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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 (“(AZWR‘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 {Tsfox» ~61 W mg: m {mm3(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 ENCZ 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)$2,thenx<1or:r22. (i5 (“4): rural/.1” ¢< Smii 3M“ 62 Continued on page 9 V
,«t w} x
i {its Aﬁv . 12 June 2009 MATH 220 UBC ID: Page 9 of 14 pages [9] 8. (a) Prove: For real numbers a and 0, one has {V5>O, aSc+€] 22> [age]. (1)) Prove: Whenever A and B are bounded nonempty subsets of IR,
sup(A U B) 2 max {sup(A)7 sup(B)}. (a) CawuafoS/‘bvﬁ. 36135": cod *3?) <3 i> o gstp a: > 6+ 6? (KW Q I“? We?» is MMJ d2 view): We,“ 40:7 / 0 4M. J H 1 I A / :>
494W 6W3 8 / <:+a:@e§:§<q La“, /A: ‘W {09;}, g 3M
@ W6 New“? \i’Vé/i, 901:”< Vxe/luré xé7W06f VJ'XEB) X415
3:) /" ’9’ W“ A EL >C7/ Mow/992; N07“ cw Mf/ﬂ’i :3 C1 (\l ’ZA/k alv‘j
K GM“ 3: ., M»; (X. V (
We, know 3%“ 9L 0&8on mm 4W? (X WA 6%va ﬂka :w a 4:)? I I": [Vi/Wm Him/L 3 Ix sf, f4, < x , TWA
mm : W30 yer/X U 9% Continued on page 10 12 June 2009 MATH 220 UBC ID: Page 10 of 14 pages
1 2 — 1 n
[9] 9. Prove: VnEN, 130+231+332++n3"_1= H7: )3.
W J QM, %;4 F (m) I, , l V
"3 1°35; j. ) if?!) ﬂat/(E! 9s"st ) ﬁn maﬁa ﬂ my) PM.
w! T (ZW‘BV‘ PC") 3 20 ‘4‘ n p» N 9“. VCB 27:7 wer
W +7 wv~< W0, ‘ t I ‘+ I "g0 “'"~ t 7* WBVMJ; (mt/)3“ “l”: w At
7 .
SO lgzﬁi/L Skim Jf hem,” 106‘); [4’ (Zavogw + W
‘1 1520+“ H + “3"? (maﬁa7 / + 3" [‘Qmwm ~+ ﬁw—Wﬂ
M _, 4 WW
3;. f 4» .3a [émhﬂ m m“ #11 W‘H ......._. ‘zxaE a msmnw i Q Continued on page 11 12 June 2009 MATH220 UBC ID: Page 11 of 14 pages [10] 10. Decide Whether the following sequence converges or diverges:
n2(—1)" — 25
a : m. ” 4n2 +16 Prove your answer in complete detail. If you use results proved in claSS, quote them precisely. FUN: Primi Ggﬂmm 0552/ Léwmiﬁ ) jg Slimmlj 30C) : \y’A/é/V) Hwy/v sf, {exam/22:,
1 “vlleET
EXPLME; )P n M ODD, 61M: Arm/é <14) .A M one ,M A} ‘3?) mm 111+! M WVK
4/1/er wf/Z/Q) 567 2fL(M*/j2 22$, 2 lama 2s 2 Mu 4,411? : emf
ﬁlm1‘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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