Counterexamples and generalization
Decide whether examples justify a generalization and use counterexamples to check a claim's scope.
Meaning
Generalization turns shared features of several cases into a broader claim. A counterexample is an allowed case that does not satisfy the claim. One counterexample disproves a claim about all objects in its scope.
Example
2 is a counterexample to “all prime numbers are odd.” It is prime but even, so the condition “prime numbers other than 2” is needed. A counterexample must lie within the claim's scope. Failing to find one does not prove that the claim is true.
Practice for this concept
Scope and references
Topics
Examples versus proof, finding counterexamples, conjecturing patterns
Learning goal
Decide whether examples justify a generalization and use counterexamples to check a claim's scope.
Helpful background
Identify the objects and conditions covered by the claim.
Counterexamples and conditions
Does not cover the full range of proof methods.