{ "key_pair_value_system": true, "answer_rating_count": "", "question_feedback_html": { "html_star": "", "html_star_feedback": "" }, "answer_average_rating_value": "", "answer_date_js": "2024-09-13T00:04:40-04:00", "answer_date": "2024-09-13 00:04:40", "is_docs_available": "", "is_excel_available": "", "is_pdf_available": "", "count_file_available": 0, "main_page": "student_question_view", "question_id": "10521262", "url": "\/study-help\/questions\/if-you-know-how-to-prove-things-by-induction-then-10521262", "question_creation_date_js": "2024-09-13T00:04:40-04:00", "question_creation_date": "Sep 13, 2024 12:04 AM", "meta_title": "[Solved] If you know how to prove things by induct | SolutionInn", "meta_description": "Answer of - If you know how to prove things by induction, then here is an amazing fact: Theorem. All horses are the same color. Pr | SolutionInn", "meta_keywords": "prove,things,induction,amazing,fact,theorem,horses,color,proof,ll,induct,number", "question_title_h1": "If you know how to prove things by induction, then here is an amazing fact: Theorem. All horses are the same color. Proof. We'll induct", "question_title": "If you know how to prove things by induction, then here is", "question_title_for_js_snippet": "If you know how to prove things by induction, then here is an amazing fact Theorem All horses are the same color Proof We'll induct on the number of horses Base case 1 horse Clearly with just 1 horse, all horses have the same color Inductive hypothesis Assume the claim is true for arbitrary group of N 1 horses Now, for the inductive step we'll show that if it is true for any group of N horses, that all have the same color, then it is true for any group of N 1 horses N 1 horses, you can conclude that the last N horses also have the same color Therefore all N 1 horses have the same color QED What's wrong with this proof It fails on the induction step when N 1 1 Nothing, all horses are the same color It fails on the induction step when N 1 2 The inductive hypothesis is incorrect Consider the claim for all xR,nZnx nx Would you prove it is false by counter example prove it is true by contradiction prove it is false by contradiction prove it is true directly QUESTION 14 Consider the claim There exists some integer B that is the largest possible integer (HINT If B is an integer, then B 1 is also an integer ) Would you prove it is false by counter example prove it is true by contradiction prove it is false by contradiction prove it is true directly", "question_description": "

\"image\"image<\/p> If you know how to prove things by induction, then here is an amazing fact: Theorem. All horses are the same color. Proof. We'll induct on the number of horses. Base case: 1 horse. Clearly with just 1 horse, all horses have the same color. Inductive hypothesis: Assume the claim is true for arbitrary group of N>=1 horses Now, for the inductive step: we'll show that if it is true for any group of N horses, that all have the same color, then it is true for any group of N+1 horses. N+1 horses, you can conclude that the last N horses also have the same color. Therefore all N+1 horses have the same color. QED. What's wrong with this proof? It fails on the induction step when N+1=1 Nothing, all horses are the same color It fails on the induction step when N+1=2 The inductive hypothesis is incorrect Consider the claim: for all xR,nZnx=nx Would you prove it is false by counter-example prove it is true by contradiction prove it is false by contradiction prove it is true directly QUESTION 14 Consider the claim: There exists some integer B that is the largest possible integer. (HINT: If B is an integer, then B+1 is also an integer.) Would you prove it is false by counter-example prove it is true by contradiction prove it is false by contradiction prove it is true directly", "transcribed_text": "", "related_book": { "title": null, "isbn": null, "edition": null, "authors": null, "cover_image": null, "uri": null, "see_more_uri": "" }, "free_related_book": { "isbn": "", "uri": "", "name": "", "edition": "" }, "question_posted": "2024-09-13 00:04:40", "see_more_questions_link": "\/study-help\/questions\/computer-science-programming-2022-November-11", "step_by_step_answer": "The Answer is in the image, click to view ...", "students_also_viewed": [ { "url": "\/study-help\/business-communication\/describe-the-five-layers-in-the-internet-network-model-and-2002157", "description": "Describe the five layers in the Internet network model and what they do.", "stars": 0 }, { "url": "\/a-plot-the-above-data-on-a-graph-do-you", "description": "(a) Plot the above data on a graph. 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