codomain

简明释义

英[ˈkoʊdəˌmeɪn]美[ˈkoʊdəˌmeɪn]

n. [数] 上域；变程

英英释义

单词用法

同义词

反义词

例句

1.By lattice-theoretic methods we give a necessary and sufficient condition for a continuous mapping with the codomain T_3 to be extended continuously from a dense subspace to the whole space.

本文采用格论方法研究拓扑学中连续映射的扩充问题，给出陪域为T＿3空间时连续映射扩充的充要条件。

2.By lattice-theoretic methods we give a necessary and sufficient condition for a continuous mapping with the codomain T_3 to be extended continuously from a dense subspace to the whole space.

本文采用格论方法研究拓扑学中连续映射的扩充问题，给出陪域为T＿3空间时连续映射扩充的充要条件。

3.In mathematics, the function f: X → Y has a set Y known as its codomain (余域).

在数学中，函数 f: X → Y 的集合 Y 被称为其codomain（余域）。

4.When defining a function, it is important to specify the codomain (余域) to understand the possible outputs.

在定义一个函数时，指定codomain（余域）是很重要的，以理解可能的输出。

5.The codomain (余域) of a linear transformation is often a vector space of the same dimension.

线性变换的codomain（余域）通常是同一维度的向量空间。

6.In programming, the return type of a function can be seen as its codomain (余域).

在编程中，函数的返回类型可以看作是它的codomain（余域）。

7.Understanding the codomain (余域) helps in determining if a function is onto.

理解codomain（余域）有助于确定一个函数是否是满射。

作文

In the realm of mathematics, particularly in the study of functions, the concept of codomain plays a crucial role in understanding how functions operate. A function can be thought of as a machine that takes an input and produces an output. The set of all possible outputs that a function can produce is known as its codomain (余域). This is distinct from the range of the function, which is the actual set of outputs that the function produces for a given set of inputs. Understanding the difference between these two concepts is essential for anyone delving into advanced mathematics. To illustrate this concept, consider a simple function f(x) = x^2. If we define the function's domain as all real numbers, then the codomain can be defined as the set of all non-negative real numbers, since squaring any real number cannot yield a negative result. Here, the codomain is explicitly stated, but the actual outputs (or range) of the function would only include values from 0 to infinity. This distinction highlights the importance of the codomain in mathematical definitions. The codomain also has implications in other areas of mathematics, such as linear algebra and calculus. For instance, when dealing with linear transformations, the codomain helps us understand the space in which our outputs will reside. If we have a transformation that maps vectors from one vector space to another, knowing the codomain allows us to determine if the transformation is onto or one-to-one. An onto function means that every element in the codomain is mapped by at least one element from the domain, while a one-to-one function means that no two different elements in the domain map to the same element in the codomain. Moreover, the codomain is not always explicitly stated in a function's definition, which can lead to confusion, especially among students new to the subject. For example, the function g(x) = sin(x) is often understood to have a codomain of [-1, 1]. However, if we were to define the codomain as all real numbers, we would be misrepresenting the outputs of the function. This subtlety emphasizes the need for clarity in mathematical communication and the importance of defining the codomain appropriately. In conclusion, the concept of codomain is fundamental in mathematics. It provides a framework for understanding the behavior of functions and their outputs. By distinguishing between the codomain and the range, mathematicians can better analyze and interpret the relationships between different sets. As students progress in their mathematical journey, grasping the significance of the codomain will enhance their ability to engage with more complex topics and foster a deeper appreciation for the beauty of mathematics. Thus, the codomain is not just a technical term but a gateway to a more profound understanding of mathematical functions and their applications.

在数学领域，特别是在函数研究中，余域的概念在理解函数如何运作方面起着至关重要的作用。函数可以被视为一种机器，它接受输入并产生输出。一个函数可以产生的所有可能输出的集合称为它的余域（codomain）。这与函数的值域不同，值域是函数对于给定输入集实际产生的输出集合。理解这两个概念之间的区别对于任何深入研究高级数学的人来说都是至关重要的。 为了说明这一概念，考虑一个简单的函数f(x) = x^2。如果我们将函数的定义域定义为所有实数，那么余域可以定义为所有非负实数的集合，因为平方任何实数都不能产生负数。在这里，余域被明确陈述，但该函数的实际输出（或值域）仅包含从0到无穷大的值。这一区别突显了在数学定义中余域的重要性。 余域在数学的其他领域，如线性代数和微积分中也有影响。例如，在处理线性变换时，余域帮助我们理解我们的输出将驻留在哪个空间中。如果我们有一个将向量从一个向量空间映射到另一个向量空间的变换，了解余域可以让我们确定变换是否为满射或单射。满射函数意味着余域中的每个元素至少被来自定义域的一个元素映射，而单射函数则意味着定义域中的两个不同元素不会映射到余域中的同一个元素。 此外，余域并不总是在函数的定义中明确指出，这可能会导致混淆，特别是在新接触该主题的学生中。例如，函数g(x) = sin(x)通常被理解为具有[-1, 1]的余域。然而，如果我们将余域定义为所有实数，我们就会错误地表示函数的输出。这种微妙之处强调了数学交流中清晰性的重要性，以及适当定义余域的必要性。 总之，余域的概念在数学中是基础性的。它为理解函数及其输出的行为提供了框架。通过区分余域和值域，数学家可以更好地分析和解释不同集合之间的关系。随着学生在数学旅程中的进步，掌握余域的重要性将增强他们与更复杂主题的互动能力，并培养对数学之美的更深刻理解。因此，余域不仅仅是一个技术术语，而是通往更深刻理解数学函数及其应用的门户。