extensionality

简明释义

英[ɪkˌstɛnʃəˈnælɪti]美[ɪkˌstɛnʃəˈnælɪti]

n. [数] 外延性

英英释义

单词用法

同义词

反义词

例句

1.It was my first implicit appreciation of the principle of extensionality by which I have set such store down the decades: the universal is no more than the sum of its particulars.

这是我第一次对外延性原则隐隐地表现出欣赏，在此后的几十年间我一直很重视这个原则：共相不过是其殊相的总和。

2.It was my first implicit appreciation of the principle of extensionality by which I have set such store down the decades: the universal is no more than the sum of its particulars.

这是我第一次对外延性原则隐隐地表现出欣赏，在此后的几十年间我一直很重视这个原则：共相不过是其殊相的总和。

3.In set theory, the principle of extensionality states that two sets are equal if they have the same elements.

在集合论中，外延性原则指出，如果两个集合具有相同的元素，则它们是相等的。

4.The concept of extensionality is crucial in understanding how different mathematical structures can be compared.

外延性的概念对于理解不同数学结构如何进行比较至关重要。

5.In programming languages, extensionality helps define equality between functions based on their outputs.

在编程语言中，外延性有助于根据函数的输出定义函数之间的相等性。

6.A common application of extensionality is in database management, where two tables are considered equivalent if they contain the same rows.

外延性的一个常见应用是在数据库管理中，如果两个表包含相同的行，则认为它们是等效的。

7.Philosophers debate the implications of extensionality in the context of identity and reference.

哲学家们在身份和指称的背景下讨论外延性的含义。

作文

In the realm of mathematics and logic, the concept of extensionality plays a crucial role in understanding the nature of sets and functions. Extensionality refers to the principle that two sets are considered equal if they contain the same elements. This principle is foundational in set theory, which is a branch of mathematical logic that deals with collections of objects. To illustrate this, consider two sets: Set A, which contains the elements {1, 2, 3}, and Set B, which also contains the elements {3, 2, 1}. According to the principle of extensionality, Set A and Set B are identical because they consist of the same elements, regardless of the order in which those elements are presented. The implications of extensionality extend beyond mere set equality; it influences how we perceive functions as well. In functional programming and mathematical contexts, two functions are considered equivalent if they produce the same output for the same input. This notion aligns with the extensionality principle, emphasizing that the behavior of functions, rather than their internal composition, determines their identity. Furthermore, extensionality has philosophical ramifications, particularly in discussions about identity and existence. Philosophers often grapple with questions about what it means for two entities to be the same. The principle of extensionality provides a clear criterion: if two entities share all their properties, they can be deemed identical. This perspective can lead to deeper inquiries into the nature of objects and their attributes, prompting debates about whether properties are intrinsic or extrinsic. In practical applications, extensionality is vital in computer science, especially in database theory and programming languages. For instance, when designing databases, understanding that two tables can be seen as equivalent if they hold the same data allows developers to create more efficient data retrieval methods. Similarly, in programming, recognizing that two functions can be interchangeable based on their outputs fosters better code optimization and maintenance. Moreover, the concept of extensionality encourages clarity in communication, particularly in mathematical proofs and logical arguments. When mathematicians assert that two constructs are equivalent, they rely on the principle of extensionality to convey their reasoning clearly and unambiguously. This shared understanding helps to build a common language within the mathematical community, facilitating collaboration and innovation. In conclusion, extensionality serves as a foundational principle in mathematics, logic, philosophy, and computer science. Its implications reach far and wide, influencing how we understand equality, identity, and equivalence across various domains. By grasping the nuances of extensionality, one can appreciate the interconnectedness of different fields and the importance of precise definitions in advancing knowledge. As we continue to explore complex ideas, the principle of extensionality will undoubtedly remain a cornerstone of our intellectual pursuits, guiding us toward greater clarity and understanding in both theoretical and practical applications.

在数学和逻辑领域，外延性的概念在理解集合和函数的性质方面发挥着至关重要的作用。外延性指的是两个集合被认为相等的原则，如果它们包含相同的元素。这一原则是集合论的基础，集合论是处理对象集合的数学逻辑的一个分支。为了说明这一点，考虑两个集合：集合A，包含元素{1, 2, 3}，以及集合B，也包含元素{3, 2, 1}。根据外延性的原则，集合A和集合B是相同的，因为它们由相同的元素组成，无论这些元素的顺序如何。 外延性的影响超越了简单的集合相等；它影响我们对函数的看法。在函数式编程和数学上下文中，如果两个函数对于相同的输入产生相同的输出，则认为它们是等价的。这一概念与外延性原则一致，强调函数的行为，而不是它们的内部组成，决定了它们的身份。 此外，外延性在哲学上也具有重要的意义，尤其是在关于身份和存在的讨论中。哲学家们常常探讨两个实体相同的含义。外延性原则提供了一个明确的标准：如果两个实体共享所有属性，它们可以被视为相同。这种观点可能引发对对象及其属性本质的更深入探讨，从而引发关于属性是内在的还是外在的辩论。 在实际应用中，外延性在计算机科学中至关重要，特别是在数据库理论和编程语言中。例如，在设计数据库时，理解两个表可以被视为相等，如果它们保存相同的数据，这使得开发人员能够创建更高效的数据检索方法。类似地，在编程中，认识到两个函数可以基于它们的输出进行互换，有助于更好的代码优化和维护。 此外，外延性的概念促进了交流的清晰性，特别是在数学证明和逻辑论证中。当数学家声称两个构造是等价的时，他们依赖于外延性原则来清晰明了地传达他们的推理。这种共同的理解有助于在数学界建立共同的语言，促进合作和创新。 总之，外延性作为数学、逻辑、哲学和计算机科学的基础原则，具有深远的影响。它的影响广泛，影响我们对平等、身份和各种领域中的等价性的理解。通过掌握外延性的微妙之处，人们可以欣赏不同领域之间的相互联系，以及在推动知识进步中精确定义的重要性。在我们继续探索复杂思想的过程中，外延性无疑将继续作为我们智力追求的基石，引导我们在理论和实践应用中朝着更大的清晰度和理解迈进。