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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 anetoone (also known as injective). It W e A of»? mm) 41% 1V ox.) We).
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counterexample. If f: A ——> B is not injectivc, then f must be surjectivc. This is FALSE. __________._ hf A: K) B: (R) 1pm ﬁlm) «Men
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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
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so VXC’A/ {(x)<l. (In puiiuﬁm) mam1H.) 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.
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{‘(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'llgﬂlEJ la/c ”l includes +64; [A it'll $631+:
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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 realvalued 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) plan3kg. (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 _.
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