- #1
Edwinkumar
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Why do we define(by convention) that infimum of an empty set as \infty and supremum as -\infty?
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Infimun and supremum of empty set
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Discussion Overview
The discussion revolves around the definitions of the infimum and supremum of an empty set, exploring the reasoning behind defining the infimum as \infty and the supremum as -\infty. The scope includes conceptual clarification and definitions in mathematical analysis.
Discussion Character
- Conceptual clarification, Debate/contested
Main Points Raised
- One participant questions the convention of defining the infimum of an empty set as \infty and the supremum as -\infty.
- Another participant argues that this is not merely a convention, but rather a direct consequence of the definitions of supremum and infimum as least upper bound and greatest lower bound, respectively.
- A further contribution explains that since the empty set has no elements, the definition of upper bounds implies that all real numbers can be considered upper bounds for the empty set, leading to the conclusion about supremum.
- A later reply expresses gratitude for the clarifications received, indicating a better understanding of the topic.
Areas of Agreement / Disagreement
There is disagreement regarding whether the definitions are a convention or a direct result of the definitions of supremum and infimum. The discussion remains unresolved as different perspectives are presented.
Contextual Notes
The discussion highlights the dependence on definitions and the implications of those definitions in the context of the empty set, but does not resolve the underlying assumptions or interpretations.
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