Find the function:

Let N0 denote the set of nonnegative integers. Find all functions from N0 to itself such that.
\ f(m+f(n)) = f(f(m))+f(n)\qquad\text{for all}\; m, n\in\mathbb{N}_{0}. \

2 Answers

Shubhodip ·

I just need to prove that 1 is in the range of f.

Some work of mine may help you to help me..[3]

1) f(0) = 0

2) f(f(m)) = f(m) for all m

3) f(m+ f(n)) = f(m) + f(n)= f(f(m)+ f(n))

4) f(k(f(m)) = kf(m)

5)If 1 is in range of f we must have f(1) = 1

So as you can see, we just need to prove f(1) = 1 to get the non-constant solution or f(1) = 0 to get constant solution.

But thats the step where i am stuck...[2]

Shubhodip ·

Helping you some more

Let f(1) = p

We have f(mp) = mp

f(1+mp)= p(m+1) for all m

Your Answer

Close [X]