in pzm'o xs ( ! lm' va'u'h :' v hn'gnn'n In": -H. err vx' mimunh m u 'x "J" qfirm .: lu I'i,1'.u') ln- n S" ! "lmnnh'd 'm k Ac" 3.15. 'M .' Ix" R. ku- c-x'bh: .r > from R m R amd km 1'.' hms and wh ng nn'u'vrsal z, c, mm 6 sm g m S is um bonn: an 4 El; E R)(:".:- f; m' mb C- R)=(7.r c; 5' Mt ljfh e R IUJ' if S)' "H (vh -j R.)l'?;.r > urdu-las. is . hun Hou () HK") Hm m we knuw 2 H (how 0:6st > U such ' Bt! ("awful nm m akip n p;- ' ' vt A and B be. me YUHU Hun-nr in} MM wv H H J' hk) Lr / In 'lJ' Ir) " HMJ' 7 in. HM Hun smxw} - ' H'M'! 'um th'. ku H) say In! about: means HUH'H'l'h. (h him .r hu mm, {or each 0 {my st ('pS. subcoflect'x wing, som ence "Us on his, ous oi the un is Mways Hue: rim e A)(31; 6 B)P(I.y). s is. Qnmn H'u-n 35 that b In Hm! them ' r-hnw 'hm S 1:- or her head. is (w I tends (.r) funds to '. Jun) -r Llu < s. L'sexhe IS at 0 means -- fia" < 5. Use 'wcrse of discourse.I'P'v-TJ) 7" J'i "- If. ' .'(: IIL:.H' -'* fl" J'ilv'i'll 36 (hq 1 ' ' . 'IVH'HIUM pm "fr!" schit-iii;'!),";'l;flv('llrm Muiljhil' IN." i> rrm- By Eu mun "ML 'm'. "Will bmw)" ..' 'r' "'Iw "mw Hf " Un- HH'IH {filth "lin" (J J,' [hp i'm) VAN? togo}: hp! m ml" mr". IN." In". in lim! lin'l'c "Hill" M. w. i, ,{ ("M'Uurfl' Hifi! "mt Pu.) 1" "up. {La In Iiu' lllnx'('l:';)l)'f] in H'rms nfnrtiinury quantifiprs; ('H (hr puniiiii t *'-' ' ' mv mine 0 . fl L Hi" It" ( N'N ihr' wnu'uu ( (12) lih'l "'4 fi UIN'V'.' 'ill' in. whi' "mild liunc- up him tim! thru ill'Il' i- ann liNI'H'. "s'ilrpsx Ii (' Hf. .'i)r' wi}: Jew/"(IN ' HI" mimi ii. .'i in il EUiX'HHl'l'iiUl] Of "IP ' ' ' iiv'se' iii-ui in Ill). Ui (U . - . . l uniqui- mhw of J' Hl .i MK}! that ['(J') is iirlltl'lfi? "7 iIL'i IS iugimlh' (wim . int tin-rv is i Him!" "in" 'E xl.'P(.!'l mmns Ii mixing is Iiniqiw mmus {hm it "wry umquv" when they mpa" nw)". i'l thing Pixiu-r is (he only Um, Of it's Whvn liu" word "unique" is used if"! 'iuviih! mink "' "1 '"NUHZ'W. Iilt'H i:'.'.i rrm't English. in say that an" iivni's pi'Opi' " i,)('('zuinv 'HIHt-r nil rim! in ('0 t' "it"! "very uniqiw. Ui. hrs kimi, ) "waning. if'lw Sil." i'o rimpx If is uliprugninrv In i'm kimi hi raw-Int years. on (Uin' should not say "very (he only "UP inwndwi x> liu univ nm" ui its "H'H spu'ml " (hiv "will! no! My it is i-li "waning is rho nun-ua!" or Lllui ul H h "it Hi Hudiil'HMfH'N. il- ('UH'H'I Eng] iqur' i'm) uHIIlifl'l' y such thm : l) mmus "There vxish n HH 3 "'4 (ML?) : l. and if y jh a real :" Bit 3 + 'I/ 2 is ii rmi number. 3.17 Eanplc. 'Hw wniwnw (5'3/ = 5 !/ '-* "I'M 'U 1." Tili" N'UU'HH' is mw iwwmxc - hm j/ : i -.'i : --2. fl jy v- ' .. ' >; Rum}; : R)(.I T- 1/ : i) mmm i'm mri: i'm] number r fhprp exist . .. 5 Win] In J." This sum-mm is IN!"- T" "'" mi" ('m'qdor . a ' a nu] inimhm'. J'n * (1 " "U) : 1 "nd Tn} 4 . r l I (irma-in of R. Therefor: li'Hliiu'! sIN'II lim! 3 <- 1/ -- 1. l 3.18 Exmnple. '"u' M'Illt'm'l' (7.1' u nmquv rmi number I/ sum (lim ./' + :/ i* 1 Il .Z '. 3' " ' 4 i ' ' > J i' w 1 Un," IMIJ/ :: Rn..1(,4+'(/= H is H'lw. hr-('nuw I - ri, is' M an mun = ' - . ' ni Hui] lim! .r.<, +y : 1. (hon y : I - In. Now I'" is 'm 'irhin'nx . > r ( i ' ' ('.I -? RHU/ e'-' RHJ + 'i/ -2' H is mw. 3 19 EXZU " nplo. "u' wumiu -' ' ' . -. '(JJ't "- " ' LN "In", m in This WWII") ik LI ! AR)" - 1) mum Their ('XISIS il Imiqnv i'm! i i . ul N' N'CUUSP '2 and -'2 urn Hm diffbrmn I'( ii i ( mm wr 1. SUN] "wt 12 '3 X'n lim, nf' , fi - ' ' V .. i )! Which ri . - .1 03 ) I ('N' (l .I --1 . I. n l ( 1 l l (' 't CHILD!) ' ' i ' ' l ) , . ' 3' I ( . ( 'l' ' l ' ' ' I ' IC .'! K I' . I "i 1 VI] 1 v ' )( - I . .f In" Pvudl t .. ,. . 4 (. -1. [MN . . '- 5l'Hfl'nH- x f- '- FQUJIH' Hf il rp", " d 5(' boraugo l. . mimin-r ("nm " It" LN no i'm] - v - lu! In; "('",H- ( KAI)". Uv , .' l , ' M IVO. ' ' J" mimi; 1"' - . ' " '4. since th e '.'e 14. Fur ('m'il ofli ' I wf " II 'J m mw nw "mw wu" "nmwnlluumo M nH-m'ca. write out what it ' ' (a) un mm + 7 = ii "Mm" '" "'"!'!-x "mw wi m ' (b) (am 6 mn: - if L'? U , u-thtrr n. is true (if; e Z)(:'' - if + 3 <-' U)' (c (if RNJ'B - Lr + 4 . m) (i'm 6 KW - Jr + 5 :i' ii in E- BNE'y e R) " - U. (8) (VI 6 RNSQ; C R U + y : Oj, (b) (Vw 6 RW: l 0 my = U' i r then E R My = UJ. fi) (VI 5 R 0) (VI 6 R))[(3.'y E R)(rg = 0') e