DS_Lecture_5.pdf - Lecture 5 Proofs Dr Chengjiang Long Computer Vision Researcher at Kitware Inc Adjunct Professor at SUNY at Albany Email

DS_Lecture_5.pdf - Lecture 5 Proofs Dr Chengjiang Long...

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Lecture 5: Proofs Dr. Chengjiang Long Computer Vision Researcher at Kitware Inc. Adjunct Professor at SUNY at Albany. Email: [email protected]
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C. Long Lecture 5 September 7, 2018 2 ICEN/ICSI210 Discrete Structures Outline Rules of inference Fallacies Proofs with quantifiers Types of proofs Proof strategies
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C. Long Lecture 5 September 7, 2018 3 ICEN/ICSI210 Discrete Structures Outline Rules of inference Fallacies Proofs with quantifiers Types of proofs Proof strategies
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C. Long Lecture 5 September 7, 2018 4 ICEN/ICSI210 Discrete Structures Concepts A theorem is a statement that can be shown to be true (via a proof) A proof is a sequence of statements that form an argument A corollary is a theorem that can be established from theorem that has just been proven A conjecture is a statement whose truth value is unknown The rules of inference are the means used to draw conclusions from other assertions, and to derive an argument or a proof
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C. Long Lecture 5 September 7, 2018 5 ICEN/ICSI210 Discrete Structures Theorems: Example Theorem Let a , b , and c be integers. Then If a | b and a | c then a |( b + c ) If a | b then a | bc for all integers c If a | b and b | c , then a | c Corollary: If a , b , and c are integers such that a | b and a | c , then a | mb + nc whenever m and n are integers What is the assumption? What is the conclusion?
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C. Long Lecture 5 September 7, 2018 6 ICEN/ICSI210 Discrete Structures Rules of Inference: Modus Ponens Intuitively, modus ponens (or law of detachment) can be described as the inference: p implies q; p is true; therefore q holds In logic terminology, modus ponens is the tautology: (p Ù (p ® q)) ® q Note: ‘therefore’ is sometimes denoted \ , so we have: p ® q º p \ q
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C. Long Lecture 5 September 7, 2018 7 ICEN/ICSI210 Discrete Structures Rules of Inference: Addition Addition involves the tautology p ® (p Ú q) Intuitively, if we know that p is true we can conclude that either p or q are true (or both) In other words: p \ (p Ú q) Example: I read the newspaper today, therefore I read the newspaper or I ate custard Note that these are not mutually exclusive
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C. Long Lecture 5 September 7, 2018 8 ICEN/ICSI210 Discrete Structures Rules of Inference: Simplification Simplification is based on the tautology ( p Ù q ) ® p So we have: ( p Ù q ) \ p Example: Prove that if 0 < x < 10, then x ³ 0 1. 0 < x < 10 º (0 < x) Ù (x < 10) 2. (x > 0) Ù (x < 10) ® (x > 0) by simplification 3. (x > 0) ® (x > 0) Ú (x = 0) by addition 4. (x > 0) Ú (x = 0) º (x ³ 0) Q.E.D. QED= Latin word for “quod erat demonstrandum” meaning “that which was to be demonstrated” or “that which was to be shown”.
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C. Long Lecture 5 September 7, 2018 9 ICEN/ICSI210 Discrete Structures Rules of inference: Conjunction The conjunction is almost trivially intuitive. It is based on the following tautology: (( p ) Ù ( q )) ® ( p Ù q ) Note the subtle difference though: On the left-hand side, we independently know p and q to be true Therefore, we conclude, on the right-hand side, that a logical conjunction is true
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C. Long
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