2009fe-sol

2009fe-sol - This examination has 14 pages including this...

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Unformatted text preview: This examination has 14 pages including this cover The University of British Columbia Final Examination — 12 June 2009 Mathematics 220 Mathematical Proof Closed book examination - Time: 150 minutes Name Signature UBC Student Number Special Instructions: To receive full credit, all answers must be supported with clear and correct derivations. No calculators, notes, or other aids are allowed. Rules governing examinations [ 1. All candidates should be prepared to produce their library/AMS cards upon request. 2. Read and observe the‘following rules: No candidate shall be permitted to enter the examination room after the expiration of one half hour, or to leave during the first half hour of the examination. Candidates are not permitted to ask questions of the invigilators, except in cases of supposed errors or ambiguities in examination questions. CAUTION — Candidates guilty of any of the following or similar practices shall be immediately dismissed from the examination and shall be liable to disciplinary action. (a) Making use of any books, papers or memoranda, other than those authorized by the examiners. (b) Speaking or communicating with other candidates. (c) Purposely exposing written papers to the view of other candidates. The plea of accident or forgetfulness shall not be received. 3. Smoking is not permitted during examinations. 1 5 2 5 3 8 4 8 5 8 6 10 7 9 8 9 9 9 10 10 “1.; 10 i 9 Total 100 12 June 2009 MATH 220 UBC ID: Page 2 of 14 pages [5] 1. Prove: For logical statements P, Q, and R, [(P /\ Q) => R] E P :> (Q => R). (FA all"? R "E":- R v ~(FALQ> C143“ E a v (m) v m) at Wm ‘35 (R v you) v @P) womb/e. as“ or? a) v w) W E: 3:? (611» 12!) I [5] 2. The distributive law is a famous fact valid for any sets A, B , and C: Afl(BUC’) = (AflB)U(AflC’). Prove this equation from first principles. (E) PF all; my 6%" 90 =2) 6: MB] v [9% MC] g “(Amid (Anal, as? Fifi/[L 5W3 ’Y ~ fig 4%qu 0‘ I (A dug/j 0 , ‘HL’e’K Continued on page 3 12 June 2009 MATH 220 UBC ID: Page 3 of 14 pages [8] 3. Let A and B be nonempty sets, and suppose f : A —>» B. Prove: If A1, A2 g; A, then (a) f(A1 U A2) = f(A1) U f(A2)- (b) f(A1 0 A2) Q f(A1) n HA2). (0) If f is one—to—one, then f(A1 0 A2) = f(A1) fl f(A2). (a) (Q) m jéfl‘AMAQu TM WM; jack) 1%? 4w; MAMA} WAN/mm jgI‘M/)QWAJWMJ; «6/42! 75%» je (“(AZW-R‘Awflm. 1“ mm W/ Mfrs. (PM “MOW/M, View MEL/pug 7316; 0’2, jam» [m WA], If 3670,) m 3m; {297 gm :XéA‘ Ara/12) so [MN/11;), (£15 SIM/t f 1n 43cm; (b) 36 ((40/42). 7% mm rec.) 74;? (INLQL kg 4,042 I , v ). . Km‘ufi“ lye/1, 5 3 6M1); WLWT XML in)? ef/Az). SNCQ MKULZV‘W 6 (MPH/4 M 21¢}:sz (a) (Q) m 36.]{2/10nfl'xi2), “Tm MW 414W 7%? 