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 DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: C8202  DISCRETE MATHEMATICS
SEMESTER EXAMINATION 2
FALL 2007 Time Limit  Sevenﬂ Five Minutes This is a closed book, closed notes examination. The numbers in parentheses after each question is the number of points allocated to that
question. Write your answers on the examination paper in ink or legible pencil. I our answer cannot be read or understood or i our answer is vague or confused, it will be marked wrong. NAME (Print Legibly. All Cagitals): PLEDGE (Write Out In Full And Sign): Department of Computer Science CS 202 University of Virginia 1. Prove by induction that: (10 pts) n Zri = (r”+'—1)/(r—l) i=0
3:. +6
'5» Os 9 O“
,r; r _1,r1:r 1
5:9 ' \'~ r4
n+1)“
.. . «H A _
JV“\“C+WC S‘H’?‘ g rL _ (“‘4 we Wei ﬁlmA HAM Z r‘ : i
_A$§Uwuncl 1° ~ (“1 ’ [:0 r4
. .4 mu n+1 ’l 1
VEH “H+%r‘ ~ (“Hib‘q “MLl — “A “r " ’1 r AMH‘
 +  ..
1:9 r [:0 ("'1 r‘( r“ 1‘"
(32.5."D 2. A real number r is rational if there are integers p and q with q ¢ 0 such that r = p/q. Prove
using a direct proof that if r and s are rational then r + s is rational. (10 pts) Six/ma r WA 5 Ma rcuh'onaei, ﬂare exi$+ (vii26103 05b, gamdal swan Jame r: an, “an, auva new 5 MAM 3w». ‘ﬂm
{ml—egm are. clo‘SaA uvxc\e)f‘ aclciii'iew C(Mtl. Mui‘HPh'Cﬁf‘bV) $0 *‘lDC l" cw (“713W MA. \03« is cwx‘mle$2f. giwce modalLax lamaoi
is Sore. lag» is wuvjexb. m5, Lt‘ A,» I'm '00) “AH” is rodwéwki LJ,
v Q “(‘7 ‘ ‘ 
Svmc; h +5 = A, + z b; ‘ helmet \3 rqhowqi’ f+g "S rathéml, a5 was «in» be 4w“. 3. Prove using a proof by contradiction that for any integer n, if 3n + 2 is odd, then n is odd.
(10 pts) Suwose ~lo ‘l'luz. Con‘l'rwrbl JAM} We»: is $9Me ;M‘\>e¢3a¢~ W such M 313+; is 6AA lowl VLis wow. '31 MM. c\,4\'u(Jném5 0? 05M WA gum,
‘H’UUI £435" (“13% q WA l) such ‘uhaui' 3n+1=2a+l awxci m=2l>_
SubsaiﬂM'Hm (11¢,le 5(1Hi’2 =24+i’ $0, solu‘mS ¥D¢ lo, M M 5:35” W5 15 my? 4M $447939!" $0 we \Aawc rectal/mi a unhaol'mi'fcn. Page score Department of Computer Science CS 202 University of Virginia 4. How many total functions are there from a source set of three elements to a target set of two
elements? Carefully explain your answer. (5 pts) 8. «pungjiam i5 gr each twain," O? 'HALwUfa 52+, ‘0. Wylyb'ma' wad $e+ es T={€.,ée§ owl “4‘ $09M 5* “3 S'isttSaSs 3' HM “is” fumhees “‘5 “‘W ’t “NW “#03 { (Soto), been, (939:1? 5
{(54'tt); (Epic ), (531t2)} 161, ‘33,), (522%,), ($3, :1);
{(51t£1)1(52_ybz),(53)t1)% {(Shtz)’ (5mb1), (S3, , }
‘Z (sues), (51,151), (534:1); =1 (Shea), (slim), (53,12); 5. How many partial surjective functions are there from a source set of three elements to a target
set of two elements? Carefully explain your answer. (5 pts) 1.? A th'wl Suded'k 'FVwch’Oﬂ cougrs HM. c‘l’ Sci" secleMohln$ Source.
36‘, “ S={5h51’53} W ‘HM. 55+ 6 T: if“ z E, M ‘lchuc Pun/\Choﬂs km‘. (sum, (syn) E {with}, (sate)? {(5.,4:t),($,.£n_), (sate)?
