MAT 3749 Introduction to Analysis Section 0.1 Part I Methods of Proof I

Slides:

Advertisements

Similar presentations

Discrete Math Methods of proof 1.

Advertisements

Introduction to Proofs

Chapter 3 Elementary Number Theory and Methods of Proof.

Proofs, Recursion and Analysis of Algorithms Mathematical Structures for Computer Science Chapter 2 Copyright © 2006 W.H. Freeman & Co.MSCS SlidesProofs,

Write the negation of “ABCD is not a convex polygon.”

Axiomatic systems and Incidence Geometry

(CSC 102) Lecture 12 Discrete Structures. Previous Lecture Summary Floor and Ceiling Functions Definition of Proof Methods of Proof Direct Proof Disproving.

– Alfred North Whitehead,

Direct Proof and Counterexample II Lecture 12 Section 3.2 Thu, Feb 9, 2006.

1 Discrete Structures CS Johnnie Baker Comments on Early Term Test.

So far we have learned about:

Introduction to Proofs ch. 1.6, pg. 87,93 Muhammad Arief download dari

Methods of Proof & Proof Strategies

The Logic of Geometry. Why is Logic Needed in Geometry? Because making assumptions can be a dangerous thing.

Introduction to Proofs

1 Methods of Proof CS/APMA 202 Epp, chapter 3 Aaron Bloomfield.

Copyright © Cengage Learning. All rights reserved. CHAPTER 4 ELEMENTARY NUMBER THEORY AND METHODS OF PROOF ELEMENTARY NUMBER THEORY AND METHODS OF PROOF.

Methods of Proofs PREDICATE LOGIC The “Quantifiers” and are known as predicate quantifiers. " means for all and means there exists. Example 1: If we.

CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett.

1 Sections 1.5 & 3.1 Methods of Proof / Proof Strategy.

Geometry CH 4-1 Using Logical Reasoning Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz.

1.1 Introduction to Inductive and Deductive Reasoning

Inductive/Dedu ctive Reasoning Using reasoning in math and science.

Section 2.2 Conditional Statements 1 Goals Recognize and analyze a conditional statement Write postulates about points, lines, and planes using conditional.

Copyright © by Holt, Rinehart and Winston. All Rights Reserved. Warm-up 1.Proof is a logical argument. 2.A statement is either true or false. 3.If Tom.

Logical Reasoning:Proof Prove the theorem using the basic axioms of algebra.

Section 2-1 Conditional Statements. Conditional statements Have two parts: 1. Hypothesis (p) 2. Conclusion (q)

Section 2-2: Conditional Statements. Conditional A statement that can be written in If-then form symbol: If p —>, then q.

BIG IDEA: REASONING AND PROOF ESSENTIAL UNDERSTANDINGS: Some mathematical relationships can be described using a variety of if-then statements. Each conditional.

Chapter 2 Section 2-1: Conditional Statements

15) 17) 18) ) 23) 25) Conditional Statement: A conditional statement has two parts, a hypothesis and a conclusion. If-Then Form: the “if” part.

1 Discrete Structures – CNS2300 Text Discrete Mathematics and Its Applications Kenneth H. Rosen (5 th Edition) Chapter 3 The Foundations: Logic and Proof,

Methods of Proof Dr. Yasir Ali. Proof A (logical) proof of a statement is a finite sequence of statements (called the steps of the proof) leading from.

Write paragraph proofs

MAT 3749 Introduction to Analysis Section 2.3 Part I Derivatives

GEOMETRIC PROOFS A Keystone Geometry Mini-Unit. Geometric Proofs – An Intro Why do we have to learn “Proofs”? A proof is an argument, a justification,

MAT 2720 Discrete Mathematics Section 2.2 More Methods of Proof Part II

Method of proofs.  Consider the statements: “Humans have two eyes”  It implies the “universal quantification”  If a is a Human then a has two eyes.

Direct Proof and Counterexample I Lecture 12 Section 3.1 Tue, Feb 6, 2007.

2.3 Methods of Proof.

CS104:Discrete Structures Chapter 2: Proof Techniques.

