(b) The following is a proposed proof of the given statement. 1. Suppose m is any even integer and n is any odd integer. 2. If m · n is even, then by definition of even there exists an integer r such that m n = 2r. 3. Also sincem is even, there exists an integer p such that m = 2p by definition of even. 4. And sincen is odd, there exists an integer q such that n = 2g + 1 by definition of odd. 5. Thus, by substitution, m n = (2p)(2q + 1) = 2r, where r is an integer. 6. Hence, by definition of even, then, m•n is even, as was to be shown. Reorder the sentences in the following scrambled list to explain the mistake in the proposed proof. Step 5 deduces a conclusion that would be true if S were known to be true. Hence, the proposed proof is circular; it assumes what is to be proved. Let S be the sentence, "There is an integer r such that m·n equals 2r." The truth of S is not known at the point where Step 5 occurs because the assumption that m n is even has not been proved. Since the truth of Step 6 depends on the truth of the conclusion in Step 5, Step 6 is not a valid deduction. Step 2 states that the truth of S would follow from the assumption that m n is even. Thus, the conclusion in Step 5 is not a valid deduction. Select appropriate sentences from the list and put them in the correct order to explain the mistake in the proposed proof. 1. Let S be the sentence, "There is an integer r such that m - n equals 2r." 2. Step 2 states that the truth of S would follow from the assumption that m n is even. 3. -Select--- 4. Select.. 5. Step 5 deduces a conclusion that would be true if S were known to be true. 6. Select-- 7. Hence, the proposed proof is circular; it assumes what is to be proved.
(b) The following is a proposed proof of the given statement. 1. Suppose m is any even integer and n is any odd integer. 2. If m · n is even, then by definition of even there exists an integer r such that m n = 2r. 3. Also sincem is even, there exists an integer p such that m = 2p by definition of even. 4. And sincen is odd, there exists an integer q such that n = 2g + 1 by definition of odd. 5. Thus, by substitution, m n = (2p)(2q + 1) = 2r, where r is an integer. 6. Hence, by definition of even, then, m•n is even, as was to be shown. Reorder the sentences in the following scrambled list to explain the mistake in the proposed proof. Step 5 deduces a conclusion that would be true if S were known to be true. Hence, the proposed proof is circular; it assumes what is to be proved. Let S be the sentence, "There is an integer r such that m·n equals 2r." The truth of S is not known at the point where Step 5 occurs because the assumption that m n is even has not been proved. Since the truth of Step 6 depends on the truth of the conclusion in Step 5, Step 6 is not a valid deduction. Step 2 states that the truth of S would follow from the assumption that m n is even. Thus, the conclusion in Step 5 is not a valid deduction. Select appropriate sentences from the list and put them in the correct order to explain the mistake in the proposed proof. 1. Let S be the sentence, "There is an integer r such that m - n equals 2r." 2. Step 2 states that the truth of S would follow from the assumption that m n is even. 3. -Select--- 4. Select.. 5. Step 5 deduces a conclusion that would be true if S were known to be true. 6. Select-- 7. Hence, the proposed proof is circular; it assumes what is to be proved.
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter2: The Integers
Section2.2: Mathematical Induction
Problem 49E: Show that if the statement
is assumed to be true for , then it can be proved to be true for . Is...
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Please help me with 3,4,5 and 6
![(b) The following is a proposed proof of the given statement.
1. Suppose m is any even integer and n is any odd integer.
2. If m · n is even, then by definition of even there exists an integer r such that m·n = 2r.
3. Also sincem is even, there exists an integer p such that m = 2p by definition of even.
4. And sincen is odd, there exists an integer q such that n = 2g + 1 by definition of odd.
5. Thus, by substitution, m n = (2p)(2q + 1) = 2r, where r is an integer.
6. Hence, by definition of even, then, m•n is even, as was to be shown.
Reorder the sentences in the following scrambled list to explain the mistake in the proposed proof.
Step 5 deduces a conclusion that would be true if S were known to be true.
Hence, the proposed proof is circular; it assumes what is to be proved.
Let S be the sentence, "There is an integer r such that m·n equals 2r."
The truth of S is not known at the point where Step 5 occurs because the assumption that m n is even has not been proved.
Since the truth of Step 6 depends on the truth of the conclusion in Step 5, Step 6 is not a valid deduction.
Step 2 states that the truth of S would follow from the assumption that m n is even.
Thus, the conclusion in Step 5 is not a valid deduction.
Select appropriate sentences from the list and put them in the correct order to explain the mistake in the proposed proof.
1. Let S be the sentence, "There is an integer r such that m - n equals 2r."
2. Step 2 states that the truth of S would follow from the assumption thatm n is even.
3.
-Select---
4. Select..
5. Step 5 deduces a conclusion that would be true if S were known to be true.
6.
Select--
7. Hence, the proposed proof is circular; it assumes what is to be proved.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F4b3fec29-7392-4b20-b19f-838ecbe2b098%2F306d608e-f663-4482-91e9-e94412d4a8d5%2F2ddj5q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:(b) The following is a proposed proof of the given statement.
1. Suppose m is any even integer and n is any odd integer.
2. If m · n is even, then by definition of even there exists an integer r such that m·n = 2r.
3. Also sincem is even, there exists an integer p such that m = 2p by definition of even.
4. And sincen is odd, there exists an integer q such that n = 2g + 1 by definition of odd.
5. Thus, by substitution, m n = (2p)(2q + 1) = 2r, where r is an integer.
6. Hence, by definition of even, then, m•n is even, as was to be shown.
Reorder the sentences in the following scrambled list to explain the mistake in the proposed proof.
Step 5 deduces a conclusion that would be true if S were known to be true.
Hence, the proposed proof is circular; it assumes what is to be proved.
Let S be the sentence, "There is an integer r such that m·n equals 2r."
The truth of S is not known at the point where Step 5 occurs because the assumption that m n is even has not been proved.
Since the truth of Step 6 depends on the truth of the conclusion in Step 5, Step 6 is not a valid deduction.
Step 2 states that the truth of S would follow from the assumption that m n is even.
Thus, the conclusion in Step 5 is not a valid deduction.
Select appropriate sentences from the list and put them in the correct order to explain the mistake in the proposed proof.
1. Let S be the sentence, "There is an integer r such that m - n equals 2r."
2. Step 2 states that the truth of S would follow from the assumption thatm n is even.
3.
-Select---
4. Select..
5. Step 5 deduces a conclusion that would be true if S were known to be true.
6.
Select--
7. Hence, the proposed proof is circular; it assumes what is to be proved.
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