Let f N rightarrow N be the function defined via the assign

Let f : N rightarrow N be the function defined via the assignment f(n) = {n/2 n is even 3n+1 n is odd Prove that f is not an injective function. Prove that f is a subjective function. Does there exist a function g : N rightarrow N that, satisfies g(f{x)) = x for all x N? Does there exist, a function h : N rightarrow N that satisfies f(h(y)) = y for all y N?

Solution

i) f(1) = 4 ,f(8) = 4 ,since f(1) = f(8) ,

f is not an injective function

ii) for every natural number N ,2 N is even ,f(2N) = 2N/2 = N ,

hence f is a surjective function.

iii)No ,

let there there exist g(x) ,such that g(f(x)) = x

consider g(f(1)) ,g(f(8))

g(f(1)) = g(4) = 1

g(f(8)) = g(4) = 8

,this is contradiction,as g(4) cant be 1 as well as 8 , {properties of function}

hence our assumption was wrong

there does not exist g(x) ,such that g(f(x)) = x

iv) cosider h(y) = 2y

f(h(y)) = f(2y) = y { 2y is even ,f(2y) = 2y/2 = y}