MAT 3749 Introduction to Analysis

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MAT 3749Introduction to Analysis

• Section 0.1 Part II
• Methods of Proof II

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I cannot see some of the symbols!

• Download the trial version of MATH TYPE equation
  edition from design science
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Preview of More Reviews

• Indirect Proofs
• Contradiction
• Proof by Contrapositive is considered as a
  special case of proof by contradiction
• Proof by cases
• Existence proofs

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Proof by Contradiction

• Proof by Contrapositive
• Proof by Contradiction

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Example 1
Analysis Proof
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Proof by Contradiction
Analysis Proof by Contradiction of If-then Theorem Suppose the negation of the conclusion is true. Find a contradiction. State the conclusion.
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Proof by Contradiction

• The method also work with statements other then
  If P then Q

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Example 2
Analysis Proof
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Proof by Cases
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Example 3
Analysis Proof
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Proof by Cases
Analysis Proof by Cases of If-then Theorem Split the domain of interest into cases. Prove each case separately. State the conclusion. Note that the cases do not have to be mutually exclusive. They just have to cover all elements in the domain.
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Existence Proofs
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Example 4
Analysis Proof
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Existence Proofs
Analysis Existence Proof Prove the statement by exhibiting an element in the domain of interest that satisfies the given conditions. State the conclusion.
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Rolling Problems

• 1. The classwork from last time
• 2. (More will be added)

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Take a look at your HW

• You may encounter the first challenging problem.
• Start thinking about it right the way.
• Work with a concrete number for n first (say,
  n3,4)
• Do not use induction
• This problem arise in a past senior project
  related to superconcentrator