📚 Proof of the Distributive Law | 分配律的证明
The distributive law is a fundamental result in both arithmetic and set theory. In this article, we focus on the set-theoretic version, which states that intersection and union distribute over each other. We will provide rigorous element-based proofs, an alternative membership-table method, worked examples, and exam-focused tips for AQA A-Level Mathematics.
分配律是算术与集合论中的一条基本结论。本文将重点讨论集合论版本的分配律，即交集与并集相互分配的性质。我们将提供严格的基于元素的证明、隶属表这一替代方法、例题精讲，以及针对 AQA A-Level 数学的考试技巧。
1. What Is the Distributive Law? | 什么是分配律
4 + 3 × 5 = 27. In set theory, two analogous identities hold for the operations of union (∪) and intersection (∩).
在普通代数中，分配律表述为 a(b + c) = ab + ac。例如 3 × (4 + 5) = 3 × 4 + 3 × 5 = 27。在集合论中，并（∪）与交（∩）两种运算之间也存在两条类似的恒等式。
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C)
These two statements say that each set operation distributes over the other. Notice that this is stronger than in arithmetic, where multiplication distributes over addition but addition does not distribute over multiplication.
这两条陈述说明两种集合运算互相分配。注意，这比算术中的结果更强——在算术中乘法对加法分配，但加法并不对乘法分配。
2. Notation and Prerequisite Definitions | 符号与预备定义
Before proving the law, we must agree on the notation used throughout this article. Let A, B and C be arbitrary sets, all assumed to be subsets of a universal set U.
在证明之前，我们先统一本文使用的符号。设 A、B、C 为任意集合，并假设它们都是全集 U 的子集。
| Symbol 符号 | Meaning 含义 |
| ∈ | is an element of / 属于 |
| ∉ | is not an element of / 不属于 |
| ∩ | intersection / 交集 |
| ∪ | union / 并集 |
| ⊆ | is a subset of / 是……的子集 |
| ∅ | empty set / 空集 |
Recall two key definitions. The union B ∪ C is the set of all elements that belong to B or to C (or both). The intersection A ∩ B is the set of all elements that belong to both A and B.
回忆两条关键定义。并集 B ∪ C 是所有属于 B 或属于 C（或同时属于两者）的元素构成的集合。交集 A ∩ B 是所有
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