The task is to prove that $S$ is a partial order relation on $\mathbb{N}$, where $S$ is defined such that $k_1 S k_2 \implies k_1$ divides $k_2$ ($k_1 | k_2$).

To prove $S$ is a partial order relation, we need to verify the following three properties:

1. Reflexivity: $k S k$ for all $k \in \mathbb{N}$.
2. Antisymmetry: If $k_1 S k_2$ and $k_2 S k_1$, then $k_1 = k_2$.
3. Transitivity: If $k_1 S k_2$ and $k_2 S k_3$, then $k_1 S k_3$.

Proof:

1. Reflexivity:

   • For any $k \in \mathbb{N}$, $k$ divides itself ($k | k$) because $k = k \times 1$.
   • Thus, $k S k$ holds for all $k \in \mathbb{N}$, proving reflexivity.

2. Antisymmetry:

   • Suppose $k_1 S k_2$ and $k_2 S k_1$. This means $k_1 | k_2$ and $k_2 | k_1$.
   • If $k_1 | k_2$, then $k_2 = k_1 \times m$ for some $m \in \mathbb{N}$.
   • If $k_2 | k_1$, then $k_1 = k_2 \times n$ for some $n \in \mathbb{N}$.
   • Combining these, $k_2 = k_1 \times m$ and $k_1 = k_2 \times n$. Substituting, $k_1 = (k_1 \times m) \times n$.
   • Simplifying, $k_1 = k_1 \times (m \times n)$. Since $k_1 \neq 0$, this implies $m \times n = 1$.
   • The only integers $m, n$ such that $m \times n = 1$ are $m = 1$ and $n = 1$.
   • Therefore, $k_1 = k_2$, proving antisymmetry.

3. Transitivity:

   • Suppose $k_1 S k_2$ and $k_2 S k_3$. This means $k_1 | k_2$ and $k_2 | k_3$.
   • If $k_1 | k_2$, then $k_2 = k_1 \times m$ for some $m \in \mathbb{N}$.
   • If $k_2 | k_3$, then $k_3 = k_2 \times n$ for some $n \in \mathbb{N}$.
   • Substituting, $k_3 = (k_1 \times m) \times n = k_1 \times (m \times n)$.
   • Hence, $k_1 | k_3$, which implies $k_1 S k_3$, proving transitivity.

Conclusion:

Since $S$ satisfies reflexivity, antisymmetry, and transitivity, $S$ is a partial order relation on $\mathbb{N}$.

Let me know if you'd like further clarifications or details! Below are some related questions:

1. What is the significance of partial order relations in mathematics?
2. How does a partial order differ from a total order?
3. Can we represent $S$ as a Hasse diagram? If so, how?
4. Are there any real-world applications of this divisibility relation?
5. What happens if we define $S$ differently (e.g., $k_1 S k_2 \iff k_1^2 | k_2$)?

Tip: Always verify the three properties (reflexivity, antisymmetry, transitivity) systematically when proving a relation is a partial order.