Mathematical Proof | 数学证明

📚 Mathematical Proof | 数学证明

Proof is the logical engine of A Level Mathematics. In Edexcel Pure Mathematics, you are expected not only to use correct algebra but also to justify why a statement must be true, or to show that a statement is false by a single counterexample.

证明是 A Level 数学的逻辑核心。在 Edexcel 纯数学中，你不仅要正确使用代数，还要论证一个命题为什么必然成立，或者通过一个反例说明某个命题不成立。

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1. What Is Proof? | 什么是证明

A mathematical proof is a watertight chain of reasoning that starts from known facts, definitions or axioms and leads to the conclusion. Unlike science, where evidence supports a theory, a proof in mathematics establishes truth beyond any doubt.

数学证明是一条严密的推理链：从已知事实、定义或公理出发，推出结论。与科学中用证据支持理论不同，数学证明确立的结论是无可置疑的。

ery case. For example, checking x = 1, 2, 3 does not prove a claim about all real numbers.

在 Edexcel A Level 数学中，证明必须具有一般性。除非定义域是有限的并且你检查了所有情况，否则不能只靠代入几个数值来证明命题。例如，检查 x = 1、2、3 并不能证明关于所有实数的结论。

'For all n ∈ ℕ, n² − n is even' is a general statement.

'对于任意 n ∈ ℕ，n² − n 为偶数' 是一个一般性命题。

There are four main tools you need to know: proof by deduction, proof by exhaustion, proof by contradiction, and disproof by counterexample.

你需要掌握四种主要工具：演绎证明、穷举证明、反证法以及反例证伪。

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2. Proof by Deduction | 演绎证明

Proof by deduction starts with known identities or definitions and uses algebra to reach the conclusion. It is the most common method in Pure Mathematics.

演绎证明从已知恒等式或定义出发，通过代数运算得出结论。这是纯数学中最常见的方法。

Example: Prove that the sum of any three consecutive integers is divisible by 3.

例题：证明任意三个连续整数之和能被 3 整除。

d n + 2. Their sum is n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1), which is a multiple of 3.

设这三个整数为 n、n + 1 和 n + 2。它们的和为 n + (n + 1) + (n + 2) = 3n + 3 = 3(n + 1)，这是 3 的倍数。

n + (n + 1) + (n + 2) = 3(n + 1)

ful deduction: if n is even, then n² is divisible by 4. Write n = 2k, so n² = (2k)² = 4k².

另一个常用演绎：若 n 为偶数，则 n² 能被 4 整除。设 n = 2k，则 n² = (2k)² = 4k²。

n = 2k ⇒ n² = 4k²

In a deduction, every step must be a logical consequence of the previous step. Use words such as 'therefore', 'hence' and 'since' to show the flow of reasoning.

在演绎证明中，每一步都必须是前一步的逻辑结果。使用 'therefore'、'hence'、'since' 等词来展示推理过程。

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3. Proof by Exhaustion | 穷举证明

Proof by exhaustion works by checking every possible case in a finite domain. It is valid only if the list of cases is complete and no case is missed.

穷举证明通过检查有限定义域中的每一个可能情况来证明命题。只有当情况列表完整且没有遗漏时才有效。

Example: Prove that for n ∈ {1, 2, 3, 4}, n² + 1 is not divisible by 3.

例题：证明对于 n ∈ {1, 2, 3, 4}，n² + 1 不能被 3 整除。

n	n² + 1	Divisible by 3?
1	2	No
2	5	No
3	10	No
4	17	No

All four possible values are checked, so the statement is proved by exhaustion.

所有四个可能值都已被检查，因此命题通过穷举法得证。

Edexcel questions sometimes ask for exhaustion when there are only a few cases, such as small integers or a limited number of shapes. Make a clear table or list to show every case.

Edexcel 考题有时在只有少数情况时要求使用穷举法，例如较小的整数或有限数量的图形。用清晰的表格或列表列出每一种情况。

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4. Disproof by Counterexample | 反例证伪

To disprove a universal claim, it is enough to produce one allowed value for which the statement fails. This value is called a counterexample.

要证伪一个全称命题，只需给出一个符合条件的值使结论不成立。这个值称为反例。

Example: Disprove 'for all positive integers n, n² + n + 1 is prime'.

例题：证伪 '对于所有正整数 n，n² + n + 1 都是质数'。

3 × 7, so the expression is not prime. This single counterexample disproves the statement.

取 n = 4。则 n² + n + 1 = 16 + 4 + 1 = 21 = 3 × 7，因此该式不是质数。这一个反例就推翻了原命题。

n but fail the conclusion. If n = 4 is not allowed, it cannot be used as a counterexample.

请记住：反例必须满足已知条件，但不满足结论。如果 n = 4 不符合条件，就不能用作反例。

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5. Proof by Contradiction | 反证法

Proof by contradiction begins by assuming the opposite of what you want to prove. If this assumption leads to a logical impossibility, the original statement must be true.

反证法从假设你要证明的结论的反面开始。如果这个假设推出逻辑矛盾，那么原命题一定成立。

Classic example: prove that √2 is irrational.

经典例题：证明 √2 是无理数。

Assume, for contradiction, that √2 is rational, so √2 = a/b where a and b are integers with no common factor and b ≠ 0.

假设 √2 是有理数，即 √2 = a/b，其中 a、b 为没有公因数的整数且 b ≠ 0。

Squaring gives 2 = a²/b², so a² = 2b². Hence a²

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