mt02sol2008 - 12 June 2008 MATH 220 UBC ID Page 2 of 7...

Info icon This preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 12 June 2008 MATH 220 UBC ID: Page 2 of 7 pages [13] 1. Suppose nonempty sets A and B are given, together with a function f : A ——; B. (a) Define precisely what it means for f to be ane-to-one (also known as injective). It W e A of»? mm) 41% 1V ox.) We). 0__f_< . 1F Shof-Q’Xz) {‘5‘- [XI )O‘ZGA) M “F’Xza (b) Define precisely what it means for f to be onto (also known as semeeteee) \{JEB} 3x04) 3:10“) 9E rang}; "' B) W (W4) : {7%) : “A? (c) Say whether the statement below is true or false. If it’s true, prove it; if it’s false, give a specific counterexample. If f: A ——> B is not injectivc, then f must be surjectivc. This is FALSE. __________._ hf A: K) B: (R) 1pm film) «Men 4‘ lg flo+ flee-We (Wt, {[03 a—Ffif) ); MJ WP i5 ”0+ Sui] u-h‘dC’, (Vt 5 ¢ Mn (9). Flu ectNC a“! SWjQOlNe are. lAJQfGMJZML fm’flZIE’J, NOT SwFIe. oncoSlf'es 'J Continued on page 3 12 June 2008 MATH 220 UBC ID: _—_— Page 3 of 7 pages (d) Suppose A = [0, +00), B = [0, 1], and f: A —> B is defined by f(:t:) : cc/V 1 + :62. Decide whether or not f is injective; prove your answer with reference the definition in part (a). [Sketches may help you plan your presentation, but there will be no marks for pictures] Suave mzéA 3N3 “Wopfizlieu TAG-n «AZ 7‘: __i_._———- = 1 H x,‘ H x; 1 1 2 1’ ~. «1"sz 50 Mommas», fig 1 a” 3 sing; xfi/Oflgzo knw) ThIS inpii?) lyt|rlxz IUC haw-C ’X,=’X2.. ms (e) Taking A, B, and f as in part ((1), decide whether or not f is surjective; prove your answer with reference to the definition in part (b). Tflfs [5 NOT SuRTECTNE . ROM (L)- ‘fitrt om}; “@115 “(5), Re, 33(8) We A) 3757500 mm “mm iéB mosh fin 5 Indeed) \ixéA) x1<7<2+l So V‘XéA’ fi— <| UScJ “‘dxeA,xZo:’ VXéA ,L—~ <l 4/ so VXC’A/ {(x)<l. (In puiiufim) mam-1H.) Continued on page 4 12 June 2008 MATH 220 UBC 1D: m Page 4 of 7 pages [8] 2. Suppose nonempty sets A and B and a function f: A —> B are given. Prove: Iff is injective, then f—1(f(U)) = U for every set U Q A. [Recall these standard definitions: f(V) = {f(v) : v E V}, f_1(W) : {x E A : fix) E W}.] 86% eiudxg MTLGJA 4180 50% indium“, We lame 135011. Fix m mind: UQA. so VuéU) “('(nw). M136) {mi/(9211, (9.) Pick Mg m {UH/O). 33 42f"; Jmfl) 4% Wu; «MA M4 mafia). flw EMU, 39(16):?(v). New {A}; INjECTM/E) so M has; 13:6,... {in Patagonian) MéU, . so weflm), «MU. W 42;, {‘(RmFU. Continued on page 5 12 June 2008 MATH 220 UBC 1D: _——_ Page 5 of 7 pages 3. Let K be the collection of symbols on a standard computer keyboard. The set K is finite; its elements include A, a, B, b, ), (, #, @, [space], $, %, etc. For this problem only, define “a story” as follows: a story is an ordered list containing a finite number of elements, each chosen from K. (Repetition is allowed.) Stories do not need to be in any recognizable language, past or future. Here are two stories according to the definition above: it : (M7 aflt'lh," rJUTl'leJ S): ’11) : (auqaeay:>[email protected]$1H3a:ma]—aeat1~i4al)‘ ‘ Let S denote the set of all stories. (a) IS the set S finite or infinite? Prove your answer. (b) Is the set S countable 0r uncountable? Prove your answer. (CK) fie 59—4, Bi'llgfll-EJ la/c ”l includes +64; [A it'll $631+: 3(0) 012)) (1)212)) (52);») ”.2; In“ (L) The Q [3 Jenumuatle [LMC COLLAWC , M m Co‘aZClim o I n- charqcm S‘l‘ories, 7001 gal“ MN. Each Sol 8;, I's Continued on page 6 12 June 2008 MATH 220 UBC ID: —_ Page 6 of 7 pages [10] 4. Let 3 denote the set of all real numbers in the interval (0, 1) that have a. decimal representation involving no odd digits. 13.9., 246/999 : 0246246246 . . . is an element of S but l/rr = 0318309886 . . . is not. Is the set S countable or uneonntable? Prove your answer. The gei [g uncaunl’able: Continued on page 7 12 June 2008 MATH 220 UBC ID: —______— Page 7 of 7 pages [10] 5. (a) A real-valued sequence a = ((11, a2, a3, . . .) and a. real number 3 are given. State precisely what it means to say, “The sequence mu)”6N converges to E.” V00, ENQN, WON) plan-3kg. (b) Consider the particular sequence {ix/771 + 13 sin(n) — 9 cos(n) 5w: ’ Find 5 so that (“Unetv converges to E, then prove that the definition from (a) holds. A 3 A _. a2? -— I‘nSPQL‘hO‘A F0) leach film/KN :l(%+ BSA-Kn flawsn)_é NR 5 : [I3Sinn—::Snl 5471 an: HEN. NOHLQ 5»- <E <‘—> 2739“: TM fmsen‘l’ m‘ceIJ: Gwm 270 ‘3“0059 my We?“ N23 £2" 11% 10-03 a?“ erg/4‘35 fi>— 22) 50 %<£' RS 5km?”A W/m mph"; {an‘3/§l<£. The End ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern