📚 Laws of Boolean Algebra | 布尔代数法则
Boolean algebra is a branch of algebra in which the values of variables are the truth values true and false, usually denoted 1 and 0 respectively. In the IB Mathematics curriculum, Boolean algebra appears in the study of logic, sets, and circuits, providing essential techniques for simplifying logical expressions and designing digital systems. Understanding its fundamental laws is crucial for mastering topics related to mathematical reasoning and computer science.
布尔代数是代数的一个分支，其中变量的取值是真值“真”和“假”，通常分别用1和0表示。在IB数学课程中，布尔代数出现在逻辑、集合和电路的学习中，为简化逻辑表达式和设计数字系统提供了基本方法。理解其基本法则对于掌握与数学推理和计算机科学相关的主题至关重要。
1. Introduction to Boolean Algebra | 布尔代数简介
Boolean algebra was introduced by George Boole in the mid-19th century as a symbolic system for logical reasoning. It operates on binary variables and uses three basic operations: conjunction (AND), disjunction (OR), and negation (NOT). Today it forms the foundation of digital electronics and logical problem-solving in the IB mathematics course, especially in the HL options or core topics on logic and proof.
布尔代数由乔治·布尔在19世纪中叶提出，作为逻辑推理的符号系统。它处理二元变量，并使用三种基本运算：合取（与）、析取（或）和否定（非）。如今它为数字电子学和IB数学课程中的逻辑问题求解奠定了基础，特别是在高级水平（HL）的选修内容或逻辑与证明的核心主题中。
The laws of Boolean algebra allow us to manipulate and simplify logical expressions without having to rely on truth tables every time. These rules are analogous to the algebraic laws of real numbers but with some unique properties, such as idempotence and absorption, which do not hold in ordinary algebra.
布尔代数的法则使我们能够在不每次都依赖真值表的情况下操作和简化逻辑表达式。这些规则类似于实数的代数法则，但具有一些独特的性质，例如幂等律和吸收律，这些性质在普通代数中不成立。
2. Boolean Variables and Basic Operations | 布尔变量与基本运算
In Boolean algebra, every variable can take only one of two possible values: 0 (false) or 1 (true). The basic operations are defined by their truth tables. The AND operation, denoted by the symbol ∧ or by multiplication (·), yields 1 only if both operands are 1; otherwise it gives 0.
在布尔代数中，每个变量只能取两个可能值之一：0（假）或1（真）。基本运算由其真值表定义。与运算，用符号 ∧ 或乘号（·）表示，仅当两个操作数都为1时才得到1；否则得到0。
For example, if A = 1 and B = 1, then A ∧ B = 1; if A = 1 and B = 0, then A ∧ B = 0. This is similar to the logical ‘
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