# Register now to access 7 million high quality study materials (What's Course Hero?) Course Hero is the premier provider of high quality online educational resources. With millions of study documents, online tutors, digital flashcards and free courseware, Course Hero is helping students learn more efficiently and effectively. Whether you're interested in exploring new subjects or mastering key topics for your next exam, Course Hero has the tools you need to achieve your goals.

3 Pages

### MIT6_042JS10_lec03

Course: CS 6.042J, Spring 2011
School: MIT
Rating:

Word Count: 389

#### Document Preview

for 1/30/10 Mathematics Computer Science MIT 6.042J/18.062J The Well Ordering Principle This work is licensed under a Creative Commons Attribution-Noncommercial-Share Alike 3.0 Unported License. Albert R Meyer February. 8, 2010 Lec 2M.1 Well Ordering principle Well Ordering principle Every nonempty set of nonnegative integers has a least element. Familiar? Now you mention it, Yes. Obvious? Yes. Trivial?...

Register Now

#### Unformatted Document Excerpt

Coursehero >> Massachusetts >> MIT >> CS 6.042J

Course Hero has millions of student submitted documents similar to the one
below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.

Course Hero has millions of student submitted documents similar to the one below including study guides, practice problems, reference materials, practice exams, textbook help and tutor support.
Find millions of documents on Course Hero - Study Guides, Lecture Notes, Reference Materials, Practice Exams and more. Course Hero has millions of course specific materials providing students with the best way to expand their education.

Below is a small sample set of documents:

MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 8revised January 26, 2010, 74 minutesIn-Class Problems Week 2, Mon.Problem 1.The proof below uses the Well Ordering Princi
MIT - CS - 6.042J
Massachusetts Institute of Technology6.042J/18.062J, Spring 10: Mathematics for Computer ScienceProf. Albert R. MeyerFebruary 8revised January 26, 2010, 73 minutesSolutions to In-Class Problems Week 2, Mon.Problem 1.The proof below uses the Well Or
MIT - CS - 6.042J
Propositional (Boolean) LogicA proposition is either True or FalseThe Logic ofPropositionsExample:There are 6 regular solids.5TrueFalseNon-examples:Wake up!Where am I?lec 2W.1February 10, 2010Albert R MeyerEnglish to Mathlec 2W.2February
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