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2.24. In simpler language, describe the meaning of the following two statements and their negations. Which one implies the other, and why? a) There is a number M such that, for every x in the set S, |x| ≤ M. b) For every x in the set S, there is a number M such that |x| ≤ M.

2.24. In simpler language, describe the meaning of the following two statements and their negations. Which one implies the other, and why? a) There is a number M such that, for every x in the set S, |x| ≤ M. b) For every x in the set S, there is a number M such that |x| ≤ M.

Advanced Engineering Mathematics
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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**2.24** In simpler language, describe the meaning of the following two statements and their negations. Which one implies the other, and why? a) There is a number \( M \) such that, for every \( x \) in the set \( S \), \( |x| \leq M \). b) For every \( x \) in the set \( S \), there is a number \( M \) such that \( |x| \leq M \). **Explanation:** - Statement (a) suggests that there exists a fixed number \( M \) that acts as an upper bound for the absolute values of all elements \( x \) in the set \( S \). - Statement (b) implies that for each element \( x \) in the set \( S \), you can find some number \( M \) such that the absolute value of \( x \) is less than or equal to \( M \). Understanding this will help identify which statement logically follows from the other.
Transcribed Image Text:**2.24** In simpler language, describe the meaning of the following two statements and their negations. Which one implies the other, and why? a) There is a number \( M \) such that, for every \( x \) in the set \( S \), \( |x| \leq M \). b) For every \( x \) in the set \( S \), there is a number \( M \) such that \( |x| \leq M \). **Explanation:** - Statement (a) suggests that there exists a fixed number \( M \) that acts as an upper bound for the absolute values of all elements \( x \) in the set \( S \). - Statement (b) implies that for each element \( x \) in the set \( S \), you can find some number \( M \) such that the absolute value of \( x \) is less than or equal to \( M \). Understanding this will help identify which statement logically follows from the other.
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