To prove that the divisibility relation | is a partial ordering on the set $\mathbb{Z}^+$ (the set of positive integers), we need to show that it satisfies the three properties of a partial order: reflexivity, antisymmetry, and transitivity.

1. Reflexivity

For all $a \in \mathbb{Z}^+$, $a | a$. This is true because any integer divides itself.

2. Antisymmetry

If $a, b \in \mathbb{Z}^+$ and $a | b$ and $b | a$, then $a = b$. This holds because if $a | b$, then $b = ak$ for some $k \in \mathbb{Z}^+$, and if $b | a$, then $a = bm$ for some $m \in \mathbb{Z}^+$. Substituting $b = ak$ into $a = bm$, we get $a = akm$, which implies $km = 1$. Since $k, m \in \mathbb{Z}^+$, the only possibility is $k = m = 1$, so $a = b$.

3. Transitivity

If $a, b, c \in \mathbb{Z}^+$, $a | b$ and $b | c$, then $a | c$. If $a | b$, then $b = ak$ for some $k \in \mathbb{Z}^+$, and if $b | c$, then $c = bj$ for some $j \in \mathbb{Z}^+$. Substituting $b = ak$ into $c = bj$, we get $c = akj$, so $a | c$.

Thus, the divisibility relation | is a partial order on $\mathbb{Z}^+$.

Would you like further clarification or examples?

Related Questions:

1. How does divisibility differ from a total ordering?
2. Can we extend this partial order to a total order on $\mathbb{Z}^+$?
3. What is an example of a set where divisibility is not antisymmetric?
4. How does the divisibility relation compare to other types of partial orders like subset inclusion?
5. How can we visualize the divisibility relation using a Hasse diagram?

Tip:

To verify partial ordering, always focus on the three core properties: reflexivity, antisymmetry, and transitivity.