power set

简明释义

动力装置发电机组

英英释义

例句

1.In mathematics, the power set (幂集) of a set is the set of all its subsets.

在数学中，power set（幂集）是一个集合的所有子集的集合。

2.In computer science, the power set (幂集) can be used to generate all possible combinations of items.

在计算机科学中，power set（幂集）可以用来生成所有可能的项目组合。

3.To find the power set (幂集) of {a, b}, we list: {}, {a}, {b}, {a, b}.

为了找到{a, b}的power set（幂集），我们列出：{}、{a}、{b}、{a, b}。

4.The concept of a power set (幂集) is essential in set theory and combinatorics.

在集合论和组合数学中，power set（幂集）的概念是至关重要的。

5.When analyzing data, the power set (幂集) helps in understanding different groupings of variables.

在分析数据时，power set（幂集）有助于理解变量的不同分组。

作文

In the realm of mathematics, particularly in set theory, one of the fundamental concepts is the power set. The power set of a given set is defined as the set of all possible subsets of that set, including the empty set and the set itself. Understanding the concept of a power set is crucial for various fields such as computer science, probability, and logic. To illustrate this concept, let’s consider a simple set: A = {1, 2}. The power set of A, denoted as P(A), would include all subsets of A. These subsets are: the empty set {}, the set containing only the first element {1}, the set containing only the second element {2}, and finally, the set A itself {1, 2}. Therefore, the power set P(A) can be expressed as P(A) = {{}, {1}, {2}, {1, 2}}. The number of subsets in a power set is determined by the formula 2^n, where n is the number of elements in the original set. For our example set A, which has 2 elements, the power set contains 2^2 = 4 subsets. This exponential growth becomes more apparent as the size of the original set increases. For instance, if we take a set B = {1, 2, 3}, the power set P(B) will have 2^3 = 8 subsets: {}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, and {1, 2, 3}. The significance of the power set extends beyond mere enumeration of subsets. In computer science, it plays a vital role in algorithms that involve combinations and permutations. For instance, when designing algorithms for data analysis or machine learning, understanding how to generate and manipulate power sets can lead to more efficient solutions. Additionally, in database theory, the power set can help in determining all possible combinations of data entries, which is essential for queries and data retrieval processes. Moreover, the power set concept finds applications in probability theory as well. When calculating probabilities of events, one often needs to consider all possible outcomes, which can be represented by the power set. Each subset of a sample space corresponds to a potential event, and analyzing these subsets allows statisticians to derive meaningful conclusions from their data. In conclusion, the power set is not just a theoretical construct but a practical tool that aids in understanding complex systems in mathematics and its applications. By grasping the definition and implications of the power set, students and professionals alike can enhance their analytical skills and apply these principles across various disciplines. As we continue to explore deeper into the world of mathematics and its applications, the power set remains a cornerstone concept that exemplifies the beauty and utility of set theory. In Chinese, the power set is translated as "幂集". A power set is a vital concept that helps us understand the structure and relationships within sets, making it an essential topic for anyone studying mathematics or related fields.

在数学的领域中，特别是在集合论中，一个基本概念是幂集。给定集合的幂集被定义为该集合所有可能子集的集合，包括空集和集合本身。理解幂集的概念对于计算机科学、概率论和逻辑等多个领域至关重要。为了说明这一概念，让我们考虑一个简单的集合：A = {1, 2}。集合A的幂集，记作P(A)，将包括A的所有子集。这些子集包括空集{}、仅包含第一个元素的集合{1}、仅包含第二个元素的集合{2}，以及集合A本身{1, 2}。因此，幂集P(A)可以表示为P(A) = {{}, {1}, {2}, {1, 2}}。幂集中的子集数量由公式2^n决定，其中n是原始集合中的元素数量。对于我们的例子集合A，它有2个元素，因此幂集包含2^2 = 4个子集。当原始集合的大小增加时，这种指数增长变得更加明显。例如，如果我们取一个集合B = {1, 2, 3}，则幂集P(B)将有2^3 = 8个子集：{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}和{1, 2, 3}。幂集的重要性不仅限于对子集的枚举。在计算机科学中，它在涉及组合和排列的算法中发挥着重要作用。例如，在设计用于数据分析或机器学习的算法时，理解如何生成和操作幂集可以导致更高效的解决方案。此外，在数据库理论中，幂集可以帮助确定数据条目的所有可能组合，这对于查询和数据检索过程至关重要。此外，幂集概念在概率论中也有应用。在计算事件的概率时，人们通常需要考虑所有可能的结果，这可以通过幂集来表示。样本空间的每个子集对应一个潜在事件，分析这些子集使统计学家能够从他们的数据中得出有意义的结论。总之，幂集不仅仅是一个理论构造，而是一个实用工具，有助于理解数学及其应用中的复杂系统。通过掌握幂集的定义和含义，学生和专业人士都可以增强他们的分析能力，并在各个学科中应用这些原理。在我们继续深入探索数学世界及其应用时，幂集仍然是一个基石概念，体现了集合论的美丽和实用性。在中文中，幂集翻译为“power set”。幂集是一个重要的概念，帮助我们理解集合内部的结构和关系，使其成为任何学习数学或相关领域的人必不可少的主题。