50 sample documents related to CS 103A

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  01 Course Information

  Stanford CS 103A
  

  Handout #1 January 9, 2008 CS103A Robert Plummer CS103A Course InformationWinter 2008 Course Title: Discrete Mathematics for Computer Science Units: 3 Lectures: MWF 11:00 11:50 AM in Gates B3 Instructor: Dr. Robert Plummer Email: Office: Office hours: Off
   
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  02 Syllabus

  Stanford CS 103A
  

  Handout #2 January 9, 2008 CS103A Robert Plummer CS103A Syllabus Please note the frequency of the problem sets in the first half of the quarter. We have learned through experience that frequent practice is the key to learning formal logic and proof skills
   
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  03 Introduction

  Stanford CS 103A
  

  CS103A HO#3 Introduction 1/9/08 Questions for today CS103A What is Discrete Mathematics? Discrete Mathematics for Computer Science Why study it? Instructor: Bob Plummer plummer@cs.stanford.edu TAs: johnnyw@cs.stanford.edu kehymes@cs.stanford.edu tiyuwang@
   
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  05 Intro to FOL

  Stanford CS 103A
  

  CS103A HO# 5 Intro to FOL 1/11/08 SCPD Students - Submit your assignments electronically using the Submit program - Fax or email written portions to SCPD; they will forward them to us - Be sure you have an exam monitor if you are not coming to campus for
   
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  06 Intro to Proofs

  Stanford CS 103A
  

  CS103A HO #6 Slides-Intro to Proofs What is a proof? 1/14/08 Do we ever use inductive reasoning in mathematics? Science: \"prove\" a hypothesis using inductive reasoning, i.e., the scientific method. Yes, we use it to form hypotheses. We gather bits of spec
   
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  08 Slides--Boolean Connectives

  Stanford CS 103A
  

  CS103A HO# 8 Boolean Connectives 1/16/06 Truth Tables for Boolean Connectives Negation Disjunction Conjunction P P TF FT Truth Tables for Boolean Connectives P is true if and only if P is false. Conjunction P T T F F Q T F T F PQ T T T F P Q is true if an
   
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  09 Slides--Consequence

  Stanford CS 103A
  

  CS103A HO #9 Consequence 1/18/08 Tautologies Tautologies are sentences that cannot be false, due to their structure and the meanings of the truth-functional connectives they contain. Tautological Equivalence We can use test two sentences for tautological
   
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  11 Slides--Fitch Proofs

  Stanford CS 103A
  

  CS103A HO# 11 Fitch Proofs 1/23/08 Conjunction Elimination ( Elim) LPL Websites http:/www-csli.stanford.edu/LPL/ http:/ggww2.stanford.edu/GUS/lpl/ Go to Student Resources for solutions and hints Go to Related Pages for a link to a page that identifies the
   
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  12 Slides--Fitch Examples

  Stanford CS 103A
  

  CS103A HO#12 Slides-Fitch Examples 1/25/08 Example Example Example Example Example 1 CS103A HO#12 Slides-Fitch Examples 1/25/08 Proof of Resolution Principle Proof of Resolution Principle (Proof 6.19) AB BC (Proof 6.19) AB BC A AC B AC AC AC Proof of Reso
   
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  13 Proof Strategies

  Stanford CS 103A
  

  Handout #13 January 25, 2008 CS103A Robert Plummer Proof Strategies Where To Start? As you begin a proof, convince yourself that the conclusion is indeed true by studying the premises and understanding their meaning: Analyze the structure of each premise
   
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  15 Slides--Conditionals

  Stanford CS 103A
  

  CS103A HO#15 Slides-Conditionals 1/28/08 Conditionals P T T F F Q T F T F Conditionals P?Q T F T T P T T F F Q T F T F P?Q T F T T Observation 1: The ? connective is pretty weak. Knowing that P ? Q is true only eliminates one possible combination of P and
   
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  15A Biconditional rules

  Stanford CS 103A
  

  CS103A: Missing Slide from 1/28/08 This slide shows the inference rules for biconditionals. The Elimination rule says that if you have a biconditional and you have one side, you can assert the other side in your proof. The Introduction rule shows that, ju
   
