'Inductive step' diaporamas de présentation

CDT314 FABER Formal Languages, Automata and Models of Computation Lecture 4

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The Bernstein Basis and Bezier Curves

The Bernstein Basis and Bezier Curves. Dr. Scott Schaefer. Problems with Interpolation. Problems with Interpolation. Bezier Curves. Polynomial curves that seek to approximate rather than to interpolate. Bernstein Polynomials. Degree 1: (1- t ), t Degree 2: (1- t ) 2 , 2(1- t ) t , t 2

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Induction and recursion

Induction and recursion. Chapter 5. With Question/Answer Animations. Chapter Summary. Mathematical Induction Strong Induction Well-Ordering Recursive Definitions Structural Induction Recursive Algorithms Program Correctness ( not yet included in overheads ). Mathematical Induction.

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Relations

Relations. Chapter 9. Chapter Summary. Relations and Their Properties n - ary Relations and Their Applications ( not currently included in overheads ) Representing Relations Closures of Relations ( not currently included in overheads ) Equivalence Relations Partial Orderings.

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CPSC 121: Models of Computation

CPSC 121: Models of Computation. Unit 9b: Mathematical Induction - part 2. Outline. Strong Mathematical Induction. Pattern and Examples More examples using induction. Further exercises. Strong Mathematical Induction.

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Introduction to Finite Automata

Introduction to Finite Automata. Languages Deterministic Finite Automata Representations of Automata. Alphabets. An alphabet is any finite set of symbols. Examples : ASCII, Unicode, {0,1} ( binary alphabet ), {a,b,c}. Strings.

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General Approach for Generating Stochastic Bit Streams

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Discrete Mathematics 6 th edition, 2005

Discrete Mathematics 6 th edition, 2005. Chapter 4 Algorithms. Introduction Examples of Algorithms Analysis of Algorithms Recursive Algorithms. 4.1 Introduction. An algorithm has the following characteristics: Input : the algorithm receives input

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The Binomial Theorem

The Binomial Theorem. Lecture 34 Section 6.7 Wed, Mar 28, 2007. The Binomial Theorem. Theorem: Given any numbers a and b and any nonnegative integer n , . The Binomial Theorem. Proof: Use induction on n . Base case: Let n = 0. Then ( a + b ) 0 = 1 and

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Advanced Topics in FOL

Language, Proof and Logic. Advanced Topics in FOL. Chapter 18. 18.1. A first-order structure (sometimes called a model , or interpretation ) is a function M defined on the predicate and function symbols of the language, the names (constants),

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Abstract Syntax

Abstract Syntax. Language of l -expressions. <exp> ::= <identifier> | ( lambda ( <identifier> ) <exp> ) | ( <exp> <exp> ) E.g., concrete syntax Scheme S-expressions ( lambda (x) ( f ( f x ) ) ). Abstract Syntax ( vs Concrete Syntax ). lambda-exp. id. body.

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Mathematical Induction II

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Induction Practice

Induction Practice. CS1050. Prove that whenever n is a positive integer. Proof: Basis Case: Let n = 1, then . Prove that whenever n is a positive integer. Inductive Case: Assume that the expression is true for n, i.e., that Then we must show that:.

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Relations and Their Properties

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Mathematical Induction

Mathematical Induction. An inductive proof has three parts: Basis case Inductive hypothesis Inductive step Related to recursive programming. Theorem: For all n>=1. Proof #1: (by induction on n ) Basis: n = 1 1 = 1. Inductive hypothesis:

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Discrete Structures

Discrete Structures. Chapter 5: Sequences, Mathematical Induction, and Recursion 5.4 Strong Mathematical Induction and the Well-Ordering Principle for the Integers.

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Mathematical Fundamentals

Mathematical Fundamentals. Properties of Logarithms: 1. Log b is strictly increasing If x<y then log b x < log b y Log b is one-to-one if log b x = log b y then x = y Log b 1 = 0 Log b b a = a (Log b b = 1). Properties of Logarithms: 2. Log b (xy) = Log b x + Log b y

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Strong Mathematical Induction

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Principle of Strong Mathematical Induction

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