Methods of proof

form of mathematical proof

A counterexample is a specific example that contradicts a claim, hypothesis, or generalization. In logic a counterexample disproves a universally stated claim, and does so rigorously in the fields of mathematics and philosophy. For example, the statement that "student John Smith is not lazy" is a counterexample to the generalization "students are lazy", and both a counterexample to, and disproof of, the universal quantification "all students are lazy."

set of axioms from which some or all axioms can be used in conjunction to logically derive theorems

kind of proof calculus

proof by examining all individual cases

fundamental concept in automated theorem proving

formal proof

result demonstrating that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general