question_answer 26)

Let f : N  N be defined by             for all , state whether the function f is bijective. Justify your answer.  

Answer:

Given,       Let x1 =1 and x2 = 2 be two element of N             and f(x2) = f(2)              is not one.       It is clear that             f(1) = 1 = f(2(1) ? 1)             f(2) = 1 = f(2 × 1)             f(3) = 2 = f(2(2) ? 1)             f(4) = 2 = f(2 × 2)             f(5) = 3 = f(2 × 2)             f(6) = 3 = f(2(3) ? 1)             f(7) = 4 = f(2(4) ? 1)             f(8) = 4 = f(2 × 4)                          f(9) = 5 = f(2(5) ? 1)             f(10) = 5 = f(2×5)                        n = f(2n ? 1)             n = f(2n)              f(2n ? 1) = n, when n is odd             f(2n) = n, when n is even       Let  (Co-domain of f)       If y is odd, of f(x) = y = f(2y ? 1)             Now         If y is even, f(x) = y = f(2y)       Now,        Corresponding to every  there exists .       is onto but not one-one.