5. In this question, we will prove that all non-empty subsets of Z>o has a smallest element. Start your proof as follows, and finish the proof. Proposition. If S is a non-empty subsets of Z>o, then S has a smallest element. Proof. We will prove by contrapositive. Suppose S is a subset of Zo which has no smallest element. We will prove that S is empty. To do so, we will show by strong induction on n that for all n ≥ 1, we have n & S. (Finish the proof. Hint: you can use the fact that 1 is the smallest number in Z>0. ) 6. Provide a counterexample to the previous question if we replace Z>o by the closed interval [0, 1].
5. In this question, we will prove that all non-empty subsets of Z>o has a smallest element. Start your proof as follows, and finish the proof. Proposition. If S is a non-empty subsets of Z>o, then S has a smallest element. Proof. We will prove by contrapositive. Suppose S is a subset of Zo which has no smallest element. We will prove that S is empty. To do so, we will show by strong induction on n that for all n ≥ 1, we have n & S. (Finish the proof. Hint: you can use the fact that 1 is the smallest number in Z>0. ) 6. Provide a counterexample to the previous question if we replace Z>o by the closed interval [0, 1].
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Please answer question 6
![5. In this question, we will prove that all non-empty subsets of \( \mathbb{Z}_{>0} \) has a smallest element. Start your proof as follows, and finish the proof.
**Proposition.** If \( S \) is a non-empty subset of \( \mathbb{Z}_{>0} \), then \( S \) has a smallest element.
**Proof.** We will prove by contrapositive. Suppose \( S \) is a subset of \( \mathbb{Z}_{>0} \) which has no smallest element. We will prove that \( S \) is empty. To do so, we will show by strong induction on \( n \) that for all \( n \geq 1 \), we have \( n \notin S \).
(Finish the proof. Hint: you can use the fact that 1 is the smallest number in \( \mathbb{Z}_{>0} \).) □
6. Provide a counterexample to the previous question if we replace \( \mathbb{Z}_{>0} \) by the closed interval \([0, 1]\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F382d69cf-ffe2-43c0-99c4-21bedf550518%2F0df0dbfa-192e-4616-be2a-d34909a36a51%2Fv59qfca_processed.png&w=3840&q=75)
Transcribed Image Text:5. In this question, we will prove that all non-empty subsets of \( \mathbb{Z}_{>0} \) has a smallest element. Start your proof as follows, and finish the proof.
**Proposition.** If \( S \) is a non-empty subset of \( \mathbb{Z}_{>0} \), then \( S \) has a smallest element.
**Proof.** We will prove by contrapositive. Suppose \( S \) is a subset of \( \mathbb{Z}_{>0} \) which has no smallest element. We will prove that \( S \) is empty. To do so, we will show by strong induction on \( n \) that for all \( n \geq 1 \), we have \( n \notin S \).
(Finish the proof. Hint: you can use the fact that 1 is the smallest number in \( \mathbb{Z}_{>0} \).) □
6. Provide a counterexample to the previous question if we replace \( \mathbb{Z}_{>0} \) by the closed interval \([0, 1]\).
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