Proof Techniques
📋 What it is
Proof techniques are the standard tools — direct proof, contradiction, counterexample, induction — you pick the right one for the job.
🗣️ Coach says
Just like a toolbox, different claims need different proof tools. Part of being good at proof is recognising “this one wants a counterexample” or “this needs contradiction”. Pick the tool before you start hammering.
🧠 Memory hook
Match the tool to the claim. “For all” → often induction; “there is no” → often contradiction; “this is false” → counterexample.
😂 Giggle
Why did the indirect proof take the long way around?
Because it proved the truth by showing the opposite was impossible!
😲 Whoa!
The choice of technique can turn a page-long mess into a two-line proof — expert mathematicians spend as much time CHOOSING the tool as using it.
✅ Quick check: You want to DISPROVE “every odd number is prime.” Which tool fits best?
Say your answer out loud first — then reveal.
A counterexample — 9 is odd but not prime, so a single case disproves the whole claim.
Disproving a universal (“every”) statement needs just one counterexample.
🧪 Try it! (2 minutes)
For each claim, name the tool you’d reach for: “there is no biggest number” / “the sum of two evens is even” / “not all shapes tile the plane.”