Universal programs and universal functions

1. 1. Show that there is a decidable predicate Q(x,y,z) such that
      1. if and only iff
      2. if , and Q(x,y,z), then
   2. Deduce that there is a computable function g(x,y) such that
      1. g(x,y) is defined if and only if
      2. if , then and ; i.e.
   3. Deduce that if f is a computable injective function (not necessarily total or surjective) the is computable.
2. Show that there is total computable function k(e) such that for any e, if is the characteristic function for a decidable predicate M(x), then is the characteristic function for .
3. Show that there is a total computable function k(x) such that for every x, .
4. Show that there is a total computable function s(x,y) such that for all x,y .
5. Suppose that f(x) is computable; show that there is a total computable function k(x) such that for all x, .