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week13b

Course: CSC 310, Fall 2009
School: University of Toronto
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Notes for CSC 310, Radford M. Neal, 2004 13b.2 )Xv6UtH5tS@ 9I2 4E 5bp bIs6UH5a@ G$ 4 E X R E2 B % F W( @ X( @ HX 4E XRE B %F 8 y 4E 5txY@ tPrtyb1sPqh 5 ) D X R E B @ ( 4 E Q Ip E 8 ( @ Q Xp E Q XBp X j E 0 c&I8I XRE 8 Pa& )SP15t}y 4E 2 6(S5a`)Y1pf 4 UXRE hX7@Q(X 7(& 8 0 B@ 82Q 5)y 4E SPSY c@ )eHX x FPw5{b1)3 X2&E 88 Xp y U @ Q X 7 @ 7p X 7 @ Q h 7 X 7 @ Q X 7 hD@ )Y1z919Y11)pf 8 ab}y 4E Q 1Hyyy35t1)t1TC0 4 R E U @pB 8 4E& UXE808 0 XRE EI yEI B Xp I 7 X Q @p y X 7 @ Q W 4 U U & ))b11Y5)Y1`56('{)" c&I8I XRE 8 )SPau0 4E 2 F5a`)Y1e'hA X8XRE hX 7 @QX f TgDqvQ t1HSFy1y{6(T)Y156(uu)@97 2 6& Up @B 4E & U XE8 08 4 X 7 @ Q W 4 U U & " 8 0 5 UbuF1D35 S1wSy 4E ea5SFPCh @pB & E & 7 7 X 4 X Q Xp Xp & 8 X 8 X R E 7 ( X E Xp Q @ 8 X R U @pB 8 E I y )X1p)IbFv5aE by 4E 19)6(4 2 P5{5aE w E X ( X R &p 4 X W &p X & ( 8 X R E B % # c ( @ W 4p X c 9)5w51Tb@ x y 4E 5a@ 3$ 0 9b36(gSFE )& X 6c9R t'X D)56(YW qDu5tS@ 3# 1FHw)Ye8 4 X 4 8 c | c & 4 4p @ X R E B 2 0 @E X Qp I @ 0 pS&S6(4 PeFSub1}1StHv5Sex89wS$" & 8 X 8 8 Xp y U @ Q E & R E X U X R Q 8 & Xp X S "# s) ' # #!C(D5SP (X8@R !e5Q ( 4 E Q ( IB ( 4 Ep @ E 4 7 X R E ( @ 7 ( & 8 4 b@ P5)uY@ tDyy8 Yu5tub35)X E 46c4 & w5y @p X Qp I @ 8 X R E R E ( @ 8 7 ( X y X 7 1)beH5aut@ YA959S) 15aHFS E&RE XE@ )St$F11wb353b@ )F8 Y7 ( & R E p X E & Xp W @ ( ( 4 Ep @ E 4 X W &p X & 8 & R E & R E X U X R Q 8 ( 4 8 8 Xp y U @ )5w515s1Stsse{b@ tFSub1Q o w p @B c U 08 E I y 4 p X y 8 X W &p X & XE & a Dqd Y@ )y1))6(9H 4E )5S51p`F1wp c @ n ~ r XRE 4 (XRE @7 (&Q 1tuV~'Pn 5a}8 E!5t`9x)uX VcSX 5R 8I cX s8 VcyE x DDm p5)w Dmppg1 vn r)qt 5H x w v q w vt m u X R x E '6U4vbe{)St}FX 3b@ )F8 Y7 4 c X U @ 8 ( & R E 8 8 c ( 4 Ep @ E 4 X W &p X & 8 & R E & R E E I y E I @ 0 ( & X Q I 7 @p )5w51Sv1St1)t1b{1)x1bYw5y @ E 7 c & 4 ( 4 8 8 Xp y U @ Q X 7 4 E & E I F1X @ cD8 `Y@ aF5{b1)CB v1SR 1 XE &p E F1ws1& x 4X 4E G 8 @y16(4v8 b@ 9yAF5{b1x0 aFX Py9@ dFs cX {X E c U 08 8 8 Xp y U @ Q c 8 8 c 88 c @ E & @ydEa1X9y)P'X x 8 x1)Ye9b{tba1'aB E Q X 4 X Qp I @ 8 p I @ B @ 0 y @p E ( X X R E # # $"tPtb3b9s{gtH R Qp & X 8 Xp p @ !SePwCDqB X W ( c & R Q E ( Xpp I Q & 4 8 X I 8 4 X 8 X R E W 4 U @ Qp X 95X Vc)5r9T8 sPF8 TeP5aT6(ubwsB 7 X 4 R Q & ( X ( 8 & R ( 4 88 Xp y U @ h15aX 95TX E !P5zY@ aFPw5{b1Q c)&sU6tbCS`Y5aash)Q P5wS3zy)15aE 4 E y @ B @ 7 @ R E X U c & 4 E Q &p y & @E U Xp @ X R (Y@ tDyy8 Y'y1aH5Sdy1eH)Y1Q 4 Ep @ E 4 7 X E &p X R E X @p y @ E 7 X 8 I 8 X 7 @ U @ 7 ( &p c & 4 E Q &p y 4 X R E U @pB y XE8 X R Y)5)w)Q tP5wSu6UT5abuyyu'$h c I Q 4 7 4 8 X Q ( &p c @E 7 ( & 8 4 PE )` YA8 sP15)wX by5)X tE 46c4 & c & I E y X Qp X y ( & U I R B @ E ( I @ Q Q & W 4 2 & )Sa193)s))d}1)b15z56(5)eE ( 4 E Q ( IB ( 4 Ep @ E 4 7 & 4 I 8 & W 4 c & Up @ Y@ tP){b@ DF8 Yu cX eE Px356(5 4)sr)YC 4 R E p @B 8 ( @ 8 & Xp 18 aDq}Yed@ 7 X 8 I Xp & 8 7 @ R E X q m 1eSA9YaPvU b9 b@ qS5 x 5Fy1q 8 I 4p & 7 & XE8 ( h ( 4E & 4E ( & I p @EQ X f B @ 28 &E p 4 4 Db@ 1 t1)SY)FP1gS'PeS& 6c'6Ua8 Xa$DrzY@ aFPw5{bxF1DxyF9@ )Q P553V(TX bp R E p @ ( 4 88 Xp y U @ Q &E & 7 088 c c & 4 E Q &p y 4 c @ X caE 4Ha& 930 t1r9x)15t)D1X A8!b))S E c 8 0 c y c E ( Xpp I Q 0p @ X R E E ( & W c X ( @ ( ( & R b"te#D5S3 P5C # Notes for CSC 310, Radford M. Neal, 2004 13b.3 X 7 4 7 X 4 X Q Xp X R E @ E W 4 7 ( @ y 8 Xpp @ !P)6(1 S115tF`6(Y5bPr)1Q 7p X 7 @ Q X R E E I y E I j f e D@ )Yu5t1)t1TB gf )rDgDhD B @ XE &p & p @ 8 # Suy1wz)qB x 4E t)F)Vc6c4 a)56(s8 aE X2 &E X7 4 4R W 4 7 ( X 8 7p X 7 @ Q X 8 @ R Q 56(!5e8 x 9D@ )Yue95SuX Bw)@ )Y1Q 7p X 7 @ E&RE B@ X 7 4 ( & 7(X8 (XRE HX W&88X U c &IEQ 1Saxa)56()i5F{5tb9)5FPsh)StP5& X R E @ E ( 4 Ep @ E 4 7 5aySb@ )Fy8 Yx0 1)asHpHye@ Q 7 Xp I 8 & X U 8 & E8 X 8 c 7p X 7 @ Q X R E 7 4 j f e D@ )Yu5t56( rf )rDgDhY 8 4 1X tE 46c4 & Ss19a1Yue5tyD)5b1X @p y E I y E I @ X 8 X R E @E c & I 8PX tE 46c4 & SYY@ )yHt 4E b)5)wA1w)@ )Y1Q 4 @p y c U 08 R U @ 7 ( &p E & 8 7p X 7 @ 2 4 y ( X R E 7 ( c X ( ( & R Q R Q I 8 p @B 8 4 Q 93a5)& x ))Sx5PDq}X E 46c4 & w5y @p EI)a1Yu5a5H )StyX d56(SX 1)ab@ yEI@ XRE 7( | X ( &RE 88 c W 4 EI yEI 7 ( & E I y 4 ( X E ( 4 Ep @ E 4 7 X R E @ E E Q X I 5)A1)96(`1X aX b@ tDyy8 YuayP1 P8 $VV X u6U6(uv1SaE S 59)Sx97 ab@ R ` Y 8 X 4 4 4 U E & R c X ( ( & R Q & p X 4 8 ( U @ 7 ( &p E & ( X8 @ R Q 8 X 7 @ Tb))v1F95AP)Y1Q W 8 I 7 X @p y 56(4a1d15SX )Sw)15tub@ t)Fy8 !uF1wp ( & Q U Xp @ X R E ( 4 Ep @ E 4 7 X E & 8!(b@59)S x Sw)15t`56(!Y1xy8 YH8 9Dq}8 ( ( & R U Xp @ X R E W 4 7 @ Q 0 4 @ ( 4 R p @B P 9 "$Y` H) #
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University of Toronto - CSC - 310
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University of Toronto - CSC - 310
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University of Toronto - CSC - 310
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University of Toronto - CSC - 310
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University of Toronto - CSC - 310
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University of Toronto - CSC - 310
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University of Toronto - CSC - 310
CSC 310 Solutions to Mid-term Test1. You would like to encode a sequence of symbols that come from an alphabet with d + 3 symbols. You want to encode symbols a1 , a2 , and a3 using codewords that are three bits long. You want to encode symbols a4 , a5 ,
University of Toronto - CSC - 310
CSC 310, Spring 2004 - Assignment #4Due 4:00pm April 8 (drop it off in my office, SS 6016A). You're also welcome to hand it in at the start of lecture on April 7. Worth 10% of the course grade. Note that this assignment is to be done by each student indi
San Diego State - BUS - 315
Dakota State University BUS 315 Credit and Collection Unit Three: Business Credit I Reading Assignment Chapter Chapter Chapter Chapter 12: 13: 14: 15: All All except pp. 279-280 All except p. 306 Pp. 308-328 Objectives Chapter 12: Business Use of Merchand
National Taiwan University - PS - 536
Journal of Contemporary China (2000), 9(24), 291308New Interventionism in the Making: interpreting state interventions in Hong Kong after the change of sovereigntyANTHONY B. L. CHEUNG*After the change of sovereignty in Hong Kong in July 1997, there has
