section03

section03 - MAT 300 Spring 2009 Chad Awtrey SCHOOL OF...

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Unformatted text preview: MAT 300 Spring 2009 Chad Awtrey SCHOOL OF MATHEMATICAL AND STATISTICAL SCIENCES Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning In the first two sections we introduced some of the vocabulary of logic and mathematics. MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning In the first two sections we introduced some of the vocabulary of logic and mathematics. Our aim is to be able to read and write mathematics, and this requires more than just vocabulary. It also requires syntax. MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning In the first two sections we introduced some of the vocabulary of logic and mathematics. Our aim is to be able to read and write mathematics, and this requires more than just vocabulary. It also requires syntax. That is, we need to understand how statements are combined to form proofs. MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning In the first two sections we introduced some of the vocabulary of logic and mathematics. Our aim is to be able to read and write mathematics, and this requires more than just vocabulary. It also requires syntax. That is, we need to understand how statements are combined to form proofs. We first consider the two main types of logical reasoning: inductive and deductive reasoning MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning Consider the function f ( n ) = n 2 + n + 17 MAT 300 Awtrey MATHEMATICS AND STATISTICS 3 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning Consider the function f ( n ) = n 2 + n + 17 Lets evaluate this number for various positive integers. MAT 300 Awtrey MATHEMATICS AND STATISTICS 3 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning Consider the function f ( n ) = n 2 + n + 17 Lets evaluate this number for various positive integers. f ( 1 ) = 19 f ( 2 ) = 23 f ( 3 ) = 29 . . . f ( 7 ) = 73 . . . f ( 11 ) = 149 . . . f ( 15 ) = 257 MAT 300 Awtrey MATHEMATICS AND STATISTICS 3 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning All of these values (even the ones we skipped) are prime numbers. MAT 300 Awtrey MATHEMATICS AND STATISTICS 4 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning All of these values (even the ones we skipped) are prime numbers. On the basis of this experience, we might conjecture that the function f ( n ) = n 2 + n + 17 will always produce prime numbers when n is a positive integer. MAT 300 Awtrey MATHEMATICS AND STATISTICS 4 / 34 Section 3 TECHNIQUES OF PROOF: I EXAMPLES Inductive Reasoning All of these values (even the ones we skipped) are prime numbers....
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section03 - MAT 300 Spring 2009 Chad Awtrey SCHOOL OF...

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