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Stanford | CS 103A
Discrete Mathematics For Compu
Professors
• Plummer, R

#### 50 sample documents related to CS 103A

• 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

• 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

• 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@

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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),

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

• 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

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