In this exercise, you will prove that PMI is logically equivalent to PSMI; in other words, given PMI, show that you can deduce the statement PSMI, and vice versa. 3. Induction (a) Assume that PSMI is true. To prove that PMI is true, let P(n) be a statement about the positive integer n. Assume that (i) P(1) is true and (ii) for all m E Z+, if P(m) is true, then P(m + 1) is true. Your goal is to prove that for all n e Z+, P(n) is true. You should do this using PSMI. This means that you should prove (iii) P(1) is true and (iv) for all m e Z+, if for all integers k with 1 < k < m, P(k) is true, then P(m + 1) is true. You can then conclude from PSMI that for all n e Z+, P(n) is true. (b) Assume that PMI is true. To prove that PSMI is true, let P(n) be a statement about the positive integer n. Assume that (i) P(1) is true and (ii) for all m e Z+, if for all integers k with1 < k < m, P(k) is true, then P(m + 1) is true. Your goal is to prove that for all n e Z+, P(n) is true. You should do this using PMI applied to a slightly different statement. Let Q(n) be the statement (Vk < n)P(k). Use PMI to prove that for all n E Z+, Q(n) is true. This means that you should prove (iii) Q(1) is true and (iv) for all m E Z+, if Q(m) is true, then Q(m +1) is true. You can then conclude from PMI that for all n E Z+, Q(n) is true. Finally, explain why for all n e Z+, P(n) is true.
In this exercise, you will prove that PMI is logically equivalent to PSMI; in other words, given PMI, show that you can deduce the statement PSMI, and vice versa. 3. Induction (a) Assume that PSMI is true. To prove that PMI is true, let P(n) be a statement about the positive integer n. Assume that (i) P(1) is true and (ii) for all m E Z+, if P(m) is true, then P(m + 1) is true. Your goal is to prove that for all n e Z+, P(n) is true. You should do this using PSMI. This means that you should prove (iii) P(1) is true and (iv) for all m e Z+, if for all integers k with 1 < k < m, P(k) is true, then P(m + 1) is true. You can then conclude from PSMI that for all n e Z+, P(n) is true. (b) Assume that PMI is true. To prove that PSMI is true, let P(n) be a statement about the positive integer n. Assume that (i) P(1) is true and (ii) for all m e Z+, if for all integers k with1 < k < m, P(k) is true, then P(m + 1) is true. Your goal is to prove that for all n e Z+, P(n) is true. You should do this using PMI applied to a slightly different statement. Let Q(n) be the statement (Vk < n)P(k). Use PMI to prove that for all n E Z+, Q(n) is true. This means that you should prove (iii) Q(1) is true and (iv) for all m E Z+, if Q(m) is true, then Q(m +1) is true. You can then conclude from PMI that for all n E Z+, Q(n) is true. Finally, explain why for all n e Z+, P(n) is true.
College Algebra (MindTap Course List)
12th Edition
ISBN:9781305652231
Author:R. David Gustafson, Jeff Hughes
Publisher:R. David Gustafson, Jeff Hughes
Chapter8: Sequences, Series, And Probability
Section8.5: Mathematical Induction
Problem 42E
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Transcribed Image Text:5. In this exercise, you will prove that PMI is logically equivalent to PSMI; in
other words, given PMI, show that you can deduce the statement PSMI, and
vice versa.
66
3. Induction
(a) Assume that PSMI is true. To prove that PMI is true, let P(n) be a
statement about the positive integer n. Assume that
(i) P(1) is true and
(ii) for all m E Z+, if P(m) is true, then P(m + 1) is true.
Your goal is to prove that for all n e Z+, P(n) is true. You should do this
using PSMI. This means that you should prove
(iii) P(1) is true and
(iv) for all m e Z+, if for all integers k with 1 < k < m, P(k) is true,
then P(m + 1) is true.
You can then conclude from PSMI that for all n e Z+, P(n) is true.
(b) Assume that PMI is true. To prove that PSMI is true, let P(n) be a
statement about the positive integer n. Assume that
(i) P(1) is true and
(ii) for all m e Z+, if for all integers k with 1 < k < m, P(k) is true,
then P(m + 1) is true.
Your goal is to prove that for all n e Z+, P(n) is true. You should do
this using PMI applied to a slightly different statement. Let Q(n) be the
statement (Vk < n)P(k). Use PMI to prove that for all n e Z+, Q(n) is
true. This means that you should prove
(iii) Q(1) is true and
(iv) for all m E Z+, if Q(m) is true, then Q(m + 1) is true.
You can then conclude from PMI that for all n E Z+, Q(n) is true. Finally,
explain why for all n e Z+, P(n) is true.
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