8.5. Injective Mapping and Uncountability

Try these first

1. 1.

   What is a computable function?

   Hint

   Think about programs that can calculate values.

2. 2.

   Define an uncomputable function.

   Hint

   Consider limitations of programming.

3. 3.

   What characterizes a computable function?

   • Can be computed by a program
   • Cannot be computed
   • Requires infinite resources

   Hint

   Remember the definition of computable functions.

4. 4.

   True or False: All functions are computable.

   • True
   • False

   Hint

   Think about the Halting problem and similar examples.

5. 5.

   Establish a proof demonstrating why the Halting problem is an example of an uncomputable function.

   Hint

   Consider creating a scenario where P evaluates itself and analyze the outcome.

6. 6.

   Create an injective function for a different set, like integers to letters, and prove its injectivity.

   Hint

   Ensure that your function consistently maps integers to unique letters without overlaps.