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### MIT6_042JS10_lec04

Course: CS 6.042J, Spring 2011
School: MIT
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Word Count: 843

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(Boolean) Propositional Logic A proposition is either True or False The Logic of Propositions Example: There are 6 regular solids. 5 True False Non-examples: Wake up! Where am I? lec 2W.1 February 10, 2010 Albert R Meyer English to Math lec 2W.2 February 10, 2010 Albert R Meyer English to Math Greeks carry Swords or Javelins Greeks carry Bronze or Copper swords True even if a Greek carries both a...

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(Boolean) Propositional Logic A proposition is either True or False The Logic of Propositions Example: There are 6 regular solids. 5 True False Non-examples: Wake up! Where am I? lec 2W.1 February 10, 2010 Albert R Meyer English to Math lec 2W.2 February 10, 2010 Albert R Meyer English to Math Greeks carry Swords or Javelins Greeks carry Bronze or Copper swords True even if a Greek carries both a Sword and a Javelin Bronze or Copper but not both G (S J) lec 2W.3 February 10, 2010 Albert R Meyer G (B C) lec 2W.4 February 10, 2010 Albert R Meyer Definition of XOR Definition of OR The value of (P OR Q) is T iff P is T, or Q is T, or both are T. The value of (P XOR Q) is T iff exactly one of P and Q is T. Truth Table for XOR Truth Table for OR P P OR Q P Q P XOR Q T T T T T F T F T T F T F T T F T T F Albert R Meyer Q F F F F F February 10, 2010 F iff both P,Q are F lec 2W.5 Albert R Meyer February 10, 2010 lec 2W.6 1 Definition of AND Definition of NOT The value of (P AND Q) is T iff both P and Q are T. Truth Table for AND P T T T F F F T F F F Truth Table for NOT (P) P AND Q T Albert R Meyer Q The NOT(P) is T iff P is F. F February 10, 2010 P T Truth Assignments February 10, 2010 lec 2W.9 Example: Suppose environment, v, assigns v(P) = T, v(Q)= T, v(R) = F. Truth value of ( (P AND Q) ) OR (R XOR ( Q)) FT FF FT T T F lec 2W.10 February 10, 2010 Albert R Meyer DeMorgans Law is equivalent to Two propositional formulas are equivalent iff they have the same truth value in all environments. February 10, 2010 lec 2W.8 February 10, 2010 Albert R Meyer Equivalence Albert R Meyer T Evaluation in an Environment A truth assignment assigns a value T or F to each propositional variable. Computer scientists call assignment of values to variables an environment. If we know the environment, we can find the value of a propositional formula. Albert R Meyer F F lec 2W.7 NOT(P) lec 2W.11 P Q T T F F T F T F Albert R Meyer (P Q) F F F T T F T T F T February 10, 2010 F T F F F T F T F T lec 2W.13 2 Definition of IMPLIES DeMorgans Law The value of (P IMPLIES Q) is F iff P is T and Q is F. Truth Table for IMPLIES () is equivalent to P Q T T F F T F T F (P Q) F F F T T F T T F T F T F F F T F P Albert Meyer February R 10, 2010 lec 2W.14 A True Implication February 10, 2010 lec 2W.16 T T F T February 10, 2010 lec 2W.15 Albert R Meyer February 10, 2010 lec 2W.17 A True Implication (1=-1) IMPLIES (I am Pope) We reasoned correctly to reach the false conclusion from the false hypothesis. February 10, 2010 F (1=-1) IMPLIES (I am Pope) We reasoned correctly to reach the false conclusion A True Implication Albert R Meyer F A True Implication (1=-1) IMPLIES (I am Pope) We reasoned correctly to reach the false conclusion Albert R Meyer Albert R Meyer T F Same final column, so equivalent -- proof by Truth Table T F T PQ T F Q T T (1=-1) IMPLIES (I am Pope) We reasoned correctly to reach the false conclusion from the false hypothesis. lec 2W.18 Albert R Meyer February 10, 2010 lec 2W.19 3 A True Implication Satisfiability & Validity (1=-1) IMPLIES (I am Pope) The whole implication is true, even though both conclusion & hypothesis are false. lec 2W.20 February 10, 2010 Albert R Meyer Verifying Valid, Satisfiable A formula is valid iff it is true in all environments. Albert R Meyer February 10, 2010 lec 2W.21 Efficient Test for Satisfiability? Truth table size doubles with each additional variable --exponential growth. Makes truth tables impossible when there are hundreds of variables. (In current digital circuits, there are millions of variables.) February 10, 2010 Albert R Meyer A formula is satisfiable iff it is