【学习笔记】离散数学（Discrete Math） - 证明 Proof 3

离散数学（Discrete Math）

                                                         - 证明proof

目录

离散数学（Discrete Math）

                                                         - 证明proof

 

Terminology 术语

How to prove conditional statements?

Direct proof直接证明法

Proof by contraposition对位证明法

Proof by contradiction 矛盾法/反证法

Vacaous and Trival proofs

Proof by cases 列举法

Uniqueness proofs 唯一性证明

名词补充

参考课程

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Terminology 术语

Theorem 定理 ：A theorem is a statement that can be shown to be true.

Proposition 命题：Less important theorems are called propositions.

Proof 证明：A proof is a valid argument that established the truth of a theorem.

Axiom 公理：Statements that we assume to be true.

Lemma 引理：A less important throrem that is helpful in the proof of other results.

Corollary 推论：A theorem that can be established directly from a theorem that has been proved.

Conjecture 推测（猜想）：A statement that is being proposed to be a true statement.

How to prove conditional statements?

p->q

1. Direct proof 直接证明法
2. Proof by contraposition 对位证明法
3. Proof by contradiction 矛盾法/反证法
4. Vacaous and Trival proofs 空真和平凡证明

Direct proof 直接证明法

例：

P(n): n is odd.

Q(n): n^2 is odd.

∀n∈Z, P(n)-> Q(n)

证：

设n=2k+1，则有

n^2 = 2(2k^2+2k)+1

证毕

 

Proof by contraposition 对位证明法

p->q ≡┐q->┐p

Assume q is false by applying rule of inference, we conclude that p is false.

例：

Prove that if n is an integer and 3n+2 is odd, then n is odd.

证：

Assume n is not odd.

Then, n is