Finding a Counterexample
Disprove a claim by producing one case where it fails, and see why a single case is enough to sink a general statement.
What a learner can do afterwards
- Produce a counterexample to a plausible claim such as every odd number being prime
- Explain the asymmetry: one case disproves, no number of cases proves
- Decide, for a given claim, whether to hunt for a counterexample or attempt an argument
1 · Read
Disprove a claim by producing one case where it fails. The claim that every odd number is prime dies at 9, because 9 equals 3 times 3. Note that 1 is a special case: neither prime nor composite.
See the asymmetry: one case disproves, while no number of cases proves. Dan tests 3, 5, and 7, finds all prime, and declares every odd number prime. He has not proved it. The untested 9 breaks his claim.
Hunt where claims look weakest. Against every number ending in 3 is prime, try 33: it equals 3 times 11. Against all multiples of 3 are odd, try 6. Against every factor of 12 is even, try 3. Against every prime is odd, try 2, the only even prime. Multiples of 4 are even integers because each one is 4 times an integer, so 2 divides each one.
Some hunts use even numbers. 6 is a multiple of 3 and it is even, so all multiples of 3 are odd is false. 3 divides 12 and 3 is odd, so every factor of 12 is even is false.
One hunt uses the number 2. 2 is prime and even, so every prime is odd is false. Multiples of 4 are even integers because each one is 4 times an integer.
One failing case sinks every, while passing cases never prove it.
Three readings are original in-repo writing; prime, divisibility and factor items follow Portland Community College problem records with corpus-recorded CC0.
2 · Watch
به انگلیسیTake it off screen
Where it sits
Learn first
Optional connections
This opens up
Where this leads
Jobs that lean on this skill. Follow one to see everything it is built on.
17 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.