By de Weger B.M.M.

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Extra info for Algorithms for Diophantine Equations [PhD Thesis]

Example text

A related and equally tricky issue in teaching proof is how to specify an acceptable range of proof styles for students' work. Students need reassurance that acceptable proofs may be written in many different styles, but they also need encouragement to write coherently. To motivate students to write in sentences and in adequate detail, you might suggest that they imagine writing their proofs for an intelligent classmate who has missed the last few days of the course or in a style that they themselves would have been able to understand when they were first learning the subject matter.

True e. False 4. a. Q(2): 22 < 30 - true because (-2)2 =4and4<30 Q(7): 72 < 30 - false because (-7)2 = 49 and 49 S 72 f. True 22 = 4 and 4 < 30, Q(-2): (-2)2 < 30 = 49 and 49 : 30, Q(-7): (-7)2 < 30 - true because - false because 30 c. truth set = In c Z+In 2 < 30} = {1,2,3,4,5} 5. b. Let x =-1 and y = 0. Then x < y because -1 < 0 but x2 4 y2 because (_1)2 = 1 5 02 = 0. Thus the hypothesis x < y is true and the conclusion x2 < y2 is false, so the statement as a whole is false. d. Here are examples of three kinds of correct answers: (1) Let x = 2 and y = 3.

H. The statement says that there is a positive real number u whose product with any positive real number v is less than v. This is true. For example, let u be any positive real number between 0 and 1. Then u < 1, and if v is any positive real number we may multiply both sides of the inequality by v to obtain uv < v. i. The statement says that no matter what positive real number v might be chosen, it is possible to find a positive real number u so that uv < v. This statement is also true. For any positive real number v, u can be taken to be any real number between 0 and 1.

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