Final answer:
Preconditions and post-conditions are criteria that must be true before and after function execution. To test them, one sets up the required preconditions, executes the function, then checks if the post-conditions are met. Counterexamples are used to verify the necessity and sufficiency of these conditions.
Step-by-step explanation:
Preconditions and post-conditions are concepts used in the field of software engineering to define what must be true before a function (or method) is executed and what should be true after the function has completed its execution, respectively. To test these conditions, one typically writes test cases that set up the necessary environment or context to satisfy the preconditions, executes the function, and then checks if the post-conditions hold.
For example, suppose we have a condition where 120 credit hours are necessary to graduate. Here, the precondition is having completed the 120 credit hours prior to graduation. The post-condition could be the issuance of a bachelor's degree. To test this in a real-world hypothesis, one may track the credit hours of students and verify whether those who have met the precondition of completing 120 hours indeed receive their degrees.
Identifying necessary and sufficient conditions is also crucial. A sufficient condition guarantees that something else will occur, while a necessary condition must be met for something to happen. Using counterexamples can help assess the truth of conditionals by considering situations where the expected outcome does not occur despite the precondition being satisfied. This leads to a refinement of our hypothesis or understanding of the underlying conditionals.