MAT 2720 Discrete Mathematics Section 2.1 Mathematical Systems, Direct proofs, and Counterexamples

Direct Proof and Counterexample I Lecture 11 Section 3.1 Fri, Jan 28, 2005.

Bellwork Write if-then form, converse, inverse, and contrapositive of given statement. 3x - 8 = 22 because x = 10.

EXAMPLE 3 Write an indirect proof Write an indirect proof that an odd number is not divisible by 4. GIVEN : x is an odd number. PROVE : x is not divisible.

Chapter 2, Section 1 Conditional Statements. Conditional Statement Also know as an “If-then” statement. If it’s Monday, then I will go to school. Hypothesis:

Conditional Statements A conditional statement has two parts, the hypothesis and the conclusion. Written in if-then form: If it is Saturday, then it is.

Section 2.1 Conditional Statements Standards #1&3 Wednesday, July 06, 2016Wednesday, July 06, 2016Wednesday, July 06, 2016Wednesday, July 06, 2016.

Indirect Argument: Contradiction and Contraposition

Introduction to Deductive Proofs

Section 2.1 Conditional Statements

Chapter 4 (Part 1): Induction & Recursion

Direct Proof by Contraposition Direct Proof by Contradiction

Contrapositive, Inverse, and Converse

Methods of Proof CS 202 Epp, chapter 3.

Objectives Identify, write, and analyze the truth value of conditional statements. Write the inverse, converse, and contrapositive of a conditional statement.

Conditional Statements

Methods of Proof A mathematical theorem is usually of the form pq

2.2 Analyze Conditional Statements

Conditional Statements

Conditional Statements

Indirect Proof by Contradiction Direct Proof by Cases

Section 2.1 Proof Techniques Introduce proof techniques: o Exhaustive Proof: to prove all possible cases, Only if it is about a.

MAT 3100 Introduction to Proof

MAT 3100 Introduction to Proof

Copyright © Cengage Learning. All rights reserved.

1.1 Introduction to Inductive and Deductive Reasoning

MAT 3100 Introduction to Proof

Agenda Proofs (Konsep Pembuktian) Direct Proofs & Counterexamples

Presentation transcript:

MAT 3749 Introduction to Analysis Section 0.1 Part I Methods of Proof I

I cannot see some of the symbols in HW! Download the trial version of MATH TYPE equation edition from design science type/trial.asp type/trial.asp

Announcement Major applications and OMH 202

Preview of Reviews Set up common notations. Direct Proof Counterexamples Indirect Proofs - Contrapositive

Background: Common Symbols and Set Notations Integers This is an example of a Set : a collection of distinct unordered objects. Members of a set are called elements. 2,3 are elements of, we use the notations

Background: Common Symbols Implication Example

Goals We will look at how to prove or disprove Theorems of the following type: Direct Proofs Indirect proofs

Theorems Example:

Theorems Example: Underlying assumption:

Example 1 AnalysisProof

Note Keep your analysis for your HW Do not type or submit the analysis

Direct Proof Direct Proof of If-then Theorem Restate the hypothesis of the result. Restate the conclusion of the result. Unravel the definitions, working forward from the beginning of the proof and backward from the end of the proof. Figure out what you know and what you need. Try to forge a link between the two halves of your argument.

Example 2 Analysisproof

Counterexamples To disprove we simply need to find one number x in the domain of discourse that makes false. Such a value of x is called a counterexample

Example 3 AnalysisThe statement is false

Example 4

Indirect Proof: Contrapositive To prove we can prove the equivalent statement in contrapositive form : or

Rationale Why?

Background: Negation Statement: n is odd Negation of the statement: n is not odd Or: n is even

Background: Negation Notations Note:

Contrapositive The contrapositive form of is

Example 4 AnalysisProof: We prove the contrapositive:

Contrapositive AnalysisProof by Contrapositive of If-then Theorem Restate the statement in its equivalent contrapositive form. Use direct proof on the contrapositive form. State the origin statement as the conclusion.

Classwork Very fun to do. Keep your voices down…you do not want to spoil the fun for the other groups.