i want to show that the multiplicative semigroup of the natural numbers without zero are cancellative, that is: For all $h,x,y$, $hx=hy \implies x=y$
I tried proving this by induction over $h$ and letting $x,y$ fixed.
For $h=1$ it is trivial.
But now I’m stuck in the induction step, I gotta show that $(h+1)x=(h+1)y \implies x=y$ But I don’t know how to do that, I tried to write it like this $hx+x=hy+y$ but that doesn’t help :(
Can anyone help me?
Performing a proof by contraposition.
Without loss of generality, suppose that we have $x \gt y$. It means that it exists $a \in \mathbb N$ such that $x=y+a$. But then $hx = h y + h a$ and therefore $hx \neq hy$. And we're done.