bijection

简明释义

英[baɪˈdʒekʃən]美[baɪˈjekʃn]

n. [数] 双射

英英释义

单词用法

同义词

反义词

例句

1.In Seam documentation, you sometimes see the term "bijection".

在Seam文件中，有时你就会看到术语“双向映射”。

2.In Seam documentation, you sometimes see the term "bijection".

在Seam文件中，有时你就会看到术语“双向映射”。

3.Bijection is a cornerstone of stateful component development.

双射是有状态组件开发的基础。

4.A function is a bijection if every element in the domain maps to a unique element in the codomain.

如果一个函数在定义域中的每个元素都映射到值域中的唯一元素，则该函数是一个双射。

5.In set theory, a bijection demonstrates that two sets have the same cardinality.

在集合论中，双射证明了两个集合具有相同的基数。

6.In computer science, we often use bijections when designing algorithms for data compression.

在计算机科学中，我们在设计数据压缩算法时经常使用双射。

7.To prove that a mapping is a bijection, you must show it is both injective and surjective.

要证明一个映射是双射，你必须表明它既是单射又是满射。

8.The concept of bijection is crucial in understanding functions in higher mathematics.

在高等数学中，双射的概念对理解函数至关重要。

作文

In the realm of mathematics, particularly in set theory, the concept of a bijection is of paramount importance. A bijection is defined as a function that establishes a one-to-one correspondence between two sets. This means that every element in the first set is paired with exactly one unique element in the second set, and vice versa. The significance of a bijection lies in its ability to demonstrate that two sets have the same cardinality or size, even if they are infinite. Understanding bijection can be quite enlightening, especially when exploring the foundations of mathematics.To illustrate the concept, consider two finite sets: Set A = {1, 2, 3} and Set B = {a, b, c}. A possible bijection between these sets could be defined as follows: 1 maps to a, 2 maps to b, and 3 maps to c. In this case, every element in Set A corresponds uniquely to an element in Set B, fulfilling the criteria of a bijection. Conversely, if we were to try to create a mapping where two elements from Set A mapped to the same element in Set B, it would no longer be a bijection because it would violate the one-to-one requirement.The concept of bijection extends beyond finite sets and plays a crucial role in understanding infinite sets as well. For example, consider the set of natural numbers, N = {1, 2, 3, ...}, and the set of even natural numbers, E = {2, 4, 6, ...}. At first glance, it may seem that the set of even numbers is smaller than the set of natural numbers. However, we can define a bijection between these two sets by pairing each natural number n with the even number 2n. This mapping shows that there is a one-to-one correspondence between the two sets, indicating that they have the same cardinality despite one being a subset of the other.Understanding bijection is not only essential for mathematicians but also has implications in computer science, particularly in algorithms and data structures. For instance, when designing a hash table, ensuring that the hashing function creates a bijection can help avoid collisions, thereby improving efficiency. Moreover, in combinatorial problems, recognizing bijections can simplify counting arguments and lead to elegant solutions.In conclusion, the concept of bijection serves as a fundamental building block in the study of mathematics. It provides insight into the relationships between different sets and their sizes, whether finite or infinite. By grasping the idea of bijection, one can appreciate the elegance of mathematical structures and their applications in various fields. As we continue to explore the vast landscape of mathematics, the notion of bijection will undoubtedly remain a key tool in our analytical arsenal.

在数学领域，尤其是在集合论中，bijection的概念至关重要。bijection被定义为在两个集合之间建立一一对应的函数。这意味着第一个集合中的每个元素都与第二个集合中的一个唯一元素配对，反之亦然。bijection的重要性在于它能够证明两个集合具有相同的基数或大小，即使它们是无限的。理解bijection可以非常启发人心，特别是在探索数学基础时。为了说明这一概念，可以考虑两个有限集合：集合A = {1, 2, 3}和集合B = {a, b, c}。这两个集合之间的一个可能的bijection可以定义如下：1映射到a，2映射到b，3映射到c。在这种情况下，集合A中的每个元素都唯一地对应于集合B中的一个元素，满足bijection的标准。相反，如果我们试图创建一个映射，其中集合A中的两个元素映射到集合B中的同一个元素，那么它将不再是bijection，因为这违反了一对一的要求。bijection的概念不仅限于有限集合，还在理解无限集合方面发挥着关键作用。例如，考虑自然数集合N = {1, 2, 3, ...}和偶数自然数集合E = {2, 4, 6, ...}。乍一看，似乎偶数集合比自然数集合要小。然而，我们可以通过将每个自然数n与偶数2n配对来定义一个bijection。这个映射表明这两个集合之间存在一一对应的关系，尽管一个是另一个的子集，但它们具有相同的基数。理解bijection不仅对数学家至关重要，而且在计算机科学中也有影响，特别是在算法和数据结构方面。例如，在设计哈希表时，确保哈希函数创建一个bijection可以帮助避免冲突，从而提高效率。此外，在组合问题中，识别bijection可以简化计数论证并导致优雅的解决方案。总之，bijection的概念作为研究数学的基本构件，提供了对不同集合及其大小之间关系的洞察，无论是有限还是无限。通过掌握bijection的思想，人们可以欣赏数学结构的优雅及其在各个领域的应用。随着我们继续探索广阔的数学领域，bijection的概念无疑将继续作为我们分析工具箱中的关键工具。