Find the set C (or S in the online notes) and the ultimate bijection h given by the proof of the CBS theorem.?

Let A be the integers and let B be the even integers. Define the injections f:A -%26gt; B by f(a)=4a and g:B -%26gt; A by g(b)=b.

Find the set C (or S in the online notes) and the ultimate bijection h given by the proof of the CBS theorem.?
The conditions of the Cantor-Bernstein-Schroeder theorem are satisfied. So, A = Z, and B= {x in Z | even(x)}, where Z are the integers.





g maps an even number into itself. So





C0 = A \ g(B) = the set of odd numbers since g(B) = B.


= {x in A | x = 2n+1 for some integer n}





Now,


C1 = g(f(C0)) = {x in A | x = 4(2n+1) for some int n }


C2 = g(f(C1)) = { x in A | x = (4^2)(2n+1) for some int n}


C3 = g(f(c2)) = { x in A | x = (4^3)(2n+1) for some int n}





Ck = g(f(c(k-1))) = { x in A | x = (4^k)(2n+1) for some n }





C = U Ci = {x in A| odd (x) } U {x in A| even(x) and x is of the form (4^k)(2n+1) for some k%26gt;0 and integer n}





Then the bijection h : A -%26gt; B is defined on A as follows:





h(x) =


f(x) if x in C


g(x) if x not in C





That is





h(x) = 4x if x in C


x if x not in C