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M122F02T2Key

# M122F02T2Key - Mathematics 122 Section F02 Test 2 November...

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Unformatted text preview: Mathematics 122, Section F02 Test 2 November 2, 2007 Last Name, First Name: Mtg Student Number: Instructor: M. Schurch You have 50 minutes to complete this test. The only calculator allowed is the Sharp EL—510 RV (But no calculator is needed!) Do not simplify your answers. In each case to receive any credit at all you must show enough work that it is possible to determine where your answers came from. 1. For the universe of the real numbers, determine the truth value of each statement. Give a short justiﬁcation or counterexample as appropriate. [4 marks] (a) V33, Ey, 33y : 1 FkiS/L) if” 120 1 R795, O") :0 W (b) 3\$,Vy,\$y aé 1 True) {0 1:0, V‘ﬁ) Cj‘ﬁ:6) 7-K “Mic iiTALLSY Ugh )5 (L; “3&3 Ri‘iovx 0i C60 30 WW3 l’VLca [A C/(ﬁqCH/vﬂi liJ‘Wi/ix V’rciwi . ' 2. For a given universe, prove that Wm, np(:1:)] V [393, q(a:)] : 39:, [p(:r) —> q(1‘)].[4 marks] SVSXﬁa E’VlivpbA} V (37¢) ("ﬂ/CIA} \S ifm m“ V1,," PLQ {‘3 5m. LSV 3L 7-573 ‘5 5“ :9 \m .j {95’13 '5 ill/q. Iii/an 105%» Is gauiv; £7 ck,“ CL) 50 PLEA «77%» [5 ins. gar 4/! 7C ‘ Thence: 33L @051) ‘7 ﬂcj‘sg i5 iW‘k. I; V' VFW.» /S gni Le [(13:/:%(x> {3 irm‘ CLLHAF Mans 31L FLI) ”75/53” /8 iV‘M ﬁlm; “Janet/er 19(79)an %\19‘lc-,Vi;7}\ lair) 39(3L) i) {\l’V—x T ,\ ll' , f... ,, l“, .1 .x in (inch I) ”Z («5L\< ) .3. Write the following statement using symbolic notation where the unviverse is the rational numbers. “For all :2: and y such that 1‘ is less than y, there exists 3 such that :r is less than 2 and z is less than y.” [3 marks] \ﬁ) whims) #7 (Newﬂ (If V1 “W51 (149]) f) Bellaﬂl ) 4. Let A : {@,1,2,{2}}. Circle all true statements. No justiﬁcation is necessary. [4 marks] @194 @ecA c)Aﬂ73(A)={{2}} @{1,®}E'P(A) 5. Prove that if A g B and C' Q F, then C Q X. [3 marks] Slummc Ar 9 V5 67qu C Q l7?) . (Al 16C Hat act/Y5 (em @9913 ) ,ﬁ 3: g lnD whit/L Mild» 1 ¢ /] (arm; A ‘2 l6) 6. Prove using the laws of set theory: A — B Q B = B — Al (HintzA — B = A H B) [4 marks] E5 0 (S : Ame fix") which A Awe : (mm M: {tram W : (,4? A vs) Wm M mam Low : (B0930 Ufa Imam La, : (BYTE) rhehg uh 1 ((5 n A?) ': Y5” ﬂ @WWw‘iOJIM Luv 5. n€]l/.L&] 144/5. 7. How many of the integers from 1 to 1000 inclusive, are divisible by 4 or 5. [3 marks] QUE-'11.“,l‘w‘] (VJ—Lit A M m Mk] Ug l/xi‘iw) H’M‘Il M OLA/“it"(lz 1”? (’{n (Air l“) LL MIL Lg] ox “vi-(9kg Mam] (M crib/mix] h? 5 . The lﬁum]lhilﬂﬂl“lﬂne) : [ciao . (om [m1 , l V.“ l ‘T ‘ ,f : [2105/ Ll S M 8‘ If a fair sided die is rolled three times, what is the probability that at least two of the rolls Show the same number? [3 marks] 9. Prove using mathematical induction that for all n 6 2+, 20 + 21 + 22 + - - - + 2””1 : 2" ~ H4 marks] WW 6‘1“" (AIL: 0 1 :1 f} :5 I 1‘ 3' ”MA gm" VL‘“ l. ...
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