𝒫 幂集计算器（大小、列出所有子集、按元素个数统计）Power Set Calculator (Size, List Every Subset, Count by Size)
输入集合元素（如 a, b, c）或只输入元素个数 n，求幂集的大小 2ⁿ（大整数精确值），按子集大小列出组合数 C(n, k)，元素不超过 10 个时列出全部子集。重复元素会自动去重。本地处理。Enter the elements of a set (such as a, b, c) or just the number of elements n to get the size of the power set 2ⁿ exactly with big integers, the number of subsets of each size C(n, k), and, for up to 10 elements, every subset listed. Duplicate elements are removed. Processed locally.
使用说明与常见问题How-to & FAQ
「幂集计算器（大小、列出所有子集、按元素个数统计）」是什么？What is the Power Set Calculator (Size, List Every Subset, Count by Size) tool?
输入集合元素（如 a, b, c）或只输入元素个数 n，求幂集的大小 2ⁿ（大整数精确值），按子集大小列出组合数 C(n, k)，元素不超过 10 个时列出全部子集。重复元素会自动去重。本地处理。Enter the elements of a set (such as a, b, c) or just the number of elements n to get the size of the power set 2ⁿ exactly with big integers, the number of subsets of each size C(n, k), and, for up to 10 elements, every subset listed. Duplicate elements are removed. Processed locally.
如何使用「幂集计算器（大小、列出所有子集、按元素个数统计）」？How do I use the Power Set Calculator (Size, List Every Subset, Count by Size) tool?
在「幂集计算器（大小、列出所有子集、按元素个数统计）」中，填写「输入」、「元素个数 n」、「元素（逗号或空格分隔）」等参数，结果会随输入实时计算并显示在下方。In the Power Set Calculator (Size, List Every Subset, Count by Size): Fill in "Input", "Element count n", "Elements (comma or space separated)"; results are recalculated instantly as you type.
幂集是什么？What is a power set?
一个集合所有子集组成的集合，包括空集和它本身。例如 {a, b} 的幂集是 {∅, {a}, {b}, {a, b}}。The set of all subsets of a set, including the empty set and the set itself. The power set of {a, b} is {∅, {a}, {b}, {a, b}}.
为什么是 2ⁿ 个？Why are there 2ⁿ subsets?
每个元素都有“在子集中”或“不在子集中”两种选择，n 个元素共 2 × 2 × … × 2 = 2ⁿ 种组合。Each element is either in or out of a subset, so n elements give 2 × 2 × … × 2 = 2ⁿ combinations.
按大小怎么统计？How are subsets counted by size?
含 k 个元素的子集有 C(n, k) 个（组合数），所有 k 的 C(n, k) 加起来正好等于 2ⁿ。There are C(n, k) subsets with k elements, and adding C(n, k) over every k gives exactly 2ⁿ.
能举个计算实例吗？Can you show a worked example?
例如在「幂集计算器（大小、列出所有子集、按元素个数统计）」中填 输入 = 元素个数，元素个数 n = 5，结果区显示：幂集大小 2ⁿ：32；n：5For example, entering Input = Number of elements, Element count n = 5 in the Power Set Calculator (Size, List Every Subset, Count by Size) gives: Power set size 2ⁿ: 32; n: 5
哪些输入会被检查？Which inputs are validated?
「幂集计算器（大小、列出所有子集、按元素个数统计）」会检查这些输入——「元素个数 n」：所选方式用到时必须填写；绝对值超过 10¹⁵ 的数也会被拒绝。不符合时结果区显示具体提示，不会给出数字。The Power Set Calculator (Size, List Every Subset, Count by Size) checks its inputs — "Element count n": required when the chosen option uses it; numbers above 10¹⁵ in absolute value are rejected too. When a value fails, the result area names the problem instead of showing a number.
有哪些选项可以调整？What options can I adjust?
「幂集计算器（大小、列出所有子集、按元素个数统计）」的设置：「输入」可选 元素个数、列出元素（默认 元素个数）；「元素个数 n」默认 5；「元素（逗号或空格分隔）」默认 a, b, c。每次生成结果都按当前设置处理。Settings in the Power Set Calculator (Size, List Every Subset, Count by Size) — "Input": Number of elements, List the elements (default Number of elements); "Element count n" default 5; "Elements (comma or space separated)" default a, b, c. The result always uses the current settings.
计算精度如何？How precise are the results?
「幂集计算器（大小、列出所有子集、按元素个数统计）」使用 JavaScript 双精度浮点数（约 15–16 位有效数字），部分功能使用 BigInt 处理大整数。The Power Set Calculator (Size, List Every Subset, Count by Size) uses double-precision floating point (about 15–16 significant digits); some features use BigInt for large integers.
「幂集计算器（大小、列出所有子集、按元素个数统计）」免费吗？需要注册吗？Is the Power Set Calculator (Size, List Every Subset, Count by Size) free? Do I need an account?
「幂集计算器（大小、列出所有子集、按元素个数统计）」完全免费，无需注册或安装，打开网页即可使用，手机和电脑都能用；输入的内容只在你的浏览器本地处理。The Power Set Calculator (Size, List Every Subset, Count by Size) is completely free with no sign-up or install and works on phones and desktops; what you enter is processed only in your browser.