4 w. an {Ts-fox» ~61 W mg: m {mm-3(2) ms wag do A: W W Continued on page 4 12 June 2009 MATH 220 UBC ID: Page 4 of 14 pages [8] 4. Prove that lim (V712 + 1 — n) = 0. Use the a—based definition, not limit laws! n—>oo 72 1 2 Not" V?“ m V‘ 1 Wm «Ml +14 1: (A H) Mn I War “A \‘MLH + 1/1 - .,_,J..,.__ A) z “1 w :t ::T”””‘ $ "7M’ Tl m1 m 1! V\ m! M in I ML, OLsM. ‘Zm <& {:7 22 < ‘4 Continued on page 5 12 June 2009 MATH 220 UBC ID: Page 5 of 14 pages [8] 5. Let A = {a1,a2,...}. Define B : A~ {an2 : n E N}. (a) Assume, for this part only, that j 7é k implies aj 75 ak. Prove that |A| : |B (b) Prove or disprove: If |A| : [N], then IA] 2 (0") UM W: W‘“ mflx B SA. AISU 6214/: I V p I v z (wig Mama? M: BQ§4541§@0).,,7mm} m4 Ms as Mode) ‘ V ‘ y.(m2VjFWAL SM; is Mme 4.0 «2] rg/clavdlamfi/ MK QWVW%% a QAWEQA A&L‘Wfiz . w“ a M I dgwp) ) u wwmmw ww‘)q/énlf) @dfie) Continued on page 6 12 June~2009 MATH 220 UBC ID: Page 6 of 14 pages [10] 6. In each sentence below, write a precise definition for the term or statement in the box. (a) The function f: A —> B is \“ijé B : j “QM, (b) Statements P and Q are given, and (9 vi“ P). (c) A sequence (an) is given, and \”/i >0 QNQN; \Z/VW'NJ Mfg/<8, (d) A bounded nonempty set S Q R is given, and m ‘l (0 films) My (if) \i€>O 3M3 : ><<o<+z (e) Nonempty sets A, B, and C are given, with C’ Q A. Define g(C), where g is a relation from A (j "3—: Ajé ; KEG 693)?) k. Continued on page 7 12 June 2009 MATH 220 UBC ID: Page 7 of 14 pages (f) The set of real numbers, R, has {the least upper bound propertfl If SLR {Gm/tied ajflwe cm/ Mum 1? r6. walk a Ma Moan F: 5‘7) (g) The seriesw M . “ e (v Saw/M? _.5w>NeN ~ 3 W \iefl) gum! grt- VN>NQJ (91W “4 0 (h) The function f: A —> B is x2; VLo‘Il lw/ 0’1 4:} M97“ £10770, (\meiMfiVQ of” T; agmwflcl ,> (i) Statements P and Q are given, and I P :> Q is true but vacuous. I Pt (j) Max’s favourite integers are the ones satisfying :3 E 2 (mod 7). MJZ 7C 0le ENC-Z 9%, ’X‘: ‘7WFL. Continued on page 8 12 June 2009 MATH 220 UBC ID: Page 8 of 14 pages [9} 7. For each statement given below, do two things: (1) Write the negation as a simple direct statement. (For example, just writing “The following is false” in front of the statement is not acceptable.) (ii) Prove either the given statement or its negation, as appropriate. (a) For each n E N, 3" + 2 is prime. 0» Ma was me A e N 5“.ch mil 3"‘+2 (Ware (iii, W25 orbital/M ("M-)2 5/4 322::21/37'2 32% A; 1/(7 ‘6: (b) There exists an integer n such that n2 + 371/2 2 1. 2 . (V L1): F01 Each. {Aiéj/UL. in, r? + “in iii) 8w (2) t M: n: -—2. (C) lfm/(a:~1)$2,thenx<1or:r22. (i5 (“4): rural/.1” ¢< Smii 3M“ 62 Continued on page 9 V ,«t w} x i {its Afiv .