Libretext Kwong Example 7.4.3

Exercise.md

Show that "divides" (|) is a partial ordering of the positive integers $\mathbb{Z}^+$. Explain why $(\mathbb{Z}^*, |)$ is not a posit.

Let S = ($\mathbb{Z}^+$,|)

Reflexive:

Every number divides itself with quotient 1. So xRx for all x in $\mathbb{Z}^+$.

Antisymmetric:

1. Let xRy and yRx be in S. Then x divides y and y divides x.
2. $x \leq y$ since it divides y.
3. $y \leq x$ since it divides x.
4. If $x \lt y$, this contradicts (3).
5. If $y \lt x$, this contradicts (2).
6. So $x = y$.

Transitive

1. Let xRy and yRz be in S. Then $q_1 = y/x$ and $q_2 = z/y$.
2. $q_1 x = y$
3. $q_2 = z/(q_1x)$
4. $q_2 q_1 = z/x$
5. So x divides z with the quotient $q_2 q_1$. So xRz.

$(\mathbb{Z}^*, |)$ is not a posit because 0 doesn't divide itself, violating the reflexive property.