Proof Methods
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Learn how to construct rigorous mathematical proofs using direct proof, contradiction, contrapositive, and mathematical induction.
Concepts in this topic
- Introduction to Proofs — Understand what constitutes a valid mathematical proof
- Direct Proof — Prove statements by assuming the hypothesis and deriving the conclusion
- Proof by Contradiction — Prove statements by assuming the opposite and finding a contradiction
- Mathematical Induction — Prove statements about all natural numbers using the induction principle
Key formulas & identities
| Name | Formula | Meaning |
|---|---|---|
| Direct Proof Structure | Assume P. Derive Q. Therefore P → Q. | To prove P → Q, assume P is true and logically deduce Q |
| Proof by Contrapositive | Prove ¬Q → ¬P instead of P → Q | Since P → Q ≡ ¬Q → ¬P, proving the contrapositive is equivalent |
| Proof by Contradiction | Assume ¬S. Derive a contradiction. Therefore S. | Assume the negation of the statement and derive a logical impossibility |
| Mathematical Induction | P(n₀) ∧ (∀k ≥ n₀: P(k) → P(k+1)) ⟹ ∀n ≥ n₀: P(n) | Prove the base case and the inductive step to prove a statement for all natural numbers |
| Strong Induction | P(n₀) ∧ (∀k ≥ n₀: P(n₀) ∧ ... ∧ P(k) → P(k+1)) ⟹ ∀n ≥ n₀: P(n) | Assume all previous cases (not just the immediately preceding one) in the inductive step |
| Definition of Even | n is even ⟺ n = 2k for some integer k | An integer is even if and only if it is divisible by 2 |
| Definition of Odd | n is odd ⟺ n = 2k + 1 for some integer k | An integer is odd if and only if it leaves remainder 1 when divided by 2 |
Sample practice problem
From the free practice sample — 76 guided problems cover this topic in interactive practice.
What is the hypothesis in the statement "If n is odd, then n² is odd"?
- n² is odd
- n is odd
- n is even
- n² is even
Show answer
Answer: n is odd
In the conditional statement "If n is odd, then n² is odd," the hypothesis is "n is odd" (the part after "if") and the conclusion is "n² is odd" (the part after "then").