📚 GCSE Edexcel Computer Science: Boolean Algebra – Key Points & Revision | GCSE Edexcel 计算机：布尔代数 考点精讲
Boolean algebra is the foundation of digital logic and computing. In the Edexcel GCSE Computer Science course, you must understand how logic gates combine binary inputs to produce outputs, how to read and construct truth tables, write and simplify Boolean expressions, and apply key laws such as De Morgan’s laws. This guide walks you through every examined concept, from basic gates to expression simplification, with clear explanations and paired examples.
布尔代数是数字逻辑和计算机运算的基础。在 Edexcel GCSE 计算机科学课程中，你需要理解逻辑门如何组合二进制输入并产生输出，会阅读和构建真值表，会书写与化简布尔表达式，并运用德摩根定律等关键法则。本文带你逐一攻克所有考点，从基础门电路到表达式化简，配以清晰中英对照讲解。
1. Binary Logic & Why It Matters | 二进制逻辑及其意义
Computers process data at the lowest level using binary signals: 1 represents TRUE (high voltage) and 0 represents FALSE (low voltage). Boolean algebra allows us to describe and manipulate these signals mathematically, giving us a tool to design and simplify digital circuits. Every logical operation you see inside a CPU or memory chip can be expressed as a Boolean function.
计算机在最底层用二进制信号处理数据：1 代表真（高电平），0 代表假（低电平）。布尔代数让我们能够用数学方法描述和操控这些信号，从而设计和化简数字电路。你在 CPU 或存储芯片中看到的每一种逻辑运算都可以用一个布尔函数表达。
2. The Three Fundamental Gates: AND, OR, NOT | 三种基本门：与门、或门、非门
An AND gate outputs 1 only when all inputs are 1. Its symbol is a D-shaped gate with a flat back. The Boolean expression is written as Q = A AND B, or more commonly Q = A·B. An OR gate outputs 1 if at least one input is 1; its symbol is a curved shape like a shield, and the expression is Q = A OR B, often written as Q = A + B. A NOT gate has a single input and simply inverts the signal: Q = NOT A, shown as Q = Ā or A’. These three gates form the basis of all digital logic.
与门（AND）只在所有输入均为 1 时才输出 1。它的符号是一个平背的 D 形门。布尔表达式写作 Q = A AND B，更常见的是 Q = A·B。或门（OR）在至少一个输入为 1 时输出 1；其符号类似盾形，表达式为 Q = A OR B，常写成 Q = A + B。非门（NOT）只有一个输入，将信号取反：Q = NOT A，记作 Q = Ā 或 A’。这三种门是所有数字逻辑的基础。
3. Truth Tables for AND, OR, NOT | 与门、或门、非门的真值表
A truth table lists every possible combination of inputs and the corresponding output. For a two-input AND gate (inputs A, B), the output column shows 1 only for A=1,B=1; otherwise 0. For a two-input OR gate, the output is 0 only when both inputs are 0. For a NOT gate, a single input of 1 yields output 0 and vice versa. Mastering these tables is essential for building more complex logic.
真值表列出所有可能的输入组合及其对应输出。对于一个二输入与门（输入 A、B），输出列仅在 A=1,B=1 时为 1，其余为 0。二输入或门的输出只在两个输入均为 0 时才为 0。非门中，输入 1 则输出 0，反之亦然。掌握这些表格是构建更复杂逻辑的前提。
| A | B | A AND B | A OR B |
|---|---|---|---|
| 0 | 0 | 0 | 0 |
| 0 | 1 | 0 | 1 |
| 1 | 0 | 0 | 1 |
| 1 | 1 | 1 | 1 |
4. NAND, NOR and XOR Gates | 与非门、或非门与异或门
NAND (NOT AND) outputs the opposite of an AND gate. Its truth table is the inverted version of AND: output is 0 only when both inputs are 1. NOR (NOT OR) outputs 1 only when both inputs are 0. XOR (exclusive OR) outputs 1 when the inputs are different. These gates are widely used because NAND and NOR are functionally complete – any logic circuit can be built using only NAND or only NOR gates.
