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(d) If A is not spottily decreasing and ASZ, then A is naturally increasing. This statement is O True O False

(d) If A is not spottily decreasing and ASZ, then A is naturally increasing. This statement is O True O False

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter2: Systems Of Linear Equations
Section2.4: Applications
Problem 27EQ
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In this problem, we will deal with subsets AS R. Here, N= {0, 1, 2,...}. Let's define two new concepts. (i) We say that A is naturally increasing if VxE A, 3y € A s.t. x‡ y AND y-xEN (ii) We say that A is spottily decreasing if 3x € A s.t. Vy E A,x-y EN Below are six claims. Which ones are true and which ones are false? If a claim is true, prove it. If a claim is false, provide a counterexample and a justification of how the counterexample shows the claim is false. Hint: can you find a set A such that A is naturally increasing? can you find a set B such that B is spottily decreasing? (a) If A is spottily decreasing, then A is naturally increasing. This statement is O True O False (b) If A is naturally increasing, then A is not spottily decreasing. This statement is O True O False (c) If A is not naturally increasing, then A is spottily decreasing. This statement is O True O False (d) If A is not spottily decreasing and A = Z, then A is naturally increasing. This statement is O True O False (e) If we have two sets A, BCR, A and B are both naturally increasing, then AU B is also naturally increasing. This statement is O True O False (f) If we have two sets A, BCR, A and B are both spottily decreasing, then AU B is also spottily decreasing. This statement is O True O False
Transcribed Image Text:In this problem, we will deal with subsets AS R. Here, N= {0, 1, 2,...}. Let's define two new concepts. (i) We say that A is naturally increasing if VxE A, 3y € A s.t. x‡ y AND y-xEN (ii) We say that A is spottily decreasing if 3x € A s.t. Vy E A,x-y EN Below are six claims. Which ones are true and which ones are false? If a claim is true, prove it. If a claim is false, provide a counterexample and a justification of how the counterexample shows the claim is false. Hint: can you find a set A such that A is naturally increasing? can you find a set B such that B is spottily decreasing? (a) If A is spottily decreasing, then A is naturally increasing. This statement is O True O False (b) If A is naturally increasing, then A is not spottily decreasing. This statement is O True O False (c) If A is not naturally increasing, then A is spottily decreasing. This statement is O True O False (d) If A is not spottily decreasing and A = Z, then A is naturally increasing. This statement is O True O False (e) If we have two sets A, BCR, A and B are both naturally increasing, then AU B is also naturally increasing. This statement is O True O False (f) If we have two sets A, BCR, A and B are both spottily decreasing, then AU B is also spottily decreasing. This statement is O True O False
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