This preview shows pages 1–6. Sign up to view the full content.
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: C90 Discrete Mathemiiesfor Computer Science, Faii 20G? Midterm 1
6:ﬂﬂ~3:ﬂﬂpm, 3 October Notes: men” are [our questions on this midterm. Answer each question pan in the space beiow if, using
the back of the sheet to continue your answer ifnecessary. {fyou need more space. use the blank sheet at
the end. None of the quosﬁons requires a very long answer, so avoid writing too much! Unclear or
longwinded solutions may be penalized. The approximate oredii‘for each question part is shown in the
margin {Ioiai {it} poinm. Points are no: necessarin an i'ndioorion ofdiﬁcniry! Your Name: Your Section: For oﬁicial use; please do not Write below this line! Total [exam starts on next page] 1. Quick Questions (:1) Write down the truth tables for (i) P =e Q; {ii} [.2 => F; {iii} P {= Q. 6pm {in} Let and Gus} be propositions about integers I and 3; respectively. and suppose you want to prove 3pm:
the theorem {Elm 13(3)) : {lr'y Qliyjj. Which of the following proof strategies would be a valid way
to proceed? (Note: There may be no valid strategies+ or more than one 1relid strategy. You do not need
to explain your answers.]
{i} Assume that there is some 3,; for which Q[y} is false, and deduce that 13(3) is false for all I.
lit} Assume that Q[y] is true for all y. and deduce that is true for some 2:.
till] Assume that. for some a: and some 1;, PLr) is true and Qlﬁy} is false. and deduce a contradiction. (iv) Assume that Plrl is true for all :r and Qﬁy} is false for all y, and deduce a contradiction. [continued on next page] [Q1 continued] {c} Consider an instance of the Stable Maning problem in which the men are {1, 2, 3, 4}, the wcmen are 4pm
{31, Bic. D}, and the preference lists are Men (14} Wurnen (A—B} l: B A D C A: 2 l 4 3
2: C A D B B: 3 4 2 l
3: A C B D C: l 4 3 2
4: D C B A D: 2 3 1 4 List the stable pairing given by the traditional proposeandreject algorithm on this instance. [You need
no: Sl'IDW the execution of the algorithm] (d) Compute the multiplicative inverse of ID modulo T43 using the extended GED algorithm diecuseed in {Spits
class. Shim your workingl Your answer should be an integer in the range [0,?42], [continued on next page] 2 A Tiling Problem You are given three kinds of tiles. A, B, and C, of dimensions 1 x 2, 2 x 1, and 2 X 2 respectively [as shown
in the ﬁgure below). Note that rotations of the tiles are not ctllowe', so tiles A and B are not the some! E
112 Bi 93:2 Your goal is to tile a board of dimension 2 x n. and you are interested in the number of tiling conﬁgurations
possible [using the three types of tiles given}. The ﬁgure below shows an example of three different tiling
conﬁgurations for a 2 >< 5 (i.e. n. = 5) bosud. Let Tn denote the number of tiling conﬁgurations for a board of dimension 2 X n. (a) Compute T1 and T2. 2pm {in} Explain carefully whj.r ’1“rt = T114 + 2Tﬂ_2. [HINT: The conﬁgurations in the example given above 4pm
for n = 5 may be helpﬁil.]  . 2n+1 + {_l}l’t (c) Prove by induction that TnL = 3 for all :1 1‘3 1. Show cleaer the structure of your proof! 9pm [continued on next pagel 3. Modular Arithmetic: Which of the following staternents is mm? In each case, if the statement is true give a brief explanation; if
it is false, give a simple counterexample. ' {i} For all1,!) E Z, {r}. + 1:313 = u."1+ 153 mod 3. [Nata {a + b}3 = a3 + 3:12b + Hub: + b3.] 3m:
{ii} For ail a, b e z. {a + b)“ = a," + 5‘1 mod r1.[Notc: (a + m4 = a4 + Mb + Enab?‘ + 4gb3 + 33.4.] 3pm
(iii) For all :11 b e Z. (a.+b]5 = (15+? mod 5. [Note: {a+b]5 = a5+5a4h+lﬂa3bg+lﬂa2b3+ﬁaﬁ4+b5j 3pm [continued on next page] 4. Stable Marriage For each of the following properties, sag.r whether the propertyI is true or false. If the property is true. provide a short proof. If it is false. provide a simple counterexample. You may use without proof any facts about
Stable Marriage covered in class provided they are clearly stated. {a} In a Stable Marriage instance with at least two men and two women. if man M and woman W each 4pm
put each other at the top of their respective preference lists then there exists a stable pairing in which
fit and W are paired together. lb) In a Stable Marriage instance with at least two men and two women. if mat: .M and woman W each 4pm
put each other at the bottom of their respective preference lists then there is no stable pairing in which
M and W are paired together. to) In a Stable Maniage instance with at least two men and. two women. it is possible that there exists an 4gb:
unstable pairing in which every unmatched manwoman pair (M, W} is a rogue couple. [The end] ...
View Full
Document
 Spring '08
 PAPADIMITROU

Click to edit the document details