Numbering computable functions

1. NOTE: MISTAKE IN WHAT WAS DONE IN THE CLASS
   

   the bijection should be defined as

   

   Only then the bijection given in the class holds. Work out the complete proof that is an effective bijection. How come nobody pointed out the mistake in the class?

2. Find
   1. 
   2. 
   3. the code of the following program:

      

   4. 
3. Prove that every computable function has infinitely many indices.
4. Suppose that f(x,y) is a total computable function. For each m, let be the computable function given by . Construct a total computable function h such that for each m, .
5. Let be an enumeration of partial functions from to . Construct a function g from to such that for each i.
6. Let f be a partial function from to , and let . Construct a non-computable function g such that for .
7. Show that the set of all functions from to is not denumerable.