Initial Segment and Well-Ordering

Question

• If $(X,<)$ is an ordered set and $x\in X$, we define the initial segment of $x$ as

$$s(x)=\left\{y\in X: y<x\right\}$$

What is $s(0)$ if $0$ is the minimal element of $X$?

My answer: $s(0)=\emptyset$ because since $0$ is minimal element of $X$ there is no such $y$ which $y<0$.

Can you check my answer? If it is false, can you help? Thanks...

Answer 1

$\emptyset$ is exactly what the initial segment of the minimal element is, using that definition.