If $f:X \rightarrow Y$ is a similarity between well-ordered sets $X$ and $Y$ then $x \leqslant y$ implies $f(x) \leqslant f(y)$.
This is just by the definition.
In later my notes however it says this also implies that $x < y \implies f(x)<f(y)$.
I can’t see where this comes from. Could someone explain this please?
Suppose that $x < y$. Then $x \leqslant y$ and thus $f(x) \leqslant f(y)$. Suppose now that $f(x) = f(y)$. Since $f$ is bijective, this implies $x = y$, which contradicts the assumption that $x < y$. Thus $f(x) < f(y)$.