true in some environment. The P=NP? question is equivalent to asking if there is an efficient (polynomial rather than exponential time) procedure to check satisfiability. Albert R Meyer February 10, 2010 Digital Logic Java Logical Expression AND OR if ((x>0) || (x <= 0 && y>100)) (more code) half adder from http://en.wikipedia.org/wiki/Adder_(electronics) Albert R Meyer February 10, 2010 lec 2W.38 Albert R Meyer February 10, 2010 lec 2W.40 4 Digital Logic A B Team Problems d cin cout s c Problems 14 full adder Albert R Meyer February 10, 2010 lec 2W.41 Albert R Meyer February 10, 2010 lec 2W.42 5 MIT OpenCourseWare http://ocw.mit.edu 6.042J / 18.062J Mathematics for Computer Science Spring 2010 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 10revised February 3, 2010, 2 minutesIn-Class Problems Week 2, Wed.Problem 1.Prove by truth table that O R distributes ove
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 10revised February 3, 2010, 3 minutesSolutions to In-Class Problems Week 2, Wed.Problem 1.Prove by truth table that O R di
MIT - CS - 6.042J
surjective &amp; functionMathematics for Computer ScienceMIT 6.042J/18.062J 1 arrow outCardinalityA(the size of sets)Albert R Meyer,February 16, 2010lec 3M.1Mapping Rule (surj)February 16, 2010lec 3M.3Feb. 17, 2009BAlbert Albert R.February 16,
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 16revised February 9, 2010, 1094 minutesIn-Class Problems Week 3, Tue.Problem 1.Lemma 4.9.4. Let A be a set and b A. If A
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 12revised February 11, 2010, 172 minutesSolutions to In-Class Problems Week 2, Fri.Problem 1.Set Formulas and Propositiona
MIT - CS - 6.042J
PredicatesMathematics for Computer ScienceMIT 6.042J/18.062JPropositions with variablesPredicate LogicExample:P(x,y) := [x + 2 = y]Quantifiers ,Albert R Meyer,February 17, 2010Albert R Meyer,lec 3W.1PredicatesFebruary 17, 2010lec 3W.2Quanti
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 17revised February 11, 2010, 1155 minutesIn-Class Problems Week 3, Wed.Problem 1.For each of the logical formulas, indicat
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 16revised February 9, 2010, 1094 minutesSolutions to In-Class Problems Week 3, Tue.Problem 1.Lemma 4.9.4. Let A be a set a
MIT - CS - 6.042J
AxiomsMathematics for Computer ScienceMIT 6.042J/18.062JEqualityx[x y x z] y = zSet TheoryAlbert R Meyer,February 19, 2010Power setxps. s x s plec 3F.1Russells ParadoxFebruary 19, 2010February 19, 2010lec 3F.2Disaster: Math is broken!I am
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 19revised February 19, 2010, 1407 minutesIn-Class Problems Week 3, Fri.Problem 1.Lets refer to a programming procedure (wr
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 17revised February 11, 2010, 1187 minutesSolutions to In-Class Problems Week 3, Wed.Problem 1.For each of the logical form
MIT - CS - 6.042J
The Idea of InductionColor the integers 00, 1, 2, 3, 4, 5, I tell you, 0 is red, &amp; any intnext to a red integer is red,then you know thatInductionall the ints are red!Albert R Meyer,February 22, 2010lec 4M.1Albert R Meyer,February 22, 2010lec
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 22revised February 18, 2010, 16 minutesIn-Class Problems Week 4, Mon.Problem 1.Prove by induction:1+1111+ + + 2 &lt; 2 ,
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 19revised February 19, 2010, 1400 minutesSolutions to In-Class Problems Week 3, Fri.Problem 1.Lets refer to a programming
MIT - CS - 6.042J
!&quot; \$%&amp;!#&quot;Mathematics for Computer Scienceproper subset relationMIT 6.042J/18.062Jcfw_1,2,3,5,10,15,30cfw_1,2,5,10cfw_1,3,5,15Partial Orderscfw_1,3cfw_1,5cfw_1,2cfw_1Albert R Meyer, Feb. 24, 2010lec4W.1Albert R Meyer, Feb. 24, 2010propertie
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 24revised February 24, 2010, 756 minutesIn-Class Problems Week 4, Wed.Problem 1.Direct Prerequisites18.0118.0118.018.0
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 22revised February 18, 2010, 15 minutesSolutions to In-Class Problems Week 4, Mon.Problem 1.Prove by induction:1+1111
MIT - CS - 6.042J
2/26/10Mathematics for Computer ScienceSome Course 6 PrerequisitesMIT 6.042J/18.062J8.02 6.00218.01 6.04218.03, 6.002 6.00418.01 18.02 6.001, 6.004 6.0336.033 6.85718.01 18.036.046 6.8406.001 6.0346.042 6.046Partial Orders &amp;SchedulingAlbert