UMKC - ECON - 506
Stanford - CS - 103
Handout #68 December 7, 2008CS103A Robert PlummerFinal Exam Review SessionNumber Theory 1. (20 points) Prove that for all positive integers n: 6 | (n (n + 1) (2n +1). Induction is not required for this proof (but is allowed).Induction 2. (20 points) L
Stanford - CS - 103
Handout #67 December 7, 2008CS103A Robert PlummerPractice Final Solutions1. Number Theory (20 points) (a) (5 points) Prove or Disprove the following: The total number of distinct positive divisors of any positive integer is even. Counterexample: The di
Stanford - CS - 103
Handout 66 December 7, 2008CS103A Robert PlummerProblem Set #10 Solutions1) Product rule: 30 * 12 * 4 *6 2) a) The two kinds of numbers don't overlap (e.g., if there are exactly four 0's then there are four 1's, not three), so it's the number of ways t
Stanford - CS - 103
Handout #65 December 3, 2008CS103A Robert PlummerProblem Set #9 Solutions1) a1 = 6, an+1 = an + 6 for n 1 2) P(n): Base Case: P(0) asserts that x =0i =0nx =i1 xn+11 x1 1xSince both sides are 1, this is true. 1 xk+1 1-xInductive step: Assume P(
Stanford - CS - 103
Handout #64 December 3, 2008CS103A Robert PlummerPractice FinalExam Rules: 1) You have 3 hours to complete this exam. 2) This is an open-note exam no textbooks allowed. Handouts, problem sets and solutions are allowed. You may also bring a one-page cri
Stanford - CS - 103
Handout #63 December 3, 2008CS103A Robert PlummerRevised Combinatorics ChartHere is a revised version of the chart for combinatorics formulas. It correctly shows that when we are selecting r objects from a set of n objects with repetition allowed, r ca
Stanford - CS - 103
CS103AHO #62Gdel II12/3/08Gdel NumberingGdel's Incompleteness Theorem(84x11)9(8x=11s5y)7 13 92 3 5 7 11 13 17 19 23 29This scheme allows us to represent every formula with a unique number. Given a number, we can determine whether i
Stanford - CS - 103
CS103AHO #61Gdel I12/1/08Gdel's Incompleteness TheoremCS103A12/1/08Kurt Gdel (1906 1978)Gdel's Incompleteness TheoremGdel, Kurt (1931). ber formal unentscheidbare Stze der Principia Mathematica und verwandter Systeme I. Monatshefte fr Mathematik
Stanford - CS - 103
Handout #58 November 19, 2008CS103A Robert PlummerProblem Set #10Due: Wed., December 3 (Turn in by noon, Dec. 5 at Bob's office for one late day) Combinatoric problems can be very misleading. They seem very simple, but may turn out to be surprisingly i
Stanford - CS - 103
CS103AHO #57Functions II11/19/08InverseA function f: X Y is called invertible if and only if there exists a function g : Y X such that y = f(x) x = g(y) for all x X and for all y Y. We call g the inverse of f and write g = f-1. f y0 f y1 X Y X x1 g Y
Stanford - CS - 103
Handout #56 November 17, 2008CS103A Robert PlummerProblem Set #8 Solutions1) The formula is n(n+1). Proof: We will show that property P(n) holds for all integers n > 0 using the weak mathematical induction. Here, property P(n) means that 2 + 4 + . + .
