Proof by Contradiction and Counterexample
Kill a general claim with a single counterexample, and prove one by assuming the opposite and following it until something impossible appears.
What a learner can do afterwards
- Disprove 'every prime number is odd' with one example
- Prove that √2 cannot be written as a fraction
- State clearly what is being assumed at the start of a contradiction proof
1 · Read
Just like 9 broke every odd number is prime, a for all claim breaks at its first failure. To sink every prime is odd, scan the evens and stop at 2. Check the candidate meets every hypothesis and visibly breaks the conclusion. One case sinks such a claim, but one case never proves one.
Assume root 2 is rational in lowest terms, so root 2 = a/b with no common factor. Squaring gives 2b squared = a squared, forcing a even, and writing a = 2k forces b even too. Both even contradicts lowest terms, so the assumption dies and root 2 is irrational.
Contradiction borrows the opposite claim, then throws it. State the assumption up front or nobody can follow the throw. The conclusion always targets the assumption, never the logic: valid steps plus an impossible ending mean a false start.
Long patterns can crack late: n squared + n + 41 gives primes all the way to n = 39, then f(40) = 1600 + 40 + 41 = 1681 = 41 squared. Smallest counterexamples teach most, since they show exactly where the pattern first cracks. One example disproves a for all claim but never proves one.
Sink a general claim with one well checked counterexample, or assume its opposite and follow it to nonsense.
2 · Watch
Take it off screen
Where it sits
Learn first
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.