Supremum And Infimum Absolute Values

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Introduction

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In the realm of set theory, the concepts of supremum and infimum are crucial in understanding the properties of a given set. The supremum of a set is the least upper bound, while the infimum is the greatest lower bound. When dealing with absolute values, the relationship between these two concepts becomes even more intriguing. In this article, we will explore the relationship between the supremum and infimum of a set and its absolute value, specifically when the supremum of the original set is less than the supremum of its absolute value.

Background

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Let $X \subset \mathbb{R}$ be a nonempty and bounded set containing both positive and negative numbers. We are interested in the relationship between the supremum and infimum of $X$ and its absolute value, denoted as $|X| = \{|x| : x \in X\}$. The absolute value of a set is defined as the set of absolute values of its elements.

Supremum and Infimum of a Set

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The supremum of a set $X$, denoted as $\sup(X)$, is the least upper bound of $X$. It is the smallest number that is greater than or equal to every element in $X$. Similarly, the infimum of $X$, denoted as $\inf(X)$, is the greatest lower bound of $X$. It is the largest number that is less than or equal to every element in $X$.

Supremum and Infimum of Absolute Value

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The absolute value of a set $X$, denoted as $|X|$, is the set of absolute values of its elements. The supremum of $|X|$, denoted as $\sup(|X|)$, is the least upper bound of the absolute values of the elements in $X$. Similarly, the infimum of $|X|$, denoted as $\inf(|X|)$, is the greatest lower bound of the absolute values of the elements in $X$.

Relationship Between Supremum and Infimum

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We are interested in the relationship between the supremum and infimum of $X$ and its absolute value, specifically when $\sup(X) < \sup(|X|)$. To establish this relationship, we need to consider the properties of absolute values and their impact on the supremum and infimum of a set.

Proof

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Let $X \subset \mathbb{R}$ be a nonempty and bounded set containing both positive and negative numbers. Assume that $\sup(X) < \sup(|X|)$. We need to show that $\sup(|X|) = -\inf(X)$.

Step 1: Establish the relationship between $\sup(X)$ and $\inf(X)$

Since $X$ is a nonempty and bounded set, we know that $\inf(X)$ exists. Let $m = \inf(X)$. Then, for every $\epsilon > 0$, there exists an $x \in X$ such that $m \leq x < m + \epsilon$.

Step 2: Show that $\sup(|X|) \leq -m$

Let $y \in |X|$. Then, there exists an $x \in X$ such that $y = |x|$. Since $m \leq x$, we have $-x \leq m$. Therefore, $y = |x| \leq x \leq -(-x) \leq -m$. This shows that $\sup(|X|) \leq -m$.

Step 3: Show that $\sup(|X|) \geq -m$

Let $\epsilon > 0$. Since $\sup(|X|)$ is the least upper bound of $|X|$, there exists a $y \in |X|$ such that $\sup(|X|) - \epsilon < y \leq \sup(|X|)$. Let $x \in X$ be such that $y = |x|$. Then, we have $\sup(|X|) - \epsilon < y \leq \sup(|X|)$. Since $x \in X$, we have $m \leq x$. Therefore, $-x \leq m$. This implies that $-m \leq -x \leq \sup(|X|) - \epsilon$. Since $\epsilon > 0$, we have $-m \leq \sup(|X|) - \epsilon$. Therefore, $-m \leq \sup(|X|)$.

Step 4: Conclude that $\sup(|X|) = -m$

From Steps 2 and 3, we have $\sup(|X|) \leq -m$ and $\sup(|X|) \geq -m$. Therefore, we can conclude that $\sup(|X|) = -m = -\inf(X)$.

Conclusion

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In this article, we have established the relationship between the supremum and infimum of a set and its absolute value, specifically when the supremum of the original set is less than the supremum of its absolute value. We have shown that $\sup(|X|) = -\inf(X)$ when $\sup(X) < \sup(|X|)$. This result highlights the importance of considering the properties of absolute values when dealing with the supremum and infimum of a set.

