8.6. Conclusion on Uncomputable Functions

Try these first

1. 1.

   What defines a computable function?

   Hint

   Think about programs that always give outputs.

2. 2.

   Give an example of an uncomputable function.

   Hint

   Consider a function that determines program behavior.

3. 3.

   What is a computable function?

   • A function that can be computed
   • A function no program can compute
   • A function that only runs on specific inputs

   Hint

   Remember the definition of computability.

4. 4.

   True or False: The Halting Problem is an example of a computable function.

   • True
   • False

   Hint

   Consider if there is a way to determine program halting.

5. 5.

   Argue whether all mathematical questions can be computed through algorithms, providing examples of uncomputable functions.

   Hint

   Think of functions with indefinite outcomes.

6. 6.

   Create a visual representation or diagram showing the relationship between countable and uncountable sets, and how it applies to functions and programs.

   Hint

   Focus on illustrating elements and their sets.