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Unformatted text preview: .. . ~14..o.n4.... _15_._,ug..&¢e— . . . . . . _ . +5141 9101—34-43 . _ .. . . i i 4. i | 5... 5 [ll _ ’/r5 i_.._ g i 2A; J } Ella—nus Wuem Illa-S numbafs j; 441 25 . (3'56 €01“:l’lcij-Fill'inblflaxksw‘fl‘HA'S 19mm balm 40 Mile, L‘i’awjrflojy , _ _ .mck...}g’r{t.r angel and. migmncg- . . s.§[(?=>a)A (61%)] => ___'_ ' 1.2.1:~(P=>a)1 4% 2.i[?/\ (Mml a; __ r3. EfiPVQHR] 4‘3 ___.__ 5.i[~?£> (61 M90] <=> __ H: E(PAQ)%R'} $9 _____._ a}. g[ mam] _<’e=>_ __ *5, [Péfiavmj ‘1’”) 5. {L PA (am)? <2; _ .14- _[ PéiaAR31® _ ;. ELM (am: <9 M - [1.93019 Pm] “<3? inlt Pv (awn @ __ if. Wax) Poo] <2fl> ....____. K. 11: PA (a A331 ______. lqflUEifl Mil ‘1‘""> __“_._._, “é IT.— HPx/a) 3 w m: _ $0 m (M601 {:25 __ «ark/Defianjan‘s Law. _ H. ~ (PMUJ 4:) M__ W AM 61'1..A.ND H19 ianflloézrjfillinblmks wifk: THE or FfiLSE.__ m! (311M903 za (3a) m _ 21,§{3x)__P(x) _ :3 (3%) Wm) .. Lie) goa%u%13)_ 911 m gm. wé+lrl...a_.._....._I_5._ err 15 NOT .. PéQ _,__.______._____ . c-LuiW—Qwi M Q?) 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LL?— Mnisiav‘m. m. dams Mm.) _. at. £5 gram-fiatcimss Fix} == x 5.2a... {iu».-n_ :fia—fi: Box You! firms war A” mafl’l. daSSi-‘b are 7AM”) . i s i' .i.. mqu 530.9583 aft. ‘F‘vbfl . 1.. i g E i cofisial-Je‘r‘ fie. fF‘OPgsu‘kéfld- ng‘m _ I - FfiLSE I 1-...(C‘frcL afimprfifitmma I W R )_ (3.35111) M 0.). .. GD 4.; m a; a . 3. TRULY; .. of. 1%“; WW) i I ! I s. % 09m. 14 m emu—r? afh?oPr*W-amwer\ i E a I i 1 [ll §I.5 4+ [2; (savanna, 5/5 l l E I Iii. “his Page). {not} Mt. .i-Eloofs‘aG’m 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 - i i i 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 is correct and the proof is largely a correct claim but contains one or two incorrect statements or justifications. .a F (failure), if the claim is incorrect, the main idea of the proof is incor- J rect, or most of the statements in the proof are incorrect. l i J Justify assignments of grades other than A. Whenever a proof is incorrect, _ I explain your grade. Tell what is incorrect and why. .___._._. .. _ ._ Supposex and y are integers ._ _. _ _. . . Claim. If): and y are even then x + y is even. . . “Proofi” Suppose x and y are even but x + y is odd. Then, for some integer k, x + y = 2k + 1. Therefore, x + y + (—2)k = l.The left side I l i l E the right side, 1, is odd. Since an even cannot equal an odd, we have a . _ __ ... __ ___.. . .. _ l ‘ ' of the equation is even becau'se it is the sum of even numbers. However, contradiction. Therefore, 9: + y is even. I I _. ___________ __ _. l _ _ r- _ _ J l ,_ _ _ __ _ _ . __ _ ct. _.. . ._ _. - _ __ __ ._ _ . . . _ .. l _ T _ _ _ ! i ' II ---------.... . . _ __ _ ., _ ..i ...... .....___ a ._ , _ ,_ _ __ ..._.._..__ _ _....._.... ........._ ' ' f 1 . Claim. There is a unique set of three consecutive odd numbers that are ! ail prime. “Proof.” The consecutive odd numbers 3, 5, and 7 are all prime. ' Suppose x, y, and z are consecutive odd numbers, all prime, and 2: ye 3. Then y' = 1: +2 and z = x + 4. Since it is prime, when x is divided by 3. ’ the remainder is I or 2. In case the remainder is 1. then x = 3k 4- 1 for someintegerka1.Buttheny:x+2:3k+3=3(k+1),scyis -- . - . .. u not prime. In case the remainder is 2, then x = 3!: + 2 for some integer k z 1. Butthenz = x +4 = 3k+2+4 ='3(k-i~2), soz is notprime. In either case we reach the contradiction that y or z is not prime. Thus I x = 3 and y = 5, z = 7. Therefore, the only three consecutive odd primes are 3. 5. and “i. 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