- 12 June 2009 MATH 220 UBC ID: Page 9 of 14 pages [9] 8. (a) Prove: For real numbers a and 0, one has {V5>O, aSc+€] 22> [age]. (1)) Prove: Whenever A and B are bounded nonempty subsets of IR, sup(A U B) 2 max {sup(A)7 sup(B)}. (a) CawuafoS/‘bvfi. 36135": cod *3?) <3 i> o gstp a: > 6+ 6? (KW Q I“? We?» is MMJ d2 view): We,“ 40:7 / 0 4M. J H 1 I A / :> 494W 6W3 8 / <:+a:@e§:§<q La“, /A: ‘W {09;}, g 3M @ W6 New“? \i’Vé/i, 901:”< Vxe/luré xé7W06f VJ'XEB) X415 3:) /" ’9’ W“ A EL >C7/ Mow/992; N07“ cw Mf/fl’i :3 C1 (\l ’ZA/k alv‘j K GM“ 3: ., M»; (X. V ( We, know 3%“ 9L 0&8on mm 4W? (X WA 6%va flka :w a 4:)? I I": [Vi/Wm Him/L 3 Ix sf, f4, < x , TWA mm : W30 yer/X U 9% Continued on page 10 12 June 2009 MATH 220 UBC ID: Page 10 of 14 pages 1 2 — 1 n [9] 9. Prove: VnEN, 1-30+2-31+3-32+---+n-3"_1= H7: )3. W J QM, -%;4 F (m) I, , l V "3 1°35; j. ) if?!) flat/(E! 9s"st ) fin mafia fl my) PM. w! T (ZW‘BV‘ PC") 3 20 ‘4‘ n p» N 9“. VCB 27:7 wer W +7 wv~< W0, ‘ t I ‘+ I "g0 “'"~ t 7* WBVMJ; (mt/)3“ “l”: w At 7 . SO lgzfii/L Skim Jf hem,” 106‘); [4’ (Zavogw + W ‘1 1520+“ H + “3"? (mafia-7 / + 3" [‘Qmwm -~+ fiw—Wfl M _, 4 WW 3;. f 4» .3a [émhfl m m“ #11 W‘H ......._. ‘zxaE a msmnw i Q Continued on page 11 12 June 2009 MATH-220 UBC ID: Page 11 of 14 pages [10] 10. Decide Whether the following sequence converges or diverges: n2(—1)" — 25 a : m. ” 4n2 +16 Prove your answer in complete detail. If you use results proved in claSS, quote them precisely. FUN: Prim-i Ggflmm 0552/ Léwmifi ) jg Slimmlj 30C) : \y’A/é/V) Hwy/v sf, {exam/22:, 1 “vlleET EXPLME; )P n M ODD, 61M: Arm/é <14) .A M one ,M A} ‘3?) mm 111+! M WVK 4/1/er w-f/Z/Q) 567 2fL(M*/j2 22$, 2 lama 2s 2 Mu 4,411? :- emf film-1‘02 “HQ Aflwfl)?‘ “kart/)2 R %(Wf/)A M 3:31; 2 1 M, %(M()2 W 3 n. PROD}: ? LEAL <6; : ' GNQJA an? N QM) awj W :7’ WWW A]; (ww/ 12d" WL= Vlrl, 944/117” (4wa M fin v lamrdl“ l *- QVH—I > 2 M8” Continued on page 12 [10] 12 June 2009 MATH 220 UBC ID: 11. Recall that a sequence (an)n€N is called “Cauchy” when V5>0, ENEN: Vm,n>N, [am—an]<€. Page 12 of 14 pages Prove: If (an) is a Cauchy sequence and one has lim bk 2 220, where the sequence (bk) is defined by n—‘POO b12a2yb22a47 b3:a’87 "‘7bk2a2k) "‘7 then an —> 220 as n H 00. Proof: (mm 8>O anfl); 2/: czm/ an M g: N 13% snug»? llm/n>N/> (OM new“ 21/“; / {\l”l aC/Qagm (ma ’i’lclfificlj gen/h 14>” W <29 (694“2'20/4; /6(W~612k/+/llzk—~ZZQ/ WAN I,“ FECM 07L / k) 72%,) %£ <6 Continued on page 13 12 June 2009 MATH 220 UBC ID: Page 13 of 14 pages [9] 12. Determine whether each series converges or diverges. Justify your answer. (a) Z—Ez‘fi MLQ’M “WWW 11111..“ “LL...” O‘ V‘ ’T E01711}: w mNm‘r‘HJ—U \ V3?“ 30 S’Qh/Qw CaMgQQQE‘i [MSDLM‘TELV] 7 WCP‘L’UgéOM i ”‘ 47* uni/h Whiz. rm. yen/Lg “5/; a n5“ ii AS néX/Z/éi) F u: )/S” ) W- 5% CoNv’E/ZGEE [MSQLM‘IZ‘Zyj fine (The M071 7%.?" ILLde 406W) 1m, 00 n2 (C) 2 2009712 + 1010' r I, k 1., ‘2 HM q“ I: 200?“? ~{ H3110 [adj T W “new? Tie/A mglzcyfgj fly C1{MJR,T~€6%- Continued on page 14 ...
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