Given a partial function f : N * N, let D(f) denote the set of natural numbers
Question:
Given a partial function f : N * N, let D(f) denote the set of natural numbers at which f is defined: D(f) = {x ? N | f(x)?}; and let I(f) be the set of natural numbers that are values of f where it is defined: I(f) = {y ? N | for some x, f(x) = y}.
Prove or disprove the following statements, clearly stating any results about register machine computable functions and partial recursive functions that you use.
(a) Every subset S ? N is equal to I(f) for some register machine computable partial function f. [4 marks]
(b) If f is register machine computable, then I(f) is equal to D(g) for some partial recursive function g. [7 marks]
(c) If f is register machine computable, then D(f) is equal to I(g) for some total recursive function g. [4 marks]
(d) If g is a partial recursive function, then D(g) is equal to I(f) for some register machine computable partial function f. [5 marks]
Discrete and Combinatorial Mathematics An Applied Introduction
ISBN: 978-0201726343
5th edition
Authors: Ralph P. Grimaldi