A class of total functions is a PRC class if:
- The class includes all projection functions $p_i(x_1,\ldots,x_n) = x_i$ and the initial functions $n(x) = 0$ and $s(x) = x+1$.
- The class is closed under composition and primitive recursion.
Prove that the class of all total functions is a PRC class.
Is it true if I say every total function is computable? And then say because the class of computable functions is a PRC class, so is the class of total functions.
You cannot say that every total function is computable, since that is not true. For example, the function $f(n)$ which returns $1$ if the $n$'th Turing machine halts and $0$ otherwise is not computable.
However, the question is much simpler. The definition of a PRC class specifies several closure properties, of the form: if so-and-so belong to the class, then such-and-such also belongs to the class (these include the "axioms", in which the premise is empty). Since the class of all (total) functions contains all functions, it automatically satisfies these constraints.