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  16 Inference Rules

  Stanford CS 103A
  

  Handout #16 January 28, 2008 CS103A Robert Plummer Inference Rules = Elim = Intro Elim Intro P(n) n=m P(m) a=a P1 P2 P1 P2 P1 P2 P1 P2 Intro Elim Elim Intro P PQ P1 P 2 P1 S P P P P P2 S S Intro Elim P P P Intro P Q Q P PQ Intro P Q Elim PQ P Q
   
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  17 Slides--Quantification

  Stanford CS 103A
  

  CS103A HO#17 Quantification 1/30/08 Quantification So far we have used CS103A - propositions and connectives Propositional calculus - identity 1/30/08 - predicates Midterm Exam Tues., Feb. 12 7 - 9 pm Location TBA Quantification Quantification We are afte
   
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  18 Intro to Quantification

  Stanford CS 103A
  

  Handout #18 January 30, 2008 CS103A Robert Plummer Introduction to Quantification Predicates 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
   
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  19 Slides--Logic of Quantifiers

  Stanford CS 103A
  

  CS103A Handout #19 Slides-Logic of Quantifiers 2/1/08 Describe a world for each of these x Cube(x) CS103A x Cube(x) x Cube(x) x Cube(x) x Cube(x) x Cube(x) 2/1/08 Midterm Exam Tues., Feb. 12 7 - 9 pm Location: TBA Tautologies with Quantifiers P P x (Profe
   
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  21 Slides--Multiple Quantifiers

  Stanford CS 103A
  

  CS103A 2/4/08 Review Session Midterm Exam Fri., Feb. 8 3:15 4:30 pm Location: Gates B01 Tues., Feb. 12 7 - 9 pm Location: TBA Open book (LPL), Open Notes, Crib Sheet Lewis Carroll Examples H(x) RC(x) LOH(x) L(x) x is a hummingbird x is richly colored ( Du
   
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  22 Slides--Proofs with Quantifiers

  Stanford CS 103A
  

  22 Slides-Poofs with Quantifiers 2/6/08 12.23 Proofs with Quantifiers CS103A 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 2/6/08 Review
   
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  26 Complete Inference Rules

  Stanford CS 103A
  

  Handout #26 Feb. 8, 2008 CS103A Robert Plummer Complete Inference Rules = Elim = Intro ^ Elim ^ Intro P(n) n=m P(m) a=a P1 ^ P2 P1 P2 P1 P2 P1 ^ P2 v Intro v Elim Elim Intro P PvQ P1 v P2 P1 S P P P P P2 S S Intro Elim P P P Intro P Q Q P PQ Intro P
   
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  27 Slides--More Quantifier Proofs

  Stanford CS 103A
  

  CS103A HO# 27 More Quantifier Proofs 2/8/08 CS103A x Cube(x) 2/8/08 x Cube(x) Midterm Exam Review Session Today! Fri., Feb. 8 3:15 4:30 pm Location: Gates B01 Stanford Online, but not broadcast Tues., Feb. 12 7 - 9 pm Location: Gates B01 Open book (LPL),
   
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  30 Introduction to Number Theory

  Stanford CS 103A
  

  Handout #30 Feb. 13, 2008 CS103A Robert Plummer Introduction to Number Theory Basics of Number Theory Number theory was once thought to be pure mathematics math for the sake of math. But with the evolution of computers (and particularly cryptography), num
   
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  31 Proving Real Theorems

  Stanford CS 103A
  

  Handout #31 Feb. 11, 2008 CS103A Robert Plummer Proving \"Real\" Theorems We 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 to formal logic, e
   
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  32 Slides--Introduction to Number Theory

  Stanford CS 103A
  

  CS103A HO# 32 Introduction to Number Theory 2/11/08 Outline of topics for CS103A Basic Tools CS103A Problem Set 7 will be distributed Wednesday, 2/13 and it will be due Friday, 2/22. Formal Logic and Proof Techniques Number Theory and its Applications Pro
   
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  34 Slides--More Number theory

  Stanford CS 103A
  

  CS103A HO#34 More Number Theory 2/13/08 Some 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) CS103A Theorem 30.4 (proved in the handout) 2/13/08 If you want a text: Kenneth Rosen Discrete
   
