MIT6_042JF10_notes

MIT6_042JF10_notes - mcs-ftl 2010/9/8 0:40 page i #1...

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Unformatted text preview: mcs-ftl 2010/9/8 0:40 page i #1 Mathematics for Computer Science revised Wednesday 8 th September, 2010, 00:40 Eric Lehman Google Inc. F Thomson Leighton Department of Mathematics and CSAIL, MIT Akamai Technologies Albert R Meyer Massachusets Institute of Technology Copyright 2010, Eric Lehman, F Tom Leighton, Albert R Meyer . mcs-ftl 2010/9/8 0:40 page ii #2 mcs-ftl 2010/9/8 0:40 page iii #3 Contents I Proofs 1 Propositions 5 1.1 Compound Propositions 6 1.2 Propositional Logic in Computer Programs 10 1.3 Predicates and Quantiers 11 1.4 Validity 19 1.5 Satisability 21 2 Patterns of Proof 23 2.1 The Axiomatic Method 23 2.2 Proof by Cases 26 2.3 Proving an Implication 27 2.4 Proving an If and Only If 30 2.5 Proof by Contradiction 32 2.6 Proofs about Sets 33 2.7 Good Proofs in Practice 40 3 Induction 43 3.1 The Well Ordering Principle 43 3.2 Ordinary Induction 46 3.3 Invariants 56 3.4 Strong Induction 64 3.5 Structural Induction 69 4 Number Theory 81 4.1 Divisibility 81 4.2 The Greatest Common Divisor 87 4.3 The Fundamental Theorem of Arithmetic 94 4.4 Alan Turing 96 4.5 Modular Arithmetic 100 4.6 Arithmetic with a Prime Modulus 103 4.7 Arithmetic with an Arbitrary Modulus 108 4.8 The RSA Algorithm 113 iv mcs-ftl 2010/9/8 0:40 page iv #4 Contents II Structures 5 Graph Theory 121 5.1 Denitions 121 5.2 Matching Problems 128 5.3 Coloring 143 5.4 Getting from A to B in a Graph 147 5.5 Connectivity 151 5.6 Around and Around We Go 156 5.7 Trees 162 5.8 Planar Graphs 170 6 Directed Graphs 189 6.1 Denitions 189 6.2 Tournament Graphs 192 6.3 Communication Networks 196 7 Relations and Partial Orders 213 7.1 Binary Relations 213 7.2 Relations and Cardinality 217 7.3 Relations on One Set 220 7.4 Equivalence Relations 222 7.5 Partial Orders 225 7.6 Posets and DAGs 226 7.7 Topological Sort 229 7.8 Parallel Task Scheduling 232 7.9 Dilworths Lemma 235 8 State Machines 237 III Counting 9 Sums and Asymptotics 243 9.1 The Value of an Annuity 244 9.2 Power Sums 250 9.3 Approximating Sums 252 9.4 Hanging Out Over the Edge 257 9.5 Double Trouble 269 9.6 Products 272 mcs-ftl 2010/9/8 0:40 page v #5 v Contents 9.7 Asymptotic Notation 275 10 Recurrences 283 10.1 The Towers of Hanoi 284 10.2 Merge Sort 291 10.3 Linear Recurrences 294 10.4 Divide-and-Conquer Recurrences 302 10.5 A Feel for Recurrences 309 11 Cardinality Rules 313 11.1 Counting One Thing by Counting Another 313 11.2 Counting Sequences 314 11.3 The Generalized Product Rule 317 11.4 The Division Rule 321 11.5 Counting Subsets 324 11.6 Sequences with Repetitions 326 11.7 Counting Practice: Poker Hands 329 11.8 Inclusion-Exclusion 334 11.9 Combinatorial Proofs 339 11.10 The Pigeonhole Principle 342 11.11 A Magic Trick 346 12 Generating Functions 355 12.1 Denitions and Examples 355 12.2 Operations on Generating Functions 356 12.3 Evaluating Sums 361 12.4 Extracting Coefcients 363...
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This note was uploaded on 01/15/2012 for the course ECON 101 taught by Professor N/a during the Fall '11 term at Middlesex CC.

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MIT6_042JF10_notes - mcs-ftl 2010/9/8 0:40 page i #1...

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