Proof Techniques Math Example 1

Follow the full solution, then compare it with the other examples linked below.

easy

Name four proof techniques, give a one-sentence description of each, and identify which is best suited to prove: 'For all

n \ge 1

3 \mid (n^3 - n)

1
1. Direct proof: assume the hypothesis and derive the conclusion by logical steps.
2
2. Proof by contradiction: assume the negation of the goal and derive a contradiction.
3
3. Proof by contrapositive: prove $\neg q \Rightarrow \neg p$ instead of $p \Rightarrow q$ .
4
4. Mathematical induction: prove a base case and an inductive step for statements indexed by $\mathbb{N}$ .
5
Best technique for $3 \mid (n^3-n)$ : direct proof. Factor: $n^3-n = n(n-1)(n+1)$ — three consecutive integers, so one is divisible by 3. Done.

Answer

n^3-n=n(n-1)(n+1);\text{ direct proof via consecutive integers works best}

Knowing multiple proof techniques and choosing the most efficient one for a given claim is a key mathematical skill. Factoring

n^3-n

reveals the consecutive-integer structure, making a direct proof immediate.

About Proof Techniques

Proof techniques are standard strategies for establishing mathematical claims under different structures.

Compare direct proof and proof by contrapositive for: 'If [formula] is even, then [formula] is even.

Which proof technique is most appropriate for: 'There exists a real number [formula] such that [form

Prove using mathematical induction: [formula] for all [formula].