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  35 Mathematical Proofs

  Stanford CS 103A
  

  Handout #35 Feb. 15, 2008 CS103A Robert Plummer Mathematical Proofs John C. Mitchell & Maggie Johnson Department of Computer Science Stanford University 1. Why write proofs? According to Webster\'s Unabridged Dictionary, the word prove comes from the Latin
   
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  36 Slides--Intro to Cryptography

  Stanford CS 103A
  

  CS103A HO# 36 Intro to Cryptography 2/15/08 Cryptography: Some References Cryptography The Basic Problem David Kahn. The Codebreakers (1967). Simon Singh. The Code Book (1999). Eve Niels Ferguson and Bruce Schneier. Practical Cryptography (2003). Bruce Sc
   
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  37 Proof Techniques

  Stanford CS 103A
  

  Handout #37 Feb. 15, 2008 CS103A Robert Plummer Proof Techniques In this handout we will discuss various proof techniques for mathematical proofs. Quantifiers 1: Construction Proofs We will first consider existential conditionals. They have the form: Ther
   
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  39 Slides--RSA

  Stanford CS 103A
  

  CS103A HO #39 RSA 2/20/08 RSA Cryptography: Motivation CS103A Alice Bob 2/20/08 Midterm Exam, Question 1, Part 3: We will accept answer (d) or (f). If you answered (d) and did not get credit, please bring in your paper! RSA Cryptography: Motivation Alice
   
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  40 Theorems, Examples, RSA

  Stanford CS 103A
  

  Handout #40 February 20, 2008 CS103A Robert Plummer Number Theory: Theorems, Examples, and RSA This handout will list and give examples for some key theorems of Number Theory, and it will provide details about the RSA public-key encryption algorithm. Prel
   
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  41 Sequences

  Stanford CS 103A
  

  Handout #41 February 22, 2008 CS103A Robert Plummer Sequences and Summations A mathematician, like a poet or a painter, is a maker of patterns. G.H. Hardy A Mathematicians Apology (1940) Sequences Imagine a person (with a lot of spare time) who decides to
   
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  42 Induction

  Stanford CS 103A
  

  Handout #42 February 22, 2008 CS103A Robert Plummer Introduction to Induction One 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 about sequenc
   
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  44 Slides--Sequences, Induction

  Stanford CS 103A
  

  CS103A HO# 44 Slides-Sequences, Induction 2/22/08 Sequences and Summations CS103A A sequence is an ordered list, possibly infinite, of elements. Logic and formal proofs We will use the following notation: a1, a2, a3, . . . Proving mathematical theorems (n
   
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  45 Slides--Induction II

  Stanford CS 103A
  

  CS103A HO #45 Induction II Peano\'s Axioms 2/25/08 The Principle of Mathematical Induction A proof by mathematical induction that a proposition P(n) is true for every positive integer n consists of two steps: There is a number 0. Every number has a succe
   
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  47 Slides--Induction III, Fibonacci

  Stanford CS 103A
  

  CS103A HO# 47 Induction III, Fibonacci 2/27/08 CI WOP Well Ordering Property: If A is a non-empty set of positive integers, then A has a least element. CI WOP Well Ordering Property: If A is a non-empty set of positive integers, then A has a least element
   
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  48 Slides--Recursion

  Stanford CS 103A
  

  CS103A HO#48 Recursion 2/29/08 To move n disks: First move n 1 disks to the middle peg Then move 1 disk to the right peg The move n 1 disks to the right peg To move n disks: First move n 1 disks to the middle peg Then move 1 disk to the right peg The move
   
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  50 Slides--Recursion, Combinatorics

  Stanford CS 103A
  

  CS103A HO #50 Recursion, Combinatorics 3/3/08 function R(non-negative integer n) cfw_ if (n < 3) return 2n + 1 else return R(n-1) + power(R(n-2), 2) + power(R(n-3), 3) function I(non-negative integer n) cfw_ if (n < 3) return 2n + 1 x := 1 y := 3 z := 5
   
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  51 Combinatorics

  Stanford CS 103A
  

  Handout #51 March 3, 2008 CS103A Robert Plummer Combinatorics Combinatorics is the study of counting, which is important in Computer Science in many ways: To understand the performance of algorithms, we need to count the steps they execute We also need to
   