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 26revised February 21, 2010, 1416 minutesIn-Class Problems Week 4, Fri.Problem 1.The table below lists some prerequisite i
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 24revised February 24, 2010, 756 minutesSolutions to In-Class Problems Week 4, Wed.Problem 1.Direct Prerequisites18.0118
MIT - CS - 6.042J
Mathematics for Computer ScienceMIT 6.042J/18.062JDigraphsa set, V, of verticesa set, E VVof directed edges(v,w) E notation: vwDirected GraphsvAlbert R Meyer, March 1, 2010lec 5M.1Relations and GraphsadAlbert R Meyer, March 1, 2010lec 5M.2
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 1revised February 27, 2010, 1329 minutesIn-Class Problems Week 5, Mon.Problem 1.If a and b are distinct nodes of a digraph, t
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 26revised February 21, 2010, 1418 minutesSolutions to In-Class Problems Week 4, Fri.Problem 1.The table below lists some p
MIT - CS - 6.042J
3/1/10State machinesMathematics for Computer ScienceMIT 6.042J/18.062Jstep by step processes(may step in responseto input not today)StateMachinesAlbert R Meyer, March 3, 2010lec 5W.1State machinesAlbert R Meyer, March 3, 2010lec 5W.2Die Hard
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 3revised February 26, 2010, 1410 minutesIn-Class Problems Week 5, Wed.By now you are very familiar with the 6.042 icon that ap
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 1revised March 1, 2010, 826 minutesSolutions to In-Class Problems Week 5, Mon.Problem 1.If a and b are distinct nodes of a di
MIT - CS - 6.042J
3/5/10Euclidean AlgorithmMathematics for Computer ScienceMIT 6.042J/18.062J-for GCD(a, b)1. x := a, y := b.2. If y = 0, return x &amp;terminate;3. else simultaneously:State Machines:Derived Variables(x, y) := (y, rem(x,y)4. Go to step 2.Albert R
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 5revised February 28, 2010, 1326 minutesIn-Class Problems Week 5, Fri.Problem 1.The Massachusetts Turnpike Authority is conce
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 3revised March 1, 2010, 826 minutesSolutions to In-Class Problems Week 5, Wed.By now you are very familiar with the 6.042 icon
MIT - CS - 6.042J
3/5/10Euclidean AlgorithmMathematics for Computer Science-for GCD(a, b)MIT 6.042J/18.062J1. x := a, y := b.2. If y = 0, return x &amp;terminate;3. else simultaneously:State Machines:Derived VariablesAlbert R Meyer, March 5, 2010(x, y) := (y, rem(x
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 5revised February 28, 2010, 1326 minutesIn-Class Problems Week 5, Fri.Problem 1.The Massachusetts Turnpike Authority is conce
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 5revised March 5, 2010, 859 minutesSolutions to In-Class Problems Week 5, Fri.Problem 1.The Massachusetts Turnpike Authority
MIT - CS - 6.042J
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 8revised March 2, 2010, 734 minutesIn-Class Problems Week 6, Mon.Problem 1.Four Students want separate assignments to four VI
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 8revised March 8, 2010, 683 minutesSolutions to In-Class Problems Week 6, Mon.Problem 1.Four Students want separate assignmen
MIT - CS - 6.042J
Types of GraphsMathematics for Computer ScienceMIT 6.042J/18.062JSimpleGraphSimple GraphsDegrees,Isomorphism,PathsAlbert R Meyer, March 10, 2010Directed Graphnext weekthis weeklec 6W.1A simple graph:Multi-GraphA Simple GraphDefinition:A
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 10revised March 2, 2010, 734 minutesIn-Class Problems Week 6, Wed.Problem 1.For each of the following pairs of graphs, either
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 10revised March 8, 2010, 683 minutesSolutions to In-Class Problems Week 6, Wed.Problem 1.For each of the following pairs of g
MIT - CS - 6.042J
Connected ComponentsMathematics for Computer ScienceMIT 6.042J/18.062JEvery graph consists ofseparate connectedpieces (subgraphs) calledGraph ConnectivityTreesAlbert R Meyer, March 12, 2010connected componentslec 6F.1Connected Components1312
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 12revised March 13, 2010, 1029 minutesIn-Class Problems Week 6, Fri.Problem 1.Prove that a graph is a tree iff it has a uniqu
