Prove that the monoids $(\mathbb N,+)$ and $(\mathbb N,\cdot)$ aren't isomorphic.
I tried that by assuming that there is an isomorphism f between (N,+) and (N,*). Then f(x+y)=f(x)*f(y), for every x,y from N and f(0)=1. For x=0 and y=1 we have f(1)=f(0)*f(1). But f(0)=1. That means that f(1)=f(1), so it doesn't help me with anything.
Let $f$ is the isomorphism.
Let $f(1)= c$.
Then using induction $f(n) = f(1)^n = c^n$, which can't be bijection (since it's not onto).