bijective

简明释义

英[baɪˈdʒektɪv]美[bɪˈdʒektɪv]

adj. [数] 双射的

英英释义

单词用法

同义词

反义词

例句

1.An algebraic and a bijective proof is presented.

第一章介绍几个重要的矩阵数值域基本特性。

2.An algebraic and a bijective proof is presented.

第一章介绍几个重要的矩阵数值域基本特性。

3.A function is considered bijective if it is both injective and surjective.

如果一个函数既是单射又是满射，那么这个函数被认为是双射。

4.To solve the equation, we need to find a bijective function that maps inputs to unique outputs.

要解这个方程，我们需要找到一个将输入映射到唯一输出的双射函数。

5.A bijective relationship between two variables indicates that each value of one variable corresponds to exactly one value of the other.

两个变量之间的双射关系表明其中一个变量的每个值恰好对应另一个变量的一个值。

6.The concept of bijective functions is essential in understanding invertible transformations.

理解可逆变换时，双射函数的概念至关重要。

7.In mathematics, a bijective mapping allows for a perfect pairing between two sets.

在数学中，双射映射允许两个集合之间进行完美配对。

作文

In the field of mathematics, particularly in the study of functions, the term bijective (双射) plays a crucial role in understanding how different sets relate to each other. A function is said to be bijective if it is both injective (一一映射) and surjective (满射). This means that every element in the domain maps to a unique element in the codomain, and every element in the codomain is mapped by some element in the domain. Understanding bijective functions is essential for various applications, including computer science, cryptography, and even in the analysis of algorithms.To illustrate the concept of bijective, let’s consider a simple example involving numbers. Suppose we have a set of natural numbers {1, 2, 3} and a set of letters {A, B, C}. We can create a function that pairs each number with a letter: 1-A, 2-B, and 3-C. This function is bijective because each number corresponds to exactly one letter, and vice versa. No letter is left unpaired, which satisfies the definition of being both injective and surjective.The importance of bijective functions extends beyond theoretical mathematics. In computer science, for instance, many data structures rely on bijective relationships to ensure efficient retrieval and storage of information. Hash functions, which are used to map data of arbitrary size to fixed-size values, often need to maintain a bijective nature to avoid collisions and ensure data integrity. If a hash function is bijective, it guarantees that each input has a unique output, making it easier to retrieve the original data from the hashed value.Moreover, in cryptography, the concept of bijective functions is vital for encryption and decryption processes. A secure encryption algorithm must be bijective, as it needs to allow for the original message to be recovered without loss of information. For example, the encryption function might convert plaintext into ciphertext in a bijective manner, ensuring that each unique plaintext maps to a unique ciphertext. When the recipient receives the ciphertext, they can use the corresponding decryption function to retrieve the original message, thanks to the bijective relationship established during encryption.Additionally, the study of bijective functions leads to deeper insights in various branches of mathematics, such as topology and algebra. In topology, for instance, understanding bijective functions helps in classifying spaces and understanding their properties. In algebra, bijective homomorphisms are critical for establishing isomorphisms between algebraic structures, allowing mathematicians to determine when two structures can be considered fundamentally the same.In conclusion, the concept of bijective (双射) functions is a fundamental building block in mathematics and its applications across various fields. By ensuring unique correspondence between elements of two sets, bijective functions facilitate efficient data handling, secure communication, and deeper mathematical understanding. As we continue to explore the vast landscape of mathematics and its applications, the significance of bijective relationships will undoubtedly remain a key area of focus for researchers and practitioners alike.

在数学领域，尤其是在函数研究中，术语bijective（双射）在理解不同集合之间的关系时起着至关重要的作用。如果一个函数既是单射（即一一映射）又是满射（即每个元素都有对应），那么这个函数就被称为bijective。这意味着定义域中的每个元素都映射到值域中的唯一元素，并且值域中的每个元素都由定义域中的某个元素映射。理解bijective函数对于计算机科学、密码学甚至算法分析等各种应用都是必不可少的。为了说明bijective的概念，让我们考虑一个涉及数字的简单示例。假设我们有一个自然数集合{1，2，3}和一个字母集合{A，B，C}。我们可以创建一个将每个数字与一个字母配对的函数：1-A，2-B，3-C。这个函数是bijective的，因为每个数字对应于一个唯一的字母，反之亦然。没有字母被遗漏，这满足了既是单射又是满射的定义。bijective函数的重要性超越了理论数学。例如，在计算机科学中，许多数据结构依赖于bijective关系，以确保信息的高效检索和存储。哈希函数用于将任意大小的数据映射到固定大小的值，通常需要保持bijective特性，以避免冲突并确保数据完整性。如果哈希函数是bijective的，它保证每个输入都有一个唯一的输出，从而更容易从哈希值中检索原始数据。此外，在密码学中，bijective函数的概念对于加密和解密过程至关重要。一个安全的加密算法必须是bijective的，因为它需要允许原始消息在不丢失信息的情况下被恢复。例如，加密函数可能以bijective的方式将明文转换为密文，确保每个唯一的明文映射到一个唯一的密文。当接收者收到密文时，他们可以使用相应的解密函数来检索原始消息，这要归功于加密过程中建立的bijective关系。此外，bijective函数的研究在拓扑学和代数学等数学各个分支中带来了更深刻的见解。例如，在拓扑学中，理解bijective函数有助于对空间进行分类并理解其性质。在代数中，bijective同态对于建立代数结构之间的同构至关重要，使数学家能够确定两个结构何时可以被视为根本相同。总之，bijective（双射）函数的概念是数学及其在各个领域应用的基础构建块。通过确保两个集合之间的唯一对应关系，bijective函数促进了高效的数据处理、安全的通信和更深刻的数学理解。随着我们继续探索数学及其应用的广阔领域，bijective关系的重要性无疑将继续成为研究人员和从业者关注的关键领域。