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  52 Slides--Combinatorics II

  Stanford CS 103A
  

  CS103A HO #52 Combinatorics II 3/5/08 What is |A B C| ? A B Once Once Once Once Once Once Once A survey of 200 TV viewers found that 110 watch sports, 120 watch comedy, 85 watch drama, 50 watch drama and sports, 70 watch comedy and sports, 55 watch comedy
   
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  54 Slides--Combinatorics III

  Stanford CS 103A
  

  CS103A HO #54 Combinatorics III 3/7/08 Set of size n, selecting r items, 0 r n Permutations and Combinations with Repetition Permutations (ordered) How many strings of length r can we form from the uppercase letters of the English alphabet, if repetition
   
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  56 Godel\'s Incompleteness Theorem

  Stanford CS 103A
  

  CS103A HO #56 Gdel\'s Incompleteness Theorem Gdel\'s Incompleteness Theorem 3/10/08 Gdel, Kurt (1931). ber formal unentscheidbare Stze der Principia Mathematica und verwandter Systeme I. Monatshefte fr Mathematik und Physic, 38, 173-198. On formally undecid
   
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  57 Godel II

  Stanford CS 103A
  

  CS103A HO #57 Gdel II 3/12/08 Gdel Numbering Gdel\'s Incompleteness Theorem ( x ) 8 4 11 9 ( x 8 = 11 s y 5 ) 7 13 9 2 3 5 7 11 13 17 19 23 29 This scheme allows us to represent every formula with a unique number. Given a number, we can determine whether i
   
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  HO34

  Stanford CS 103A
  

  Maggie Johnson CS103A Handout #34 Combinatorics Key Topics: * Sum Rule and Product Rule Review * The Pigeonhole Principle * Permutations and Combinations * Binomial Coefficients and the Binomial Theorem * Permutations and Combinations with Repetiti
   
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  HO31

  Stanford CS 103A
  

  Maggie Johnson CS103A Handout #31 Proving Properties of Loops Key Topics: * More on Induction * Validation vs. Verification * Loop Invariants * Induction to Verify Loops * Example Proofs Induction Challenger While waiting for class to start, check
   
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  HO32

  Stanford CS 103A
  

  Maggie Johnson CS103A Handout #32 Recursion Key topics: * Recursive Definitions * Recursive Subprograms * Proving Properties of Recursive Programs Recursive Definitions One of the most important tasks in computer science is to discover and characte
   
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  HO29

  Stanford CS 103A
  

  Maggie Johnson CS103A Handout #29 Introduction to Induction Key topics: * Introduction and Definitions * Examples of Weak Induction * Proper Proof Form * Examples of Strong Induction * A Faulty Proof and Some Interesting Proofs * Some Useful Formul
   
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  HO28

  Stanford CS 103A
  

  Maggie Johnson CS103A Handout #28 Sequences and Summations Key topics: Sequences Summation & Product Notation Arithmetic and Geometric Progressions Fractals A mathematician, like a poet or a painter, is a maker of patterns. G.H. Hardy A Mathem
   
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  HO24

  Stanford CS 103A
  

  Maggie Johnson CS103A Handout #24 Number Theory Real\" Theorems Key topics: * Why is Proof Important? * What Are We Trying to Prove? * The Art of Proving Things * A Review of Proof Strategies * Direct Proof * Basics of Number Theory * Ind
   
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  HO27

  Stanford CS 103A
  

  Handout #27 CS103A Mathematical Proofs John C. Mitchell & Maggie Johnson Department of Computer Science Stanford University 1. Why write proofs? According to Webster\'s Unabridged Dictionary, the word prove comes from the Latin verb probare which me
   
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  HO25

  Stanford CS 103A
  

  Maggie Johnson CS103A Handout #25 Applications of Number Theory Key topics: Pseudorandom Numbers Cryptology Some Additional Results: Euler\'s Theorem RSA and Public Key Cryptography Given two different positive integers a and b, their arithmeti
   
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  HO28A

  Stanford CS 103A
  

  Maggie Johnson CS103A Handout #36 Functions Key topics: * Introduction and Definitions * Types of Functions * The Growth of Functions As used in ordinary language, the word function indicates dependence of a varying quantity on another. If I tell y