(ﬁll£1), 53't‘)} { (51’ ﬁg), (52,£1) } {I (5“ E2), [Sl’eq )1 (531t1)}
{(511‘E4 )l { ($4,€( ) (Sz’tl), { (${,£2); (SZIt‘ )I (ﬁnes). (Seth)? { Ls.,4:,), (sate), (swept .{ (5,, ea), (5;:‘52’153'tﬂ? 6. Suppose f is a function whose source set is the letters of the alphabet and whose target set is
the positive integers, and g is a function whose source set is the positive integers and whose
target set is the letters of the alphabet. If the functions f and g are deﬁned as follows: f: {(a, 2), (b, 4), (Ci 1), (d, 5), (e, 3)} g = {(1,3) (2, b), (3, C), (4, d), (5, 6)}
(a) What is the deﬁnition of the function g of? (5 pts) gear : Maintain), me), (the), up}
(b) What is the deﬁnition ofthe function fo g 7 (5 pts)
tug =f(1,a),(2,‘+l,(5:13,WIS), (5,5)}
(0) What is the deﬁnition ofthe function f” o g'1 ? (5 pts)
f’18‘1={tn,e), (5,4), (52), (it), (cm;
(d) What is the deﬁnition ofthe function g'1 of" ? (5 pts) 3’ 91/413091), (4,2), (4,3), (574), (3,5)} Page score Department of Computer Science CS 202 University of Virginia 7. Using an i f statement as an example or otherwise, carefully explain what is meant by syntac
tic ambiguity in common programming languages. (5 pts) A iquuasz con‘tm’ns a 5'1quch 5
mbfau/(ixj if a val (A, warm in //l \ ‘ .. ‘ ¢\ 5
(ten/miner?ch (1n vvwd 'plc ‘i: 61%“ 5 w ng< z
m 5;: //' \\ $9 HM $4
(daup atﬁwtmb "x gwsﬂjsesz \
“if a, Haw if (Iikw 513‘52 5;”,
i2!" Welt, Couloi be. PM «$2
8. Using a for statement as an example or otherwise, careﬁally explain what is meant by seman—
tic ambiguity in common programming languages. (5 pts)
A COW‘i’hllﬂ—5 Q SOMAch mb\,8u1¥3 RS; ‘HKUL are some (Dazet Waste/MS whose vuwa «I; mi» ’pmciseiul 44"" \2,
ix» We. woeva Qraﬁumt “4:”— x, grown t +5 to be! 1 Ac 5 “A QM)",
it \s unclear whizHater, “(H—o» iln— (:rASMMt 13 wagedcar, 7: shard; log 10’ H,
0r Manama 9. Consider the following piece of program (the parenthesized numbers are line numbers): (1) i, n, factorial : integer;
<2) (3) i = 1; (4) factorial := 1; (5) read(n); (6) (7) while i < n loop (8) i := i + 1; (9) factorial := factorial * i;
(10) end loop (11) (a) What is the loop invariant for the while loop? (10 pts) I .— ’l
‘Faci’orial ‘ l. (b) Before execution of which line or lines should the loop invariant hold? (5 pts) “ﬂu. \009 \iAuaurimAt Med)»; MeiEyre \eres (5 «and to cure
8—)Lccuie¢i, Zi clo¢s V\o+ \oefofe ii“ 0! is e/yecpei’edt. (c) The use of a loop invariant facilitates proof of the correctness of program loops using what
proof method? (5 pts) (Proon \n‘ \lwiuc‘hlon. Page score Department of Computer Science CS 202 University of Virginia 10. Consider the grammar G = (V, Z S, P) where V: {0, 1, A, B, S}, T= {0, 1}, S is the start sym—
bol, and the productions P are:
S—)0A S—)1A A—)OB B—)1A B—)1 (a) By drawing the derivation tree, show that the string 10101 belongs to the language gener—
ated by G. (5 pts) \k
\
if?
I
01 101 (b) By showing that you cannot draw the derivation tree, show that the string 10110 does not
belongs to the language generated by G. (5 pts) 5\ We have no choice loul' +0 «PP‘LI Was;
A\ Freda4,415.45 im ‘HA‘lS ordzr) bwk Hare
3\ is Vlo Wcluch‘an gr A 'HAQl' jwwas‘l’tS
{ A C; 6‘1?in \OC‘3\\:\V\'\1A3M;\\'L\ a 1 . 4 O 1
(c) What is the language generated by G? (An informal answer is sufﬁcient.) (10 pts) [o1](o1)+ m $61, a 6H<n35 beginning) w'iHa a 0 w c. 1 Rllowed
lo“ M av More. reechh'mms (Zr ‘HN. Sulos‘l‘rlujg of. ((1) Draw the ﬁnite state automaton that recognizes the language generated by G. (10 pts) Page score Department of Computer Science CS 202 University of Wrginia ll. For a directed graph G, the vertices are {v1, v2, v3, v4, v5} and the edges are {e1, e2, e3, e4, e5}.
The edge—endpoint function is: (a) List the degree of each vertex. (6 pts) Ver+mfw ‘anV3IVwIV5
1
oleﬁmc 9'1 3 2 3» M V, O 1 O 0 o (b) What is the adjacency matrixA for G? (6 pts) V1 0 o 1 o 0
v3 0 o o o o air—> 4 O 1 o O i V5 4 o 0 0 o (c) Is there a cycle for this graph? If so, give it. If not, explain why not. (6 pts) q CW ‘0‘ ‘l'bpolo (CaJith sarieel, 41y. lV‘ 'HAis ‘me’ Slit/ICC. 'HAerg are
3 “0 backwmclr 1113‘s, H‘ is
v.1. > V ——a v‘( _a Va ——9 VJ C(W .HAOJ. “Ava M¢ Mo dad‘s. V 12. If two cards are dealt from a well shufﬂed card deck, what is the probability that they are both
aces? (6 pts) , 74mm 0M WA” 51CWL’3 lad(arc
52 5/ xlLa P364 Arm awc ~H4rce [L54 «ﬂatuw. 13. How many alphabetic character strings (ignoring case) of length four are there that start and (C 9’ end with a vowel (not counting y as a vowel)? (6 pts) 5.162.45 Page score ...
View
Full
Document
This note was uploaded on 10/10/2009 for the course CS 2102 taught by Professor Knight during the Spring '08 term at UVA.
 Spring '08
 KNIGHT

Click to edit the document details