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torquerotates
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Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.
torquerotates said:Does the Archimedean property work for unbounded sets? My book does a proof of the Archimedean property relying on the existence of sup which relies on the existence of a bound.
The Archimedean property for unbounded sets states that for any two unbounded sets, there exists a natural number n such that the first set contains at least n elements more than the second set.
The Archimedean property for bounded sets states that for any two bounded sets, there exists a natural number n such that the first set contains at least n elements more than the second set. The difference is that for unbounded sets, there is no limit to the number of elements in each set, while for bounded sets, there is a specific upper limit.
The Archimedean property is important because it allows us to compare the sizes of different sets. This is useful in many areas of mathematics, such as measure theory and calculus, where we need to determine the relative sizes of sets.
Yes, the Archimedean property can be extended to infinite sets. This is known as the Archimedean principle, which states that for any two infinite sets, there exists a bijection between them. This means that the two sets have the same size, even though they may contain an infinite number of elements.
Yes, the Archimedean property for unbounded sets is used in the field of economics to compare the sizes of infinite sets, such as the number of goods and the number of consumers in a market. It is also used in probability theory to calculate the likelihood of events in infinite sample spaces.