References

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• [1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• [2] Bartle, R. G. (1976). The Elements of Real Analysis. John Wiley & Sons.
• [3] Royden, H. L. (1988). Real Analysis. Prentice Hall.

Future Work

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This result can be extended to more general cases, such as when the set $X$ is not bounded or when the set $X$ contains only positive or negative numbers. Further research is needed to explore these cases and to establish the relationship between the supremum and infimum of a set and its absolute value in more general settings.

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Introduction

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In our previous article, we explored the relationship between the supremum and infimum of a set and its absolute value, specifically when the supremum of the original set is less than the supremum of its absolute value. We established that $\sup(|X|) = -\inf(X)$ when $\sup(X) < \sup(|X|)$. In this article, we will address some common questions and concerns related to this topic.

Q&A

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Q: What is the significance of the relationship between the supremum and infimum of a set and its absolute value?

A: The relationship between the supremum and infimum of a set and its absolute value is significant because it provides insight into the properties of absolute values and their impact on the supremum and infimum of a set. This relationship can be used to establish bounds on the absolute value of a set and to understand the behavior of absolute values in different mathematical contexts.

Q: Can the relationship between the supremum and infimum of a set and its absolute value be extended to more general cases?

A: Yes, the relationship between the supremum and infimum of a set and its absolute value can be extended to more general cases. For example, it can be extended to sets that are not bounded or to sets that contain only positive or negative numbers. However, further research is needed to establish the relationship in these more general cases.

Q: How does the relationship between the supremum and infimum of a set and its absolute value relate to other mathematical concepts?

A: The relationship between the supremum and infimum of a set and its absolute value is related to other mathematical concepts such as the concept of distance and the concept of metric spaces. It can also be used to establish bounds on the absolute value of a set and to understand the behavior of absolute values in different mathematical contexts.

Q: What are some common applications of the relationship between the supremum and infimum of a set and its absolute value?

A: The relationship between the supremum and infimum of a set and its absolute value has several common applications in mathematics and computer science. For example, it can be used to establish bounds on the absolute value of a set and to understand the behavior of absolute values in different mathematical contexts. It can also be used to develop algorithms for solving optimization problems and to establish bounds on the performance of these algorithms.

Q: Can the relationship between the supremum and infimum of a set and its absolute value be used to establish bounds on the absolute value of a set?

A: Yes, the relationship between the supremum and infimum of a set and its absolute value can be used to establish bounds on the absolute value of a set. For example, if $\sup(X) < \sup(|X|)$, then we can establish that $\sup(|X|) = -\inf(X)$.

Q: How does the relationship between the supremum and infimum of a set and its absolute value relate to the concept of distance?

A: The relationship between the supremum and infimum of a set and its absolute value is related to the concept of distance. For example, the distance between two points in a metric space can be defined in terms of the absolute value of difference between the two points. The relationship between the supremum and infimum of a set and its absolute value can be used to establish bounds on the distance between two points in a metric space.

Conclusion

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In this article, we have addressed some common questions and concerns related to the relationship between the supremum and infimum of a set and its absolute value. We have established that $\sup(|X|) = -\inf(X)$ when $\sup(X) < \sup(|X|)$ and have discussed some common applications of this relationship. We have also discussed how the relationship between the supremum and infimum of a set and its absolute value relates to other mathematical concepts such as the concept of distance and the concept of metric spaces.

References

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• [1] Rudin, W. (1976). Principles of Mathematical Analysis. McGraw-Hill.
• [2] Bartle, R. G. (1976). The Elements of Real Analysis. John Wiley & Sons.
• [3] Royden, H. L. (1988). Real Analysis. Prentice Hall.

Future Work

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This research can be extended to more general cases, such as when the set $X$ is not bounded or when the set $X$ contains only positive or negative numbers. Further research is needed to establish the relationship between the supremum and infimum of a set and its absolute value in more general settings.