与非门（NAND）输出与门结果的反相。其真值表是与门的取反：仅当两个输入都为 1 时输出为 0。或非门（NOR）仅当两个输入均为 0 时才输出 1。异或门（XOR）在输入不同时输出 1。这些门广泛应用，因为与非门和或非门功能完备——仅用与非门或仅用或非门就能构造出任何逻辑电路。
| A | B | NAND | NOR | XOR |
|---|---|---|---|---|
| 0 | 0 | 1 | 1 | 0 |
| 0 | 1 | 1 | 0 | 1 |
| 1 | 0 | 1 | 0 | 1 |
| 1 | 1 | 0 | 0 | 0 |
5. Logic Gate Symbols and Drawing Circuits | 逻辑门符号与电路图绘制
Edexcel GCSE requires you to recognise both traditional (IEC) and the newer ANSI/IEEE rectangular symbols, but most exam questions use the standard shape symbols. You must be able to draw a logic circuit given a Boolean expression, and vice versa. Connect gates with lines representing signals flowing from left to right, and use small circles (bubbles) to indicate inversion. Practice drawing expressions like Q = (A AND B) OR (NOT C) step by step.
Edexcel GCSE 要求你识别传统的形状符号和较新的矩形符号，但多数考题使用标准形状符号。你必须能根据布尔表达式画出逻辑电路图，也能反过来写出表达式。用线条连接门电路，信号从左向右流动，用小圆圈（气泡）表示取反。请一步步练习绘制诸如 Q = (A AND B) OR (NOT C) 这样的表达式。
6. Boolean Expressions and Notation | 布尔表达式与记法
A Boolean expression is a combination of variables (usually A, B, C), operators (AND, OR, NOT), and parentheses that describes a logic circuit. Common notations: AND is represented by a dot (·) or simply concatenation: A·B or AB. OR is represented by a plus sign: A+B. NOT is represented by an overbar Ā or a prime A’. Parentheses dictate the order of operations, much like in algebra. In the exam, you may be asked to write an expression from a truth table or a diagram.
布尔表达式由变量（通常是 A、B、C）、运算符（AND、OR、NOT）和括号组成，用来描述一个逻辑电路。常见记法：AND 用点号（·）或直接并置表示：A·B 或 AB。OR 用加号表示：A+B。NOT 用在字母上方加横线（Ā）或加一撇（A’）表示。括号决定运算顺序，和代数类似。考试中可能要求你根据真值表或电路图写出表达式。
7. Laws of Boolean Algebra | 布尔代数定律
Just like normal algebra, Boolean algebra has laws that help you rearrange and simplify expressions. Key laws include: Commutative law (A·B = B·A, A+B = B+A); Associative law (A·(B·C) = (A·B)·C); Distributive law (A·(B+C) = A·B + A·C, A+(B·C) = (A+B)·(A+C)). You must also know identity laws (A+0=A, A·1=A), annulment (A+1=1, A·0=0), idempotent (A+A=A, A·A=A), complement (A+Ā=1, A·Ā=0), and double negation ( =A). These are the tools for simplification.
与普通代数一样，布尔代数也有一套定律帮你重组和化简表达式。关键定律包括：交换律（A·B = B·A，A+B = B+A）；结合律（A·(B·C) = (A·B)·C）；分配律（A·(B+C) = A·B + A·C，A+(B·C) = (A+B)·(A+C)）。还需掌握恒等律（A+0=A，A·1=A）、湮灭律（A+1=1，A·0=0）、幂等律（A+A=A，A·A=A）、互补律（A+Ā=1，A·Ā=0）和双重否定律（Ā = A）。这些是化简的工具。
8. De Morgan’s Theorems | 德摩根定理
De Morgan’s theorems are crucial for expressing AND in terms of OR and NOT, and vice versa. The first theorem states: (A·B) = Ā + B̄. The second states: (A+B) = Ā · B̄. In words, the complement of a product is the sum of the complements, and the complement of a sum is the product of the complements. These theorems allow you to convert any logic circuit into one that uses only NAND or only NOR gates.
