'Odd integer' diaporamas de présentation

Proof Methods

Proof Methods . May 15. 1. Logical Implication . ?. #1. #2. Recall: (#1 ? #2) precisely if (#1 ?#2 ) is a tautology . 2. Proof: ( p ? q ) ? ( q ? r ) ? p ? r ?. 3. Formal Proof.

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Chapter 2 Basic Structures: Sets, Functions, Sequences and Sums (and part of Chapter 3)

Chapter 2 Basic Structures: Sets, Functions, Sequences and Sums (and part of Chapter 3). By courtesy of Prof. Cheng-Chia Chen. 2.1 Sets. Basic structure upon which all other (discrete and continuous) structures are built. A set is a collection of objects.

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Course Summary

Course Summary. CS 202 Aaron Bloomfield. Outline. A review of the proof methods gone over in class Interspersed with the most popular demotivators from CS 101 Course summary. Demotivator winners!. Methodology 1 st place vote counted for 3 points 2 nd place vote counted for 2 points

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Negating Nested Quantifiers

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The Role of Logic and Proof in Teaching Discrete Mathematics

The Role of Logic and Proof in Teaching Discrete Mathematics Summer Workshop on Discrete Mathematics Messiah College June 2006 Susanna S. Epp Why Discrete Math?

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Lecture 2: Foundations: Mathematical Logic

Lecture 2: Foundations: Mathematical Logic. Discrete Mathematical Structures: Theory and Applications. Learning Objectives. Learn about sets Explore various operations on sets Become familiar with Venn diagrams Learn how to represent sets in computer memory

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Lecture 2: Foundations: Mathematical Logic

Lecture 2: Foundations: Mathematical Logic. Discrete Mathematical Structures: Theory and Applications. Learning Objectives. Learn about sets Explore various operations on sets Become familiar with Venn diagrams Learn how to represent sets in computer memory

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Elementary Number Theory and Methods of Proof

Elementary Number Theory and Methods of Proof. Basic Definitions. An integer n is an even number if there exists an integer k such that n = 2k. An integer n is an odd number if there exists an integer k such that n = 2k+1. An integer n is a prime number if and only if n>1

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Review for CS1050

Review for CS1050. Review Questions. Without using truth tables, prove that ? (p ? q) ? ? q is a tautology. Prove that the sum of an even integer and an odd integer is always odd.

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Proof Methods: Part 2

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Mathematical Induction

Mathematical Induction. Rosen 3.3. Basics. The Well-Ordering Property - Every nonempty set of nonnegative integers has a least element. Many theorems state that P(n) is true for all positive integers.

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Geometry

Geometry. Vocabulary. Conjecture- an unproven statement that is based on observation. Inductive Reasoning- when you find a pattern in specific cases and then write a conjecture for the general case. Deductive Reasoning- Uses facts, definitions, and /or accepted properties

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Remodeling is a Lengthy Process

Remodeling is a Lengthy Process. Bea Hind is remodeling her square family room by tearing down one wall and extending the room by 3 feet. The room will then be rectangular with an area of 180 square feet. What are the current dimensions of the family room?.

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Introduction to Proofs

Introduction to Proofs. Introduction to Proofs. A proof is a valid argument that establishes the truth of a statement. Previous section discussed formal proofs Informal proofs are common in math, CS, and other disciplines More than one rule of inference are often used in a step.

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Number Theory and Methods of Proof

Number Theory and Methods of Proof. Content: Properties of integer, rational and real numbers. Underlying theme: Methods of mathematical proofs. This Lecture. Even and odd integers Prime and composite integers Constructive and nonconstructive proofs

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Section 2.4

Section 2.4. A Combinatorial Approach to Determinants. PERMUTATIONS. A permutation of the integers {1, 2, 3, . . . , n } is an arrangement of these integers in some order without omissions or repetitions. Example : (2, 5, 1, 3, 4) is a permutation of the integers {1, 2, 3, 4, 5}.

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Discrete Structures

Discrete Structures. Chapter 4: Elementary Number Theory and Methods of Proof 4.1 Direct Proof and Counter Example I: Introduction.

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Chapter 3

Chapter 3. Elementary Number Theory and Methods of Proof. 3.4. Direct Proof and Counterexample 4 Division into Cases and the Quotient-Remainder Theorem. Quotient-Remainder Theorem. Theorem 3.4.1 The Quotient-Remainder Theorem

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Chapter 3

Chapter 3. Elementary Number Theory and Methods of Proof. 3.6. Indirect Argument. Reductio Ad Absurdum. Argument by contradiction Illustration in proof of innocence Suppose I did commit the crime. Then at the time of the crime, I would have had to be at the scene of the crime.

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