Final answer:
In direct proofs, evidence is used to logically demonstrate the truth of a statement, while a counterexample is used to disprove a general statement by showing an exception to the rule.
Step-by-step explanation:
In a direct proof, evidence is used to logically demonstrate the truth of a theorem or statement. On the other hand, a counterexample is a single example that disproves a general statement by showing an exception to the rule.
The use of evidence in proofs is crucial, as it helps to build a logical argument that leads inevitably to the conclusion that the theorem or statement in question is true. In mathematical reasoning, this can include the use of definitions, previously proven theorems, and logical deductions derived from the given information.
Counterexamples, however, are used in a different way. They are an effective tool to show that a statement, which claims to apply universally, does not hold in all cases. Finding a single counterexample is sufficient to refute a general claim, thereby emphasizing the need for precise and careful definition in mathematical theorems and conjectures.