📚 IB Mathematics: Fundamental Laws of Boolean Algebra | IB数学：布尔代数基本定律
Boolean algebra is a branch of algebra in which variables can only take two values: 1 for true and 0 for false. It is the mathematical basis of propositional logic and digital circuit design. In the IB Mathematics syllabus, Boolean algebra appears in the discrete mathematics option, where students are expected to apply its laws to prove equivalences and simplify logical expressions. A clear understanding of these fundamental laws is essential for exams and for any further study of computer science.
布尔代数是代数学的一个分支，其中的变量只能取两个值：1 表示“真”，0 表示“假”。它是命题逻辑和数字电路设计的数学基础。在 IB 数学课程中，布尔代数属于离散数学选修部分，学生需要运用基本定律来证明等价关系并化简逻辑表达式。牢固掌握这些基本定律，对考试以及将来学习计算机科学都至关重要。
1. What Is Boolean Algebra? | 什么是布尔代数
A Boolean algebra is a set B together with two binary operations: AND, written as ∧, and OR, written as ∨. It also has one unary operation, NOT, written as ¬, and two distinct constants 0 and 1. For any Boolean variable A, the value of A is always either 0 or 1.
布尔代数是由一个集合 B、两种二元运算“与”（∧）和“或”（∨）、一种一元运算“非”（¬）以及两个不同常量 0 和 1 组成的代数系统。对任意布尔变量 A，A 的值只能是 0 或 1。
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A ∧ B is the AND operation; it gives 1 only when both A and B are 1.
A ∧ B 是“与”运算；仅当 A 和 B 都为 1 时，结果才为 1。
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A ∨ B is the OR operation; it gives 1 when at least one of A and B is 1.
A ∨ B 是“或”运算；只要 A 和 B 中至少有一个为 1，结果就为 1。
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¬A is the NOT operation; it changes 1 to 0 and 0 to 1.
¬A 是“非”运算；它将 1 变为 0，将 0 变为 1。
2. Basic Operations and Their Truth Tables | 基本运算与真值表
The truth table below defines the three basic Boolean operations. We use 1 for true and 0 for false.
下面的真值表定义了三种基本布尔运算。我们用 1 表示真，用 0 表示假。
| A | B | A ∧ B | A ∨ B | ¬A |
| 0 | 0 | 0 | 0 | 1 |
| 0 | 1 | 0 | 1 | 1 |
| 1 | 0 | 0 | 1 | 0 |
| 1 | 1 | 1 | 1 | 0 |
ultiplication on the values 0 and 1, but OR is not the same as ordinary addition because 1 ∨ 1 = 1, not 2.
注意：与运算在 0、1 上类似于乘法，但或运算不同于普通加法，因为 1 ∨ 1 = 1，而不是 2。
3. The Fundamental Laws of Boolean Algebra | 布尔代数基本定律
The table below summarizes the fundamental laws you must know. In each identity, A, B and C can be replaced by any Boolean expression.
下表列出了必须掌握的基本定律。在每个恒等式中，A、B、C 都可以替换为任意布尔表达式。
| Law / 定律 | AND form / 与形式 | OR form / 或形式 |
| Commutative / 交换律 | A ∧ B = B ∧ A | A ∨ B = B ∨ A |
| Associative / 结合律 | (A ∧ B) ∧ C = A ∧ (B ∧ C) | (A ∨ B) ∨ C = A ∨ (B ∨ C) |
| Distributive / 分配律 | A ∧ (B ∨ C) = (A ∧ B) ∨ (A ∧ C) | A ∨ (B ∧ C) = (A ∨ B) ∧ (A ∨ C) |
| Identity / 同一律 | A ∧ 1 = A | A ∨ 0 = A |
| Domination / 支配律 | A ∧ 0 = 0 | A ∨ 1 = 1 |
| Idempotent / 幂等律 | A ∧ A = A | A ∨ A = A |
| Complement / 互补律 | A ∧ ¬A = 0 | A ∨ ¬A = 1 |
| Double Negation / 双重否定律 | ¬(¬A) = A | ¬(¬A) = A |
| Absorption / 吸收律 | A ∧ (A ∨ B) = A | A ∨ (A ∧ B) = A |
| De Morgan's Law / 德摩根定律 | ¬(A ∧ B) = ¬A ∨ ¬B | ¬(A ∨ B) = ¬A ∧ ¬B |
Every identity in this table can be verified by
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