Show that Z+ × Z+ is countable by showing that the polynomial function f : Z+ × Z+ ? Z+ with f(m. n) = (m + n ? 2)(m + n ? l)/2 + m is one-to- one and onto.

Solution:Step-1: In this problem we need to show that is countable by showing that the polynomial function with is one -to-one and onto. One -to-one function: A function for which every element of the range of the function corresponds to exactly one element of the domain. Test for one -to-one functions : If f(a) = f(b) implies that a = b , then f is one-to-one.Onto function:A function is said to be onto if for every y in Y, there is an x in X such that f(x) = y.Note: A countable set is either a finite set (or) a countably infinite set.Step-2:Given , For our convenience let us consider , m+n = p , where p is a positive integer.If n = 0 , then...