Stanford - CS - 103
CS103A HO# 53 Slides-Combinatorics II11/14/08Sets What is |A B| ? A (A and B not necessarily disjoint) B AWhat is |A B C| ? B|A| = 8|B| = 6 C|A| + |B| = 14, which double counts the intersection |A B| = |A| + |B| - |A B| This is known as the Principl
Stanford - CS - 103
Handout #52 November 14, 2008CS103A Robert PlummerCombinatoricsThis handout presents in prose form many of the principles and examples discussed in class.Combinatorics is the study of counting, which is important in Computer Science in many ways: To u
Stanford - CS - 103
CS103A HO#51 Slides-Combinatorics I11/12/08Combinatorics The Art and Science of CountingCombinatorics The Art and Science of CountingWhy count?Even Hollywood is interested:"There's only so many hands in a deck of cards." From the movie Shane To und
Stanford - CS - 103
Handout #50 November 10, 2008CS103A Robert PlummerProblem Set #9Due: November 19Part I: 1 point each 1) Give a recursive definition of the sequence a1, a2, . an when the nth element can be found using the formula an = 6n. n 1 xn+1 2) Prove by inductio
Stanford - CS - 103
CS103A HO# 49 Slides-Recursive Functions11/10/08Recursive Definition of a SetTo define a set S recursively, we specify - one or more initial elements of S - a rule for constructing new elements of S given those already in the set Example: 3S If x, y S,
Stanford - CS - 103
Handout #48 November 10, 2008CS103A Robert PlummerRevised SyllabusDate 11/3 Day M Lecture # 19 Topic Sequences and Summations; Induction I PS8 distributed Induction II Induction III, Recursion Recursive Functions PS9 distributed Combinatorics I Combina
Stanford - CS - 103
Handout #46 November 7, 2008CS103A Robert PlummerProblem Set #7 SolutionsNote: There is more than one way in which certain questions can be proved. What we've provided here for each question is just one of those alternatives. The main purpose of these
Stanford - CS - 103
CS103AHO# 45Slides-Induction II11/5/08Peano's Axioms There is a number 0. Every number has a successor, denoted by S(a). There is no number whose successor is 0, i.e., x (S(x) 0). Two numbers with the same successor are themselves equal, i.e., x y(S(x
Stanford - CS - 103
Handout #43 November 3, 2008CS103A Robert PlummerProblem Set #8Due: 11/1088 points total; 8 points for each problem Prove all the following using either weak or strong induction. Proper form for your proofs is important! 1) Find a formula for the sum
Stanford - CS - 103
Handout #41 November 3, 2008CS103A Robert PlummerSequences and SummationsA mathematician, like a poet or a painter, is a maker of patterns. G.H. Hardy A Mathematician's Apology (1940) Sequences Imagine a person (with a lot of spare time) who decides to
Stanford - CS - 103
Handout #42 November 3, 2008CS103A Robert PlummerIntroduction to Mathematical InductionOne of the most important tasks in mathematics is to discover and characterize regular patterns or sequences. The main mathematical tool we use to prove statements a
Stanford - CS - 103
CS103AHO#39RSA10/31/08RSA Cryptography: Motivation Alice Bob AliceRSA Cryptography: Motivation BobWe need a design such that Eve can also get a supply of Bob's locks, but cannot deduce the key.RSA Cryptography: Motivation We want a function E(m, Ke
Stanford - CS - 103
Handout #38 October 29, 2008CS103A Robert PlummerMidterm Exam SolutionsThe average on the midterm was 80.0.12Midterm Histogram10Number of Students86420 35-39 40-44 45-49 50-54 55-59 60-64 65-69 70-74 75-79 80-84 85-89 90-94 95-99 100 Grade Ran
Stanford - CS - 103
CS103A HO# 37Intro to Cryptography10/29/08Cryptography: Some ReferencesDavid Kahn. The Codebreakers (1967). Simon Singh. The Code Book (1999). Niels Ferguson and Bruce Schneier. Practical Cryptography (2003). Bruce Schneier. Applied Cryptography, 2nd
Stanford - CS - 103