德摩根定理至关重要，能把 AND 用 OR 和 NOT 表示，反之亦然。第一条定理：A·B 的补是 Ā + B̄；第二条定理：A+B 的补是 Ā · B̄。通俗地说，积的补等于补的和，和的补等于补的积。利用这一定理，你就能把任何逻辑电路转化为只使用与非门或只使用或非门的电路。
9. Simplifying Boolean Expressions | 化简布尔表达式
Simplifying a Boolean expression reduces the number of gates and connections needed in a circuit, which saves cost and power. Use the laws step by step. Example: simplify Q = A·B + A·B̄. Factor out A: A·(B + B̄). Since B + B̄ = 1, we get Q = A·1 = A. Another example: Q = A + A·B. Using Absorption law: A + A·B = A. Always show your working: label which law you are applying. The Edexcel exam often includes a simplification question worth several marks.
化简布尔表达式能减少电路所需门电路和连线的数量，从而降低成本与功耗。要一步步运用定律。例如：化简 Q = A·B + A·B̄。提取公因子 A：A·(B + B̄)。由于 B + B̄ = 1，得到 Q = A·1 = A。再如：Q = A + A·B。用吸收律：A + A·B = A。一定要展示化简步骤，标注所用定律。Edexcel 考试常会有一道几分的化简题。
10. Logic Circuit Simplification Example | 逻辑电路化简实例
Suppose a circuit is described by Q = (A + B)·(A + C). Expand using distributive law: Q = A·A + A·C + B·A + B·C. Since A·A = A, and A + A·C = A (absorption), and B·A = A·B, we can rewrite as Q = A + A·B + B·C. Apply absorption again: A + A·B = A, leaving Q = A + B·C. This simpler expression uses only one OR and one AND gate instead of multiple gates, while producing the same truth table.
假设一个电路描述为 Q = (A + B)·(A + C)。用分配律展开：Q = A·A + A·C + B·A + B·C。因为 A·A = A，A + A·C = A（吸收律），B·A = A·B，可重写为 Q = A + A·B + B·C。再次应用吸收律：A + A·B = A，最终得到 Q = A + B·C。这个更简单的表达式只需一个或门和一个与门，代替原来的多个门，产生的真值表却完全相同。
11. Boolean Reduction Using Truth Tables | 利用真值表进行布尔化简
Another method to simplify is to list all input combinations for which the output is 1 and write the sum of minterms. For example, a function F(A,B,C) that outputs 1 for the combinations 001, 011, 111 can be expressed as F = Ā·B̄·C + Ā·B·C + A·B·C. Then use the laws to factor and reduce. This canonical sum-of-products form is a systematic way to derive an expression before simplification.
另一种化简方法是列出所有使输出为 1 的输入组合，并写出最小项之和。例如，函数 F(A,B,C) 在组合 001、011、111 时输出 1，可表示为 F = Ā·B̄·C + Ā·B·C + A·B·C。然后用定律进行因式分解和化简。这种规范的积之和形式是在化简前系统推导表达式的方法。
12. Exam Tips and Common Pitfalls | 应考策略与常见误区
Always draw truth tables neatly and label columns clearly. When simplifying, state the law you use at each step – this secures method marks even if a minor error occurs. Double-check that your simplified expression matches all rows of the original truth table. Watch out for the difference between XOR and OR: an XOR is true only when exactly one input is true, while OR is true when at least one is true. Finally, practice converting a circuit of mixed gates into an expression and then simplifying; this skill integrates many topic areas.
真值表要画得整洁，列标题清晰。化简时，每一步都要注明所用的定律——这样即使出现小错也能拿下方法分。做完后，务必验证化简后的表达式与原始真值表所有行一致。注意区分 XOR 和 OR：异或仅当恰好一个输入为 1 时为真，而或门在至少一个 1 时就为真。最后，多练习将混合门电路转化为表达式再化简；这个技能整合了多个知识点。
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