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# . Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify ISBN: 9780495562023 335

## Solution for problem 15 Chapter 5.3

A Transition to Advanced Mathematics | 7th Edition

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Problem 15

. Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify assignments of grades other than A. (a) Claim. If A is denumerable, then is denumerable.Proof. Assume A is denumerable.Case 1. If then which is denumerable byhypothesis.Case 2. Assume Since A is denumerable, there existsDefine g by setting Thenso Therefore, isdenumerable. (b) Claim. If A and B are denumerable, then is denumerable.Proof. Assume A and B are denumerable, but that is notdenumerable. Then is finite. Since A and B are denumerable, theyare not empty, so we can choose and Then,and Since is finite, the subsets andare finite. Therefore, A and B are finite. This contradicts thestatement that A and B are denumerable. We conclude that isdenumerable.(c) Claim. The set of positive rationals is denumerable.Proof. Consider the positive rationals in the array in Figure 5.3.1.Order this set by listing all the rationals in the first row, then the secondrow, and so forth. Omitting fractions that are not in lowest terms, wehave an ordering of in which every positive rational appears. Therefore,is denumerable.(d) Claim. If A and B are infinite, thenProof. Suppose A and B are infinite sets. Letand Define as shown:Then, since we never run out of elements in either set, f is one-to-oneand onto B, so(e) Claim. is uncountable.Proof. is uncountable and is a subset of R. Every subset ofan uncountable set is uncountable, so is uncountable.5

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##### ISBN: 9780495562023

Since the solution to 15 from 5.3 chapter was answered, more than 219 students have viewed the full step-by-step answer. A Transition to Advanced Mathematics was written by and is associated to the ISBN: 9780495562023. The answer to “. Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify assignments of grades other than A. (a) Claim. If A is denumerable, then is denumerable.Proof. Assume A is denumerable.Case 1. If then which is denumerable byhypothesis.Case 2. Assume Since A is denumerable, there existsDefine g by setting Thenso Therefore, isdenumerable. (b) Claim. If A and B are denumerable, then is denumerable.Proof. Assume A and B are denumerable, but that is notdenumerable. Then is finite. Since A and B are denumerable, theyare not empty, so we can choose and Then,and Since is finite, the subsets andare finite. Therefore, A and B are finite. This contradicts thestatement that A and B are denumerable. We conclude that isdenumerable.(c) Claim. The set of positive rationals is denumerable.Proof. Consider the positive rationals in the array in Figure 5.3.1.Order this set by listing all the rationals in the first row, then the secondrow, and so forth. Omitting fractions that are not in lowest terms, wehave an ordering of in which every positive rational appears. Therefore,is denumerable.(d) Claim. If A and B are infinite, thenProof. Suppose A and B are infinite sets. Letand Define as shown:Then, since we never run out of elements in either set, f is one-to-oneand onto B, so(e) Claim. is uncountable.Proof. is uncountable and is a subset of R. Every subset ofan uncountable set is uncountable, so is uncountable.5” is broken down into a number of easy to follow steps, and 231 words. This textbook survival guide was created for the textbook: A Transition to Advanced Mathematics, edition: 7. This full solution covers the following key subjects: . This expansive textbook survival guide covers 39 chapters, and 619 solutions. The full step-by-step solution to problem: 15 from chapter: 5.3 was answered by , our top Math solution expert on 03/05/18, 08:54PM.

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