CS103A HO# 36 Slides-Number Theory II10/27/08Some properties of gcd If a = bq + r for integers a, b, q, and r, then gcb(a, b) = gcd(b, r). gcd(66, 45) = gcd(45, 21) Theorem 32.4 (proved in the handout)10/27/08CS103AIf you want a text: Kenneth Rosen D
Stanford - CS - 103
Handout #35 October 27, 2008CS103A Robert PlummerMathematical ProofsJohn C. Mitchell & Maggie Johnson Department of Computer Science Stanford University1. Why write proofs?According to Webster's Unabridged Dictionary, the word prove comes from the La
Stanford - CS - 103
CS103A HO# 34 Slides-Introduction to Number Theory10/24/08Outline of topics for CS103A Basic ToolsCS103A10/24/08Problem Set 7 is due Monday, 11/3.This is one class period later than the date given in the Syllabus.Formal Logic and Proof Techniques Nu
Stanford - CS - 103
Handout #33 October 24, 2008CS103A Robert PlummerProblem Set #7As we have mentioned in class, it takes practice to develop the skill of doing mathematical proofs and this problem set is meant to provide that practice. Because of the nature of the probl
Stanford - CS - 103
Handout #31 October 24, 2008CS103A Robert PlummerNumber Theory & Proving "Real" TheoremsWe have just spent weeks learning about first-order logic and how to do both formal and informal proofs in FOL. For the most part, the proofs we have done pertained
Stanford - CS - 103
Handout #32 October 24, 2008CS103A Robert PlummerNumber Theory & Proving "Real" Theorems: Part IIBasics of Number TheoryNumber theory was once thought to be "pure" mathematics math for the sake of math. But with the evolution of computers (and particu
Stanford - CS - 103
CS103AHO#29Slides-ResolutionResolution PQ Q R PRResolution PQ Q R PR Resolution is a valid rule of inference, which we can use as a proof procedure. To see how we do this, we have to remind ourselves about conjunctive normal form.Conjunctive Normal F
Stanford - CS - 103
Handout #28 October 20, 2008CS103A Robert PlummerMidterm Review SessionThe following problems will be discussed in the Midterm review session, Tuesday, Oct. 21, from 9:30 10:45 am in Gates B03. You may want to look over the problems before the session.
Stanford - CS - 103
Handout #25 October 17, 2008CS103A Robert PlummerComplete Inference Rules= Elim P(n) . n=m . P(m) = Intro . a=a . ^ Elim P1 ^ P2 . P1 P2 ^ Intro P1 . P2 . P1 ^ P2 Intro P . P P2 . S S Intro P . P Elim . P Intro P . Q Q . P PQ Intro Elim PQ P . Q PQ Eli
Stanford - CS - 103
Handout #24 October 17, 2008CS103A Robert PlummerPractice MidtermExam Rules: 1) You have 2.0 hours to complete this exam. 2) Do not include your scratch work with your exam. Please work the solutions out on another sheet of paper and then write your so
Stanford - CS - 103
CS103A HO#22 Slides-Proofs with Quantifiers10/17/08Proofs with Quantifiers Every child is right-handed or intelligent No intelligent child eats liver There is a child who eats liver and onions There is a right-handed child who eats onions Midterm Exam T
Stanford - CS - 103
Handout #23 October 17, 2008CS103A Robert PlummerProblem Set #6Due: 10/22 (Must be submitted Wednesday, 10/22-late days are not allowed on this problem set due to the midterm.) Multiple Quantifiers Notes: Hints for some of these problems are available
Stanford - CS - 103
Handout #20 October 13, 2008CS103A Robert PlummerProblem Set #5Due: 10/17 in class Logic of Quantifiers 1) Exercise 10.7 on p. 266. Explain your assessment of validity. 2) Exercise 10.9 on p. 273 3) Exercises 10.17, 10.18, 10.19 on p. 274 Multiple Quan
Stanford - CS - 103
CS103A Handout #19 Slides-Logic of Quantifiers10/13/088.38.2CS103A10/13/08Midterm Exam Thurs., Oct. 23 7 - 9 pm Location TBADescribe a world for each of these x Cube(x) x Cube(x) x (Professor(x) Smart(x) x Professor(x) x Cube(x) x Cube(x) x Smart(x)
Stanford - CS - 103
Handout #18 October 10, 2008CS103A Robert PlummerIntroduction to QuantificationPredicates Revisited We have been working with predicates since the beginning of our studies in logic. Predicate symbols are used to express some property of objects or some