This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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$+10=0
r(a:) : :1: < 0 Determine, with an explanation, the truth value of the statement Elm, 11(2) —> Mr) in each of the following
cases a) [2] When the universe consists of all real numbers lﬂ’dL “Nt\ 3(2 [ JAN; g h’fl‘ qu “’é f(l\ L: “1“}be
MewLara. P (Q C; [EdiQ b) [2] When the universe consists of only the integers 2 and 5. Valid v’\7k1\awi ? (é) CM. [0040 \XML
is H13 0mg] vi 3 GM mu Carlee 3. [3] Let A = {1, 2, {1, 2}}. Circle all true statements. No justiﬁcation is necessary. a) {2}EA v(b){1,2‘}o}1(d)@eA (e)An7>(A)=(z) (f)A[=4 4. [6] For a given universe and open statements 17(33 )and q(:1: ) show that [\7’z, p(m m)]/\ [Va:,q(:v)] is logically
equivalent to V1, p(m ) /\ q($). 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 ., ...
View
Full Document
 Spring '11
 stephenlang
 Calculus

Click to edit the document details