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# mt2key - 7 Name K L’(1 MATH 108 Section 1 Second Midterm...

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Unformatted text preview: 7 Name: K L’ (1) MATH 108 Section 1 Second Midterm ‘ November 19, 2010 Answer all questions on these pages. If you need more space, write on the back of the previous page and indicate that you have done so. Since this course emphasizes the basics and rigor, your justification of an answer is more important than the answer -— so always give a clear and complete justiﬁcation unless you are instructed not to do so. There are 7 problems on 3 pages. 1. In each part below, give the deﬁnition. (15), a. The composite of two relations. “I: R; A 9B CM g -9 C / “la/76¢! (L'CJIHFL‘SITQ‘ Safe“ 2(41‘)‘ ab ith/N—y/‘él? ’* 3 [{HC, (Kb/C) S j b. The restriction of a function. hi {'A—BE (imii 0 9A ,, Mm; Hut lama/4,, h) 013 S {:19 t {m’ I M! {3% (“um/f (1 c. Aﬁnite set. . / A Set/5 704/756 1‘) A245 8 N If) A ’3» Wk] ﬁar'same kéﬁ/ 2. a. Give an example of a function f: N —> N that is onto but not 1-1. Justify your answer. (10) _ Lat {01): 7;) n" M’/ 107,1 / . . g 1( I5 mm; 5/401 Vﬂé 15/16 Magma/f VH/ﬁl’) 45 not H We ’9’ . 2—>/ b. Give an example, if possible, of a denumerable collection of ﬁnite sets whose union is ﬁnite. If it is not possible, explain why not. W5 An 1%, ne/x/ S 7’46” 90%}! An I5 {in/ﬁe ama/ ngu A" :51}, iinite 3 . For each of the following relations, indicate in the space below whether it is an equivalence relation and justify your assertion. If it is, and if 3 is in the domain of the relation, describe3/R and indicate how many distinct equivalence classes there are. (16) a ' / 4A a. OnR,nyiffx+yisaninteger. F. R’5"‘%‘ﬁ/@K/W "? ‘2; :3, 62 /L b.R={(m,n)EZxZ:m-niseven}. T (eglriivii‘g i) V méz, rmmro Mm ¢ / 7W Wt: «02k = 2(4) we“ {toms‘yt'w‘ttva “‘>_Vm,n,peZ// If m—ncgj M017 n-,0=2/< m/Zn HEP 1/. . m—WM‘ T M5, Marl/N1 (M4))4‘UPF): m-P : 2)} 2K : ZKJ‘4k) ; I We” J {.va _’I I) 3 g 1 ‘0' E. l I I TWO 0(I5H/ld'i’ ﬁght/g LIA/5,10; claggeg / Mia/ﬂan \ 0W «ENG/w m {was 4. Let R be the set of real numbers. For r varying throughout the set of all nonnegative real numbers consider the sets Q = {(x, y) I x2 + y2 = r}. Then, prove that {Cr} is a partition of RxR. Note that the nonnegative real numbers are R+ U {0}. (12) f) W20, wnsitfaiV/ﬁ)3 050793: o+r=r So (0, f7)6CY Moi +M/efqre Cricﬁ (LL) Conga/u CtrI Mo! Ch» I13 Cn/lcfl :CP 0K hf cnncrﬁcqb, 9M; 5,1; (“Wicca ml (X,/44,)EC,/Z 714w viz—n}? -; r, 30 ftr Am! C — Y)2+%Z:f,_ I 2 Cfch m We know rgocr M42. meg/Lo 34w KM? 4: UV cf Pic/k curb/hug (>07) GKVK I , ,2+ 2 Let fax \/ / Hem /x,%)é(r My] KX/gg U r 2,1} C)’ V 90 E937 1‘50.» pMHHon e/YC 52x6? 5. a. Let A, B, and C be nonempty sets and let f: A —> B and g: B -> C. Indicate ifeach ofthe following is true or false and prove any that are true in the space below: (16) 3 a. If(got): A —> C is 1-1,then g‘: B -> C is 1-1. F 3, b.If(gof):A->Cisonto,thenf:A->Bis1-1. F. /0 c.If(gof):A-»Cisonto,theng:B—*Cisonto. T FDY‘ C: VLEC WL hWDW jkkA " (“/C\é:)°‘€ Siv‘te (3" \$5 an’l‘O, Th3“ EBGB; (A,\>\é¥ an; (\o/CBéﬂ A“egim:i(°h ,L‘Q 9‘; Con/5ft)?! ‘1 The“ VceCjE’béB" (Raésqso a M, raw-“s 6. a. Prove that ifA is inﬁnite and A Q B then B is inﬁnite. (18) We domain/imam. A SSW/we P Mo! w? A is inﬁnite, A Q 8 , Md, 3 i3 {in/1‘6 7 A W W 3...... .7: . ijm‘ﬁe 5615 {5 #‘nffe' 6290, b. Prove that the set of irrational numbers is uncountable. m: m—fioma/{S MI‘OM ;\rfM/‘0MJ5 m” aQUH 6 EM 02/)1 : d) 7 Use Whaoéidlm :. MSMM: II :‘s'wwab/e II is n0+7cmfte SPACQ 77/ JZTT, 3771 f5 Inﬁnite eel/echo” lffaHqujg Hﬁ E is JWMmafnlo/e +14% QUZI IS dame/we am we know Mai Hye Mm” 0% JUN 0 dis/oh" denumerable Set f5 p/em t men: 616 7. Prove Theorem 5.21: The union of a ﬁnite pairwise disjoint family of denumerable sets {Aiz i = 1, 2, 3, . . .n} is denumerable. (12) M58 pMI. [9(1): ,4; 2 ,4, ﬂ/Wmie/mb/e Aﬁfh/mF/v’on Kw K 453mm: plk] A,“ : .U Ac‘ (/4141 1:! 71,3 7575+ is p/emmmemé/e by PM), He seww/fs o/ewwemége b7 ﬂust-o), 7h€ 41410 Are. Mtg/90% Saga file (Al/echo” /.S Pairwise JISJDI‘AZ‘: ‘ 14+: __ SQ 'Q’AI' IS dent/(WW8 0r (Ozk‘l) ...
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mt2key - 7 Name K L’(1 MATH 108 Section 1 Second Midterm...

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