To prove that a computably enumerable set of natural numbers has a computable binary relation, one considers Turing machine computation steps to define a relation that corresponds to the enumeration of the set.
The question asks to show that for a computably enumerable (c.e.) set A of natural numbers, there exists a computable binary relation RA such that A is exactly the set of natural numbers x for which there exists a y such that RA(x,y) holds. This involves understanding Turing machines and how computations are performed with respect to these enumerable sets.
To demonstrate this, one can consider the number of steps a Turing machine takes to identify if a number is in set A. The computable binary relation RA can be defined based on the computation steps of a Turing machine. If a Turing machine computes and accepts input x within y steps, then RA(x,y) would be true.
In conclusion, a computable binary relation RA exists for c.e. set A with A = {x ∈ N : ∃y, RA(x,y)}, where the computation steps of a Turing machine can serve as the second element in the relation, demonstrating the computability and enumeration of set A.