Proof by Contradiction in IB Mathematics | IB数学：反证法证明策略

📚 Proof by Contradiction in IB Mathematics | IB数学：反证法证明策略

Proof by contradiction is one of the most distinctive methods in mathematics. It asks us to suppose that the statement we want to prove is false, and then to follow the logical consequences until we reach an impossibility. In IB Mathematics, this skill appears most often in the Higher Level courses, where students are expected to produce clear, valid proofs.

反证法是数学中最具特色的证明方法之一。它要求我们先假设要证明的命题为假，然后从这一假设出发进行逻辑推理，直到得出一个不可能的结论。在 IB 数学中，反证法尤其常见于高级水平课程，学生需要能够写出清晰、有效的证明过程。

1. What Is Proof by Contradiction? | 什么是反证法

At its core, proof by contradiction begins with the opposite of the statement to be proved. If this opposite statement leads to a contradiction, then the opposite must be false, so the original statement must be true.

反证法的核心是：先假设所要证明命题的反面成立。如果这个反面命题会引发矛盾，那么反面必然不成立，于是原命题必然成立。

The general structure has three steps.

反证法的一般结构包含三个步骤。

• Assume the negation. Suppose the conclusion is false.

  假设否定：假设结论不成立。

• Reason logically. Use definitions, theorems and algebra to derive consequences.

  逻辑推理：运用定义、定理和代数运算推导后果。

• Reach a contradiction. Find a statement that cannot be true, such as 1 = 2 or the fact that an integer is both odd and even.

  导出矛盾：得到一个不可能成立的结论，例如 1 = 2，或者某个整数既是奇数又是偶数。

2. The Logical Foundation | 反证法的逻辑基础

Proof by contradiction relies on the law of the excluded middle: for any statement P, either P is true or the negation of P is true. If we can show that the negation of P leads to a false statement, then the negation of P cannot be accepted, so P must be true.

反证法依赖于排中律：对于任意命题 P，要么 P 为真，要么 P 的否定为真。如果我们能够证明 P 的否定会导致一个错误结论，那么 P 的否定就不能成立，因此 P 必然为真。

More formally, suppose we want to prove A ⇒ B. We assume A and ¬B together. If this assumption produces a contradiction, then the implication is valid. This is equivalent to proving that A and ¬B cannot both be true.

更正式地说，若要证明 A ⇒ B，我们同时假设 A 且 ¬B。如果这个假设产生矛盾，那么蕴含关系成立。这等价于证明 A 且 ¬B 不可能同时为真。

This logical basis is important in IB Mathematics because examiners expect clear reasoning, not just a final answer. You should state the contradiction explicitly at the end of the proof.

这一逻辑基础在 IB 数学中非常重要，因为考官期待的是清晰的推理，而不只是一个最终

Published by TutorHao | IB Mathematics Revision Series | aleveler.com

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