section04 - Section 4 TECHNIQUES OF PROOF II Techniques of...

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Section 4 – TECHNIQUES OF PROOF: II Techniques of Proof Mathematical theorems and proofs do not occur in isolation, but always in the context of some mathematical system. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36
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Section 4 – TECHNIQUES OF PROOF: II Techniques of Proof Mathematical theorems and proofs do not occur in isolation, but always in the context of some mathematical system. For example, if we are talking about prime numbers, the context is all positive integers. When talking about even and odd numbers, the context is all integers. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36
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Section 4 – TECHNIQUES OF PROOF: II Techniques of Proof Mathematical theorems and proofs do not occur in isolation, but always in the context of some mathematical system. For example, if we are talking about prime numbers, the context is all positive integers. When talking about even and odd numbers, the context is all integers. When dealing with quantified statements, it is particulary important to know exactly what system is being considered. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36
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Section 4 – TECHNIQUES OF PROOF: II Techniques of Proof Mathematical theorems and proofs do not occur in isolation, but always in the context of some mathematical system. For example, if we are talking about prime numbers, the context is all positive integers. When talking about even and odd numbers, the context is all integers. When dealing with quantified statements, it is particulary important to know exactly what system is being considered. For example, the statement 8 x ; p x 2 = x is true in the context of the positive numbers but is false when considering all real numbers. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36
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Section 4 – TECHNIQUES OF PROOF: II Techniques of Proof Mathematical theorems and proofs do not occur in isolation, but always in the context of some mathematical system. For example, if we are talking about prime numbers, the context is all positive integers. When talking about even and odd numbers, the context is all integers. When dealing with quantified statements, it is particulary important to know exactly what system is being considered. For example, the statement 8 x ; p x 2 = x is true in the context of the positive numbers but is false when considering all real numbers. Likewise, 9 x s.t. x 2 = 25 and x < 3 is false for positive numbers but true for all reals. MAT 300 Awtrey MATHEMATICS AND STATISTICS 1 / 36
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Section 4 – TECHNIQUES OF PROOF: II Techniques of Proof Proving Universal Statements How do we begin the proof of a universal statement 8 x ; p ( x ) MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 36
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Section 4 – TECHNIQUES OF PROOF: II Techniques of Proof Proving Universal Statements How do we begin the proof of a universal statement 8 x ; p ( x ) The answer is always the same MAT 300 Awtrey MATHEMATICS AND STATISTICS 2 / 36
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Section 4 – TECHNIQUES OF PROOF: II Techniques of Proof Proving Universal Statements How do we begin the proof of a universal statement 8 x ; p ( x ) The answer is always the same Let x be an arbitrary member from the system under consideration.
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