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Mathematics for Computer Science - MIT OpenCourseWare

Mcs 2015/5/18 1:43 page i #1 Mathematics for Computer Science revised Monday 18th May, 2015, 01:43 Eric Lehman Google Inc. F Thomson LeightonDepartment of Mathematics and the Computer Science and AI Laboratory, Massachussetts Institute of Technology; Akamai Technologies Albert R MeyerDepartment of Electrical Engineering and Computer Science and the Computer Science and AI Laboratory, Massachussetts Institute of Technology 2015, Eric Lehman, F Tom Leighton,Albert R Meyer. This work is available under the terms of theCreative CommonsAttribution-NonCommercial-ShareAl ike license. mcs 2015/5/18 1:43 page ii #2 mcs 2015/5/18 1:43 page iii #3 Contents I Proofs Introduction 3 References 4 1 What is a Proof? 5 Propositions 5 Predicates 8 The Axiomatic Method 8 Our Axioms 9 Proving an Implication 11 Proving an If and Only If 13 Proof by Cases 15 Proof by Contradiction 16 Good Proofs in Practice 17 References 19 2 The Well Ordering Principle 27 Well Ordering Proofs 27 Template for Well Ordering Proofs 28 Factoring into Primes 30 Well Ordered Sets 31 3 Logical F

“mcs” — 2015/5/18 — 1:43 — page vii — #7. vii Contents. IV Probability Introduction 665 16 Events and Probability Spaces 667 16.1 Let’s Make a Deal 667 16.2 The Four Step Method 668 16.3 Strange Dice 677 16.4 The Birthday Principle 684 16.5 Set Theory and Probability 686 16.6 References 690 17 Conditional Probability 697 17.1 Monty Hall Confusion 697

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