Proof Strategies | Mathswell

⭐ First Principle: Always Try Direct Proof First!

Direct proof should be your first instinct. Only when direct proof is genuinely difficult or impossible should you consider indirect methods.

🎯 What are Proof Strategies?

Mathematical proofs follow a hierarchy of approaches:

Direct proof (preferred): Build a logical path from hypothesis to conclusion
Indirect proof (when needed):
- Contrapositive: For implications, prove "not Q implies not P" (written as ¬Q → ¬P)*
- Contradiction: Assume the negation, derive impossibility

*The symbol ¬ means "not" or "the negation of"

🔍 When Direct Proof Works (Most Cases!)

Example: "If n is even, then n² is even"

Direct proof is straightforward:

Assume n is even, so n = 2k
Then n² = (2k)² = 4k² = 2(2k²)
Therefore n² is even ✓

Using indirect proof here would be unnecessarily complex!

💡 When Indirect Proof is Necessary

Indirect proof is needed when:

Proving irrationality: "√2 is irrational" (can't check all rationals directly)
Proving non-existence: "There is no largest prime" (can't check all possibilities)
When working backwards is easier: "If n² is even, then n is even" (hard to go from n² to n, but easy to go from n to n²)

✅

Direct Proof

Always try first!

Works for: Most implications, existence claims, constructive proofs

Example: "If n is odd, then n² is odd"

↔️

Contrapositive

Use when: Direct proof from P is hard, but ¬Q gives clear info

Only for: Implications (If P then Q)

Example: "If n² is even, then n is even"

💥

Contradiction

Last resort: When direct construction is impossible

Necessary for: Irrationality, non-existence

Example: "√2 is irrational"

⚠️ Common Mistake to Avoid

Don't use indirect proof when direct proof works! For example, proving "If n is even, then n² is even" by contradiction is poor mathematical style—the direct proof is cleaner and clearer.

🔬 Strategy Decision Tree

Select a statement below to see which proof strategies are appropriate and why.

Easy Direct Proof

"If n is even, then n² is even"

Needs Indirect

"If n² is even, then n is even"

Irrationality

"√2 is irrational"

Existence

"There exists an x such that x² = 4"