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M122F07Mt2Key

# M122F07Mt2Key - Math 122 F01 2007 Midterm 2 N us You have...

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Unformatted text preview: Math 122 F01 2007 Midterm 2 N... us] You have 50 minutes to complete this test. There are 10 questions worth marks as shown for a total of 42 marks available The only calculator allowed is the Sharp EL— 510 R. No calculator is needed! 1. For the universe of all integers, let q(z), r(:r) and 5(33) be the following open statements. (1(1) : so is even T(:c) : a: is a perfect square 3(92) : a: is divisible by 4 . Write each of the following in symbolic form. a) [2] At least one integer is even. :1] K] xix) b) [2] If I is even and m is a perfect square, then m is divisible by 4. \3L X ) (Ci/[1d /\ Y (9L\\ -§ 73 CI\ 2. Let p(:z:) and Mar) denote the following open statements. 17(56): x2—7\$+10=0 r(a:) : :1: < 0 Determine, with an explanation, the truth value of the statement Elm, 11(2) —> Mr) in each of the following cases a) [2] When the universe consists of all real numbers lﬂ’dL “Nt\ 3(2 [ JAN; g h’fl‘ qu “’é f(l\ L: “1“}be Mew-Lara. P (Q C; [Edi-Q b) [2] When the universe consists of only the integers 2 and 5. Valid v-’\7k1\awi ? (é) CM. [0040 \XML is H13 0mg] vi 3 GM mu Carlee 3. [3] Let A = {1, 2, {1, 2}}. Circle all true statements. No justiﬁcation is necessary. a) {2}EA v(b){1,2‘}o}1-(d)@eA (e)An7>(A)=(z) (f)|A[=4 4. [6] For a given universe and open statements 17(33 )and q(:1: ) show that [\7’z, p(m m)]/\ [Va:,q(:v)] is logically equivalent to V1, p(m ) /\ q(\$). Crﬁﬂ [Vx‘lyciﬂAB/w VL‘XNWT =5, ’DCQ (3qu Cali} Owe \oder—fﬁ 977v til/l >4 :5 (>03 A OVUAVL, if“ all A ”:3 \f X) @KJL\[email protected] “(:31 v; C4?) Vi, WV A eyes a“ , :3 ”ebb Gong O} (”t\ (ll/ﬂ \Ooqﬂx ‘RM iii“ all X 3W VLK 3C)” ovnol \7L5‘ O YUA 00% l 1 Q LVX P HUS) [Av/1 q/b\1fﬁpﬂ 5. 4],LetA B, andCbe sets. Prove that 1fACBandBCCthenACC N€€C A EL LCM/WA (AVZtC, .1 ."> r?) m me 4 3M Ash KS 5, ) gma gagwck 7 (426:, .§?7\QIL Egg ) \l’Qd/k L) Go QQIC "Sow/L lwtcf/l QEéE gT‘rXQL R513) Q % A . MC. 4 co 6. [4] Let A and B he sets. Use any method to prove that A n B = I U E. xe HM?» <22.» (Xe; Ham ‘ <==\ “1%746 PﬁA (Xeigl ‘éﬁ WCXe/QVWULGTQ ea (Xe/Qwé’xét'm 43> KGQQ% 7. Let X = {1, 2, . . . , 10}. Determine, with an explanation, the number of: a) [1] non—empty subsets of X \0 94 b) [2] subsets of X that contain 1, 2 or 3 ’3: *b‘l—C‘A ~e W QGYV¥CR TH I] RC: VKHUZ Cf] \ \'Z l3 :- 40 u 9:} 8. For each i E Z+ let Ai = [—i, 22'] (the closed interval or real numbers between —i and 22'). Determine each of the following: a) [2] UiEZ+ A1 : [4‘23 a ﬂag} o _: E b) [2] “5.3.2 A 2 [~2.\A;X m [A "ab": ﬂ~_ - :- £,11L\_:\4 /3 9 Consider the experiment of tossing a fair coin six times. Describe the sample space 8 using set notation and determine [8]. HM . <g’iHHHHHE1l++HHHT HHHHTHrwxiﬁTTTTj l%\ b) [2] Determine the probability of getting exactly two heads. Show your work. {3‘ : Qwea‘i 2 MAHS C€mcthﬂ iwcxm W 1 {3A 2 (a) CCHKOO‘JC Z. JKD‘ 336‘; h CCSC+ Hl‘ Q‘i’ﬁgm’f Gay—31 \ \ r._L'iL.gC-°2)AEKS “”t“ L£\ 2b /2% 10. [6] Prove that for all n 2 1,1(1!)+ 2(2!) + - - - + n(n!) = (n + 1)! — 1. d i C“ “2 m 1 Q n U \ LH3=VD=\ , 2H3 O¥\‘4: 5m CQ LRﬁ‘: QHS) ﬁ’WH «H M \Uifxa m H.— - l y \ ll)? Asiuumk \ “H 11% 3493 " iiQH\“~ “POW ﬁll/3% L? i , _, . \ ~3\$ me \w\+2zﬁ+-+<HH1¢N:(HH H Cwmsmkr MW :2 1 1» -—1 SL 13» (HVLW l W 1 t I \ . 1 u DU] \chkLsLQ/H ﬁrm \‘H ¥1~1, "F” ‘i' W A 1 H 611; M ., ...
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