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Unformatted text preview: Outline Introduction to Proofs Direct Proof Proof by Contraposition More Examples Proof by Contradiction ICS 253 Lecture 11 Dr. Adel F. Ahmed ICS  CCSE  KFUPM November 3, 2008 Dr. Adel F. Ahmed ICS 253 Lecture 11 Outline Introduction to Proofs Direct Proof Proof by Contraposition More Examples Proof by Contradiction Introduction to Proofs Direct Proof Definition Examples: Direct Proof Proof by Contraposition Examples: Contraposition Contraposition, another example Vacuous and Trivial Proofs More Examples Example 1: The sum of rational numbers Example 2 Proof by Contradiction Proof by Contradiction for Conditional Statements Example: Proof by Contradiction for Conditional Statements Dr. Adel F. Ahmed ICS 253 Lecture 11 Outline Introduction to Proofs Direct Proof Proof by Contraposition More Examples Proof by Contradiction Terminology I Theorem: A statement that can be shown to be true. Dr. Adel F. Ahmed ICS 253 Lecture 11 Outline Introduction to Proofs Direct Proof Proof by Contraposition More Examples Proof by Contradiction Terminology I Theorem: A statement that can be shown to be true. I We demonstrate that a theorem is true with a proof. Dr. Adel F. Ahmed ICS 253 Lecture 11 Outline Introduction to Proofs Direct Proof Proof by Contraposition More Examples Proof by Contradiction Terminology I Theorem: A statement that can be shown to be true. I We demonstrate that a theorem is true with a proof. I Proof: A valid argument that establishes the truth of a theorem. Dr. Adel F. Ahmed ICS 253 Lecture 11 Outline Introduction to Proofs Direct Proof Proof by Contraposition More Examples Proof by Contradiction Terminology I Theorem: A statement that can be shown to be true. I We demonstrate that a theorem is true with a proof. I Proof: A valid argument that establishes the truth of a theorem. I Lemma: A less important theorem that is helpful in the proof of other results. Dr. Adel F. Ahmed ICS 253 Lecture 11 Outline Introduction to Proofs Direct Proof Proof by Contraposition More Examples Proof by Contradiction Terminology I Theorem: A statement that can be shown to be true. I We demonstrate that a theorem is true with a proof. I Proof: A valid argument that establishes the truth of a theorem. I Lemma: A less important theorem that is helpful in the proof of other results. I Corollary: A theorem that can be established directly from a theorem that has been proved. Dr. Adel F. Ahmed ICS 253 Lecture 11 Outline Introduction to Proofs Direct Proof Proof by Contraposition More Examples Proof by Contradiction Terminology I Theorem: A statement that can be shown to be true. I We demonstrate that a theorem is true with a proof. I Proof: A valid argument that establishes the truth of a theorem....
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 Spring '10
 ARNasser
 Mathematical proof, Dr. Adel F. Ahmed, Dr. Adel F.

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