countably

简明释义

英[/ˈkaʊntəblɪ/]美[/ˈkaʊntəblɪ/]

adv. 可数地

英英释义

单词用法

同义词

反义词

例句

1.Discusses the computational problem about union of countable infinite of countably infinite sets.

讨论了可数无穷个可数无穷集合的并的计算问题。

2.Discusses the computational problem about union of countable infinite of countably infinite sets.

讨论了可数无穷个可数无穷集合的并的计算问题。

3.We can list all the elements in a countably 可数的 set without missing any.

我们可以列出一个countably 可数的 集合中的所有元素，而不遗漏任何一个。

4.The set of natural numbers is countably 可数的 infinite.

自然数的集合是countably 可数的 无限的。

5.In mathematics, a countably 可数的 set can be put into a one-to-one correspondence with the natural numbers.

在数学中，一个countably 可数的 集合可以与自然数建立一一对应关系。

6.The rational numbers are countably 可数的, unlike the real numbers.

有理数是countably 可数的，与实数不同。

7.The collection of all finite strings over a finite alphabet is countably 可数的.

有限字母表上所有有限字符串的集合是countably 可数的。

作文

In the realm of mathematics, particularly in set theory, the concept of countability plays a significant role in understanding the nature of different sets. A set is said to be countably infinite if its elements can be put into a one-to-one correspondence with the natural numbers. This means that we can list the elements of the set in a sequence, even though the sequence might go on indefinitely. For instance, the set of all integers is countably infinite because we can enumerate them as follows: 0, 1, -1, 2, -2, and so on. Each integer corresponds to a natural number, demonstrating that they can be counted, albeit infinitely.On the other hand, there are sets that are uncountably infinite, which means that they cannot be listed in such a manner. The classic example of an uncountably infinite set is the set of real numbers between 0 and 1. Cantor's diagonal argument shows that no matter how we attempt to list these real numbers, there will always be some real numbers left out. This distinction between countably infinite sets and uncountably infinite sets is crucial in various areas of mathematics, including analysis and topology.Understanding whether a set is countably infinite or uncountably infinite has profound implications. For instance, in computer science, when dealing with data structures, we often need to determine the size of a dataset. If a dataset is countably infinite, such as the set of all possible strings of finite length, we can devise algorithms that operate on this dataset effectively. Conversely, if we are faced with an uncountably infinite set, such as the set of all possible real numbers, we must approach the problem differently, as traditional counting methods do not apply.Moreover, the concept of countably infinite sets extends beyond pure mathematics into practical applications. In probability theory, for instance, when we deal with events that can occur in a countable manner, we can apply various probabilistic models. For example, if we consider the outcomes of rolling a fair die, the set of possible outcomes is countably finite, allowing us to calculate probabilities straightforwardly.In conclusion, the notion of countably infinite sets is a fundamental aspect of mathematics that helps us categorize and understand different types of infinities. By recognizing the properties of countably and uncountably infinite sets, we can better navigate through complex mathematical theories and their applications in real-world scenarios. This understanding enriches our comprehension of the universe, where infinite possibilities exist, yet some can be counted while others remain elusive. Thus, the term countably serves not only as a mathematical descriptor but also as a gateway to deeper philosophical inquiries about the nature of infinity itself.

在数学领域，特别是在集合论中，可数性的概念在理解不同集合的性质方面起着重要作用。如果一个集合的元素可以与自然数一一对应，则该集合被称为可数无限。这意味着我们可以将集合的元素按顺序列出，即使这个序列可能会无限延续。例如，所有整数的集合是可数无限的，因为我们可以按如下方式枚举它们：0、1、-1、2、-2，依此类推。每个整数都对应一个自然数，表明它们可以被计数，尽管是无限的。另一方面，有些集合是不可数无限的，这意味着它们不能以这样的方式列出。不可数无限集合的经典例子是0到1之间的实数集合。康托尔的对角线论证表明，无论我们如何尝试列出这些实数，总会有一些实数被遗漏。这种可数无限集合和不可数无限集合之间的区别在数学的多个领域中至关重要，包括分析和拓扑。理解一个集合是可数无限还是不可数无限具有深远的意义。例如，在计算机科学中，当我们处理数据结构时，通常需要确定数据集的大小。如果数据集是可数无限的，例如所有有限长度字符串的集合，我们可以有效地设计操作该数据集的算法。相反，如果我们面临的是不可数无限集合，例如所有可能的实数集合，我们必须以不同的方式处理问题，因为传统的计数方法不适用。此外，可数无限集合的概念超越了纯数学，延伸到实际应用中。在概率论中，例如，当我们处理可以以可数方式发生的事件时，我们可以应用各种概率模型。例如，如果我们考虑掷一个公平骰子的结果，可能的结果集合是可数有限的，使我们能够简单地计算概率。总之，可数无限集合的概念是数学的一个基本方面，它帮助我们对不同类型的无限进行分类和理解。通过认识可数和不可数无限集合的性质，我们可以更好地驾驭复杂的数学理论及其在现实世界中的应用。这种理解丰富了我们对宇宙的认识，在那里存在着无限的可能性，而其中一些可以被计数，而其他则仍然难以捉摸。因此，可数这一术语不仅作为一种数学描述，还作为深入探讨关于无限本质的哲学问题的入口。