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 12revised March 13, 2010, 1024 minutesSolutions to In-Class Problems Week 6, Fri.Problem 1.Prove that a graph is a tree iff i
MIT - CS - 6.042J
Mathematics for Computer ScienceMIT 6.042J/18.062Jflights need gates, buttimes overlap.how many gates needed?Graph ColoringBipartite Matchinglec 7M.1Albert R Meyer, March 15, 2010Airline Schedulelec 7M.2Albert R Meyer, March 15, 2010Conflicts
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 15revised March 2, 2010, 734 minutesIn-Class Problems Week 7, Mon.Problem 1.Let G be the graph below1 . Carefully explain why
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 15revised March 15, 2010, 675 minutesSolutions to In-Class Problems Week 7, Mon.Problem 1.Let G be the graph below1 . Careful
MIT - CS - 6.042J
Mathematics for Computer ScienceRecursive DenitionsMIT 6.042J/18.062JRecursive Definitions&amp; Structural InductionMatched Paren Strings, Mstrings in M[][][][][][][]M, thenMAlbert R Meyer, March 17, 2010lec 7W.9s=t=s = []t=s=t =[]s = []
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 17revised March 2, 2010, 733 minutesIn-Class Problems Week 7, Wed.Problem 1.The Elementary 18.01 Functions (F18s) are the set
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 17revised March 15, 2010, 675 minutesSolutions to In-Class Problems Week 7, Wed.Problem 1.The Elementary 18.01 Functions (F18
MIT - CS - 6.042J
Planar GraphsMathematics for Computer ScienceMIT 6.042J/18.062JPlanar GraphsAlbert R Meyer,March 19, 2010Albert R Meyer,lec 7F.1March 19, 2010lec 7F.2 Source unknown. All rights reserved.This content is excluded from our Creative Commons licens
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 19revised March 12, 2010, 1326 minutesIn-Class Problems Week 7, Fri.Problem 1.Figures 14 show different pictures of planar gr
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 19revised March 22, 2010, 587 minutesSolutions to In-Class Problems Week 7, Fri.Problem 1.Figures 14 show different pictures
MIT - CS - 6.042J
Arithmetic AssumptionsMathematics for Computer ScienceMIT 6.042J/18.062Jassume usual rules for +, , - :Intro toNumber Theory:Divisibility, GCDsAlbert R Meyer,March 29, 2010a (b+c) = ab + ac, ab = ba,(ab)c = a (bc), a a =0,a + 0 = a, a+1 &gt; a, .
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 29revised March 20, 2010, 738 minutesIn-Class Problems Week 8, Mon.Problem 1.A number is perfect if it is equal to the sum of
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 29revised March 29, 2010, 739 minutesSolutions to In-Class Problems Week 8, Mon.Problem 1.A number is perfect if it is equal
MIT - CS - 6.042J
Congruence mod nMathematics for Computer ScienceMIT 6.042J/18.062JDef: a b (mod n)iff n|(a - b)Congruences:arithmetic (mod n)example: 3012 (mod 9)since9 divides 30 - 12Albert R Meyer,March 31, 2010Congruence mod nexample:66666663WHY?66666
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 31revised March 30, 2010, 1426 minutesIn-Class Problems Week 8, Wed.Problem 1. (a) Use the Pulverizer to nd the inverse of 13
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerMarch 31revised March 30, 2010, 1434 minutesSolutions to In-Class Problems Week 8, Wed.Problem 1. (a) Use the Pulverizer to nd the i
MIT - CS - 6.042J
EulerMathematics for Computer ScienceMIT 6.042J/18.062J(n) :=# k 0,1,n-1 s.t.k has a (modto nrel. prime n)inverseEulers TheoremRSA encryptionAlbert R Meyer,EulerApril 2, 2010lec 8F.1functionAlbert R Meyer,April 2, 2010lec 8F.2Calculating
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerApril 2revised March 13, 2010, 108 minutesIn-Class Problems Week 8, Fri.Problem 1.Lets try out RSA! There is a complete description
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerApril 2revised March 29, 2010, 739 minutesSolutions to In-Class Problems Week 8, Fri.Problem 1.Lets try out RSA! There is a complet
MIT - CS - 6.042J
Book StackingMathematics for Computer ScienceMIT 6.042J/18.062JHarmonic SumIntegral MethodtableAlbert R Meyer,April 5, 2010lec 9M.1Book StackingAlbert R Meyer,Copyright Albert R. Meyer, 2007. All rights reserved.April 5, 2010lec 9M.10Book St
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerApril 5revised April 2, 2010, 768 minutesIn-Class Problems Week 9, Mon.Problem 1.An explorer is trying to